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welcome to the world of numerical analysis
i am professor sachin patwardhan from department
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of chemical engineering at iit bombay and
these are series of lectures delivered on
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advanced numerical analysis in nptel phase
2 s0 this is my first lecture this is an overview
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of the course and next 1 hour also i am going
to present a birds eye view of what we are
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going to study in this course
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world of numerical analysis is pretty involved
and complex and probably some of you have
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already have some introduction to this this
is meant to be a course which is advanced
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course we will introduce you many of the things
in a different light and i hope it will help
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you throughout your academic career so let
us begin our journey with the motivation in
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chemical plants it can now you have large
number of interconnected units like heat exchangers
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reactors distillation columns
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and these days the chemical plants are very
tightly integrated to achieve high energy
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efficiency or high material efficiency which
makes it very complex to handle to operate
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to design and it is not possible to do it
without doing mathematical modelling so design
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and operation of such complex plant is always
a challenging problem and mathematical modeling
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and simulation has become a very very handy
tool
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a very cost effective method of analyzing
behavior of such plants so in a real design
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problem or a real operation problem we have
to judiciously blend mathematical analysis
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with experiments it is not possible to rely
only on experiments it will be not correct
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to rely only on mathematical modelling what
we are going to do is to plan experiments
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very carefully using mathematical models
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so mathematical modelling has become a backbone
of modern chemical engineering design and
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operation now these models have to be solved
either offline or online and when you have
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to solve this models under a variety of conditions
variety of problems you need to use numerical
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tools most often you cannot solve these problems
analytically so numerical problems is at a
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or numerical solutions is at the heart of
mathematical modelling and simulation which
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is in effect used for designing and operating
chemical plants
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now what are the typical problems that we
encounter let us look at some of the problems
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that a chemical engineer would typically have
to face when he goes to a chemical plant well
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one problem is of course a design problem
you may have to design a new section of a
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plant or if you are part of a consulting firm
which design chemical plants you have to design
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a new plant under you know you are given some
desired product composition you are given
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some raw material availability
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and then you have to find out unit sizes you
have to find out flow rates you have to find
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out operating conditions so coming up with
a base design from which a mechanical engineer
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or other engineering departments can take
over is what is the job of chemical engineer
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coming up with the basic flow sheet design
so this normally involves models for different
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linear operations
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you have to connect all these models into
a giant mathematical model into a big model
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which could be 100s and 1000s of equations
they need to be solved under a variety of
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conditions so this is one of the problems
that you normally encounter the other problem
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could be that you are already employed in
a plant and then you know you have to do process
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retrofitting
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so retrofitting involves improvement in the
existing operating conditions so you have
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a plant which is operating and then some modifications
are necessary because maybe the input conditions
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have changed maybe you know the feed quality
has changed or you need to ramp up you need
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to operate the same plant at different conditions
than what it was designed for because of the
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market conditions
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so retrofitting is another problem for an
existing plant and a problem that always comes
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when your operating plant is control or online
optimization so dynamic behaviour and operability
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analysis is integral part of operating any
complex chemical plant you have to first of
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all you have to monitor and control the plant
you have to make sure that it is operating
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safely you may have to carry out hazard analysis
conduct what-if studies
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you may want to do online optimization run
the plant in an optimal way and all these
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exercises cannot be done without mathematical
modelling and subsequently solving these mathematical
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models using numerical analysis so numerical
analysis is at the heart of all these exercises
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that we have to undertake as a chemical engineer
now what are mathematical models
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mathematical models could be in different
forms we have models that give insight into
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long term behaviour so these are typically
energy and material balances and we look at
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the steady state conditions and the design
problems or in retrofitting problems you might
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want to only restrict yourself to steady state
models that means we ignore the transient
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behaviour or short term behaviour
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whereas when you are studying operation of
a plant when you are trying to control the
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plant you cannot ignore the dynamics so in
that situation the short term behaviour or
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the transients become very very important
and then we have to we have to solve the mathematical
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models in time and possibly time and space
okay so what kind of mathematical models that
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we are going to study what we need in this
particular course
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mostly these models are going to be coming
from first principles or they are from or
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they are often called as mechanistic models
or phenomenological models so these models
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come from you know mass balances component
balances this is something that you have been
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doing for as you know in your courses and
various courses at chemical engineering so
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this could be you know models are composed
out of you know rate equations mass heat momentum
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transfer
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there are constitutive equations then chemical
reaction rate equations there could be equilibrium
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principles used while doing a modelling between
different phases also you may have to use
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equations of state if for systems involve
gases or multiple phases so the models that
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you actually use for doing this design operation
dynamic simulation are quite complex they
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are constructed out of these fundamental concepts
of energy mass material balances rate equations
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and equilibrium models
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well from a mathematical viewpoint how do
i classify these models well we can have variety
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of classifications but one classification
that is relevant to this course which will
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of course show up in a different way in terms
of classes of model equations that we are
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investigate is distributed parameter models
and lump parameter models so by this classification
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you know we are looking at 2 classes
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one that deals with variation in time and
space so distributed parameter models capture
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relationship between different variables not
just in space but also in time when i say
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space it could be in multiple dimensions not
just single dimension so for example plug
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flow reactor or a packed bed column or even
a shell and tube heat exchanger can be modelled
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as a distributed parameter system
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so this will depend upon situation in some
cases you might use very very simple lumped
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parameter model for a shell and tube heat
exchanger but there are situations where you
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may want to use more complex distributed parameter
model so one class of models that we are going
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to encounter in this course are distributed
parameter models the other class of models
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which we very often study in chemical engineering
a lumped parameter models
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for example stirred tank reactors or many
stage unit operations mixers so these are
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models with ignoring you know special variation
and if necessary we only consider variation
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in time alone so see we are looking at transient
behaviour only time comes into picture if
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you are looking at steady state behaviour
you may get only algebraic equations in this
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case so there are 2 broad classes of models
that are encountered in chemical engineering
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and we are going to study these models we
are going to study how to solve different
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subclasses belonging to these 2 broad classes
of models well if we examine from a mathematical
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viewpoint what are the equation forms that
we encounter when we are going to do this
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course well when you do a course in mathematics
or let us say courses in mathematics in your
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first or second year of engineering we start
looking at only abstract equation forms
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and it is important that you relate those
at abstract equations forms to what you see
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in the mathematical models so what a kind
of equation forms that you commonly encountered
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in chemical engineering models well one is
linear algebraic equations where we study
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linear algebraic equations maybe even before
we enter an engineering program what you study
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as you enter an engineering program
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and chemical engineering is solving nonlinear
algebraic equations so very often we have
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to deal with a single variable or multi variable
nonlinear algebraic equations thermodynamic
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relationships for example many times nonlinear
equations the other class of problem that
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you encounter in modelling chemical engineering
unit operations is ordinary differential equations
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typically an initial value is given and then
we are supposed to find a solution of first
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second third or higher order ordinary differential
equation the other class of problems that
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are encountered particularly in distributed
parameter systems or differential equations
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with boundary value problems you also may
have equations which are differential and
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algebraic equations
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so differential algebraic systems daes so
this is differential algebraic systems are
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mixtures of algebraic and differential equations
while ordinary differential equations boundary
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value problems are one in which boundary conditions
are specified partially at one boundary and
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remaining at other boundary and we are expected
to solve these kind of differential equations
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so these kind of problems typically arise
we are solving say plug flow reactor models
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or distributed parameter systems
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and other models that we often encounter while
modelling chemical engineering unit operations
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or partial differential equations so these
models they may not come in isolation in real
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problem when you are actually trying to solve
a problem associated with a section of a chemical
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plant you may get mixture of all of them not
just one of them in isolation nevertheless
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when we study these equation forms we often
study them in isolation
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and then we understand how to attack more
complex problems where combination of these
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might be encountered now how do you go about
doing this how do you go about studying these
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equation forms well if you look at many of
the approaches presented in textbooks are
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written for engineers a conventional approach
is study numerical recipes for each type of
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equations
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that means you start by saying well i am going
to first look at linear algebraic equations
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and the tools for solving linear algebraic
equations then we move on to say nonlinear
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algebraic equations having studied linear
and nonlinear algebraic equations we look
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at ordinary differential equations then initial
value problems typically you begin with then
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you might want to move on to study ordinary
differential equations with boundary values
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and then typically a course would end with
set of partial differential equations so methods
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for partial differential equations methods
for ordinary differential equations and so
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on so if we look at it from this viewpoint
one can get a view that there are separate
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methods for solving linear equations and for
partial differential equations or boundary
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value problems but this is not exactly so
when you start looking at these methods from
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a different viewpoint
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so in the conventional approach how do you
where do you encounter all these applications
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so after we have studied each one of these
equation types that is numerical methods for
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solving each one of these equation types then
in exercises or in the in the sample examples
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you will encounter real engineering problems
or it could be you might come across some
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abstract problems in terms of some x y z variables
which do not make physical sense
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so these problems are then used to form of
the concept that you have studied for each
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equation type in this course on advanced numerical
analysis we are going to be different we are
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going to look at it in a completely different
manner so what i am going to do is i am interested
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in understanding what are the fundamental
steps involved in formulation of a numerical
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scheme and then how do you come up with a
receipe or a solution approach to solve a
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particular problem
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so if you take a critical view point of all
the methods then you come across certain threads
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which are common and from that you can actually
built a different way of studying numerical
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analysis so what i am going to do here is
you know look at 2 different steps separately
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if you look at all the numerical methods that
are used for solving you know different type
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of problems that are encountered
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and you know make analysis what kind of what
is the first step and what is the second step
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so what do you realize is that invariably
a first step is you know model transformation
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so many times you have models that cannot
be directly or many times your mathematical
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problems that cannot be directly solved using
existing methods when i mean to say that could
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not be directly solved i mean to say that
they cannot be analytically solved
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if they cannot be analytically solved you
have to construct approximation approximate
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solutions but to construct approximate solutions
you have to first convert a given problem
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into a computable form a computable form is
one to which known computation tools can be
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applied now this problem transformation is
carried out using tools or using approaches
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developed in approximation theory a well developed
branch of applied mathematics
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approximation theory is used to transform
a problem into computable form and then you
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actually use different tools to attack the
transform problem and construct numerical
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solution so these tools are you know linear
algebraic equation solver or nonlinear algebraic
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equation solver it could be ordinary differential
equation initial value problem solver or it
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could be a numerical optimization scheme
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so when you actually construct receive or
when you construct a numerical scheme to solve
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a problem you first transform it into a form
that can be dealt with that can be tackled
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with one of the standard tools and then you
use one or more of these tools in combination
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to come up with a solution of their transform
problem so this is if i just put this into
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pictorial form then you have an original problem
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this original problem might be a partial differential
equation then you take this original problem
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use principles developed in the approximation
theory and transform it to what i have called
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here in a standard form a standard form is
what i mean by my standard form here is a
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computable form okay so this original form
might be a partial differential equation when
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i transform it it might turn out to be set
of linear algebraic equations or set of nonlinear
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algebraic equations
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so the original problem that you want to solve
and the transform problem do not have same
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equation type you have a partial differential
equation here you have a set of nonlinear
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algebraic equations here so to solve this
nonlinear algebraic equations you may have
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to use you may have to use special tools that
are developed for solving nonlinear algebraic
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equations these tools for solving nonlinear
algebraic equations in turn might use linear
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algebraic equation solver
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so you know it is not that i am going to use
just one tool so i am going to use multiple
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of these tools to attack this transform problem
and then come up with a solution which is
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a numerical solution of my original problem
so what we are going to study in this course
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is 2 steps well how do i take the original
problem and transform it into a solvable form
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or a computable form
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now this process actually in the conventional
approach get mixed up with various you know
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receipts that are developed for a specific
equation types we are going to separate it
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and view it as a separate step so this means
unlike the conventional approach i am not
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going to look at partial differential equations
at the end of my course or boundary value
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problems at the end of my course
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i will begin right in you know attacking these
problems right in the beginning and we will
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just transform them into forms that can be
solved using one or more of these standard
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tools so that is the approach that we want
to take so what are the overall learning objectives
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for this numerical analysis course okay well
i am assuming here that you have had some
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exposure to this numerical method prior to
doing this course
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well if you have not had does not matter this
course will give you you know from scratch
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a different viewpoint of numerical analysis
but if you had some prior experience with
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numeric analysis well it will enrich your
understanding so the first thing that i want
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to do here is to clearly bring out the role
of approximation theory in the process of
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developing a numerical receipt for solving
an engineering problem
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this word is you know i am deliberately using
this word numerical receipt it is like you
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know at the end of the course you should realize
that forming a numerical scheme is like cooking
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up some dish and you know if you know the
basic ingredients you can actually combine
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and then come up with a particular dish so
you often have to be a good cook to come up
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with a numerical receipt to solve the problem
and to be a good cook you have to understand
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the foundations
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so the first step is you know problem transformation
which is based on the approximation theory
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the next step is of course solving it but
in solving it there are 2 aspects one is of
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course the algebraic aspect of the problem
how do you actually write the algorithms and
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so on but often there are very very interesting
geometric ideas associated with these numerical
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schemes
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and if you get understanding of these geometric
ideas if you understand you know if you can
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visualize some of these if you can use your
you know power of visualization then actually
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that can help you to construct solutions much
better so unlike a traditional course i would
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like to stress a lot on explaining many geometric
ideas that are associated with development
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of numerical schemes
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so this will actually help in developing a
deeper understanding of numerical recipes
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and finally an aspect that we do not try to
stress in a first course is analysis of convergence
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or convergence analysis of numerical methods
or error analysis there are also other analytical
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aspects that are associated with the numerical
computations and i would like to stress these
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numerical these aspects along with the numerical
aspects convergence aspects or along with
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numerical aspects
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though we may not get too much deep into these
but we will nevertheless study this to some
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extent so that you have a taste of what goes
in you know understanding the convergence
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behaviour of these schemes so all these 3
aspects are very very important when it comes
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to coming up or one when it comes to concocting
a new numerical scheme so if you take a critical
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look at many many numerical schemes that are
available in the most of the textbooks
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you will see that you know there are some
fundamental 2 or 3 ideas that are used in
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developing the compatible forms okay one of
them one dominant idea that is you find in
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numerical methods is using taylor series expansion
so approximations carried or carried using
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taylor series expansion is one dominant way
of doing approximation the other method or
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other approach that is used is polynomial
interpolation
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and the third you know pillar of approximation
or problem simplification problem discretization
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is least squares approximations so the problem
transformation is carried out mainly using
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these 3 fundamental tools or fundamental ideas
one is taylors series expansion other is polynomial
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00:27:58,309 --> 00:28:03,600
interpolation and least square approximation
and we are going to study them pretty much
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in detail so as to understand their role in
problem transformations
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then after that we are going to get in depth
understanding of 4 different numerical tools
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well once you transform the problems there
are a variety of ways of attacking the problem
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to get a numerical solution so if you look
at what are the tools available today well
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we can come up with 5 different classifications
i have just mentioned 4 of them here one is
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linear algebraic equations other is nonlinear
algebraic equation ordinary differential equations
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initial value problem and numerical optimizations
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so i need these 4 toolkits with me to come
up with a numerical scheme and then the fifth
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one which is not mentioned here or which is
not going to be part of this course is stochastic
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methods that goes much beyond scope of this
particular course and would probably need
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a separate course to see how stochastic methods
can be used to solve the transform problem
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we are going to concentrate mainly on linear
algebraic equations nonlinear algebraic equations
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and ordinary differential equations initial
value problems or ode ivp as they are known
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along our way we will also pick up fundamentals
of numerical optimization i do not intend
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to have a separate module on numerical optimization
but we will on our way we will pick up tools
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for numerical optimization so this course
consists of 6 learning modules the first one
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actually here i am talking of the course ideally
what it should consist of well i will then
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at the end of this slide i will tell you what
i am going to lecture on
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but this course initially should begin with
relating abstract equation forms to process
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models okay so if i am delivering this course
to finally year undergraduate students i would
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spend first 2 or 3 lectures talking about
different mathematical models that they have
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already studied and what abstract equation
forms that arise from these mathematical models
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00:30:43,849 --> 00:30:51,789
the second module is going to be completely
different from what you do in a conventional
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numerical methods or numerical analysis course
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a few lectures these few lectures are going
to be devoted to fundamentals of vector spaces
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now vector spaces we start studying vector
spaces probably even before we enter our engineering
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00:31:16,259 --> 00:31:22,200
programs so by the time we come into engineering
programs we are familiar with 3 dimensional
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vector spaces and mostly we continue using
3 dimensional vector spaces maybe you study
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00:31:29,599 --> 00:31:34,139
you know different coordinate systems which
probably you do not study when you are in
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your school
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but more or less the idea of vector space
remains confined to 3 dimensional vector spaces
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00:31:42,419 --> 00:31:53,700
but in mathematics in the field of functional
analysis the idea of vector spaces has been
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00:31:53,700 --> 00:32:05,229
very very profoundly developed into you know
a rich concept where a large subset of objects
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00:32:05,229 --> 00:32:15,859
can be looked upon as vector spaces and we
are going to get some peek some you know understanding
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00:32:15,859 --> 00:32:22,619
of these generalized vector spaces which are
not just 3 dimensional vector spaces but 4
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00:32:22,619 --> 00:32:26,799
5 or n dimensional or even infinite dimensional
vector spaces
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00:32:26,799 --> 00:32:34,039
in fact these vector spaces a fundamental
role in formulation of or in understanding
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00:32:34,039 --> 00:32:39,489
of numerical schemes and this is what i mean
when i am saying that i want to stress upon
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00:32:39,489 --> 00:32:45,639
geometric ideas the geometric idea is that
you understand in 3 dimensions can be extended
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00:32:45,639 --> 00:32:52,339
to spaces of higher dimension and that is
what we are going to have a peak at in the
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00:32:52,339 --> 00:32:59,759
second module the third module is going to
be problem discretization using approximation
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00:32:59,759 --> 00:33:00,820
theory
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00:33:00,820 --> 00:33:07,200
so significant numbers of lectures are going
to be devoted to problem transformations so
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00:33:07,200 --> 00:33:15,469
here you know i will start with the models
which could be a nonlinear set of algebraic
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00:33:15,469 --> 00:33:19,229
equations which could be a partial differential
equation which could be an ordinary differential
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00:33:19,229 --> 00:33:24,609
equation boundary value problem i am going
to transform it into a computable form so
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00:33:24,609 --> 00:33:31,339
unlike a conventional course where these pds
or boundary value problems are discussed at
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00:33:31,339 --> 00:33:35,549
the end will encounter them right in the beginning
of this course
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00:33:35,549 --> 00:33:41,639
and we will transform to the computable forms
once we have this standard compatible forms
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00:33:41,639 --> 00:33:45,239
which could be set of linear algebraic equations
which could be set of nonlinear algebraic
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00:33:45,239 --> 00:33:50,509
equations or ordinary differential equations
initial value problems then we need to know
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00:33:50,509 --> 00:34:00,140
how to solve them so module 4 is going to
look at variety of numerical tools for solving
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00:34:00,140 --> 00:34:08,760
linear algebraic equations then we move on
to tools for solving nonlinear algebraic equations
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00:34:08,760 --> 00:34:15,760
and finally we end with tools for solving
ordinary differential equations initial value
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00:34:15,760 --> 00:34:23,820
problems so ideally this course should consist
of these 6 modules well but when i am going
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00:34:23,820 --> 00:34:31,050
to deliver these set of lectures i am assuming
that you are already well familiar different
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00:34:31,050 --> 00:34:37,720
model forms that you encounter in chemical
engineering so the modules 1 though i have
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00:34:37,720 --> 00:34:41,720
mentioned here i am not going to really start
with module 1
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00:34:41,720 --> 00:34:48,460
my lectures will start in module 2 that is
fundamentals of vector spaces in the next
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00:34:48,460 --> 00:34:56,080
few slides i will very briefly touch upon
what should go into model 1 but the second
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00:34:56,080 --> 00:35:02,280
lecture onwards will start looking at vector
spaces generalized vector spaces and what
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00:35:02,280 --> 00:35:11,360
role they play in numeric analysis okay so
moving on well how long will this journey
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00:35:11,360 --> 00:35:18,001
be it is going to be a long journey we would
deal about 48 lectures 1 hour lectures to
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00:35:18,001 --> 00:35:25,900
understand this variety of aspects of numeric
analysis
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00:35:25,900 --> 00:35:34,390
so let me get into a little more details of
module 1 so the module 1 will consist of abstract
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00:35:34,390 --> 00:35:43,020
equation forms in process modelling so what
all objective would be you know mathematical
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00:35:43,020 --> 00:35:47,140
models in chemical engineering together with
variety of designer operating conditions they
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00:35:47,140 --> 00:35:54,780
give rise to different types of abstract equations
or equation forms like odes like partial differential
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00:35:54,780 --> 00:35:55,780
equations
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00:35:55,780 --> 00:36:01,640
and so we must in the beginning associate
abstract forms with a real problem because
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00:36:01,640 --> 00:36:08,680
as we go along we just start looking at abstract
forms we lose track of the engineering problems
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00:36:08,680 --> 00:36:17,540
except when we look at some you know examples
or when we look at or when we solve some exercises
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00:36:17,540 --> 00:36:23,830
apart from that we lose connection with the
engineering problems so in the beginning it
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00:36:23,830 --> 00:36:29,340
is good to have connection with these models
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00:36:29,340 --> 00:36:36,630
and then we need to know which type of equation
forms will be treated through in this course
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00:36:36,630 --> 00:36:40,570
so if you just want to have commonly encountered
examples so linear algebraic equations where
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00:36:40,570 --> 00:36:48,970
do we get linear algebraic equations in chemical
engineering systems so many many times we
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00:36:48,970 --> 00:36:55,040
have to solve steady state material balance
for a lump parameter model for a section of
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00:36:55,040 --> 00:37:05,330
a plant and this will give rise to a set of
linear algebraic equations ax equal to b
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00:37:05,330 --> 00:37:12,030
nonlinear algebraic equations of course you
must have studied in your courses in the third
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00:37:12,030 --> 00:37:19,700
year when you study you know mass transfer
heat transfer courses or linear operation
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00:37:19,700 --> 00:37:28,420
courses mainly where we encounter models which
come through energy and material balance for
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00:37:28,420 --> 00:37:35,340
one unit or a section of a plant which consists
of multiple units and these give rise to nonlinear
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00:37:35,340 --> 00:37:39,330
algebraic equations
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00:37:39,330 --> 00:37:49,560
very often we have to solve problems using
optimization tools for example estimating
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00:37:49,560 --> 00:37:55,900
some rate parameters say reaction kinetics
parameters or estimation of mass transfer
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00:37:55,900 --> 00:38:01,950
or heat transfer correlations so these problems
have to be solved using tools that are used
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00:38:01,950 --> 00:38:06,990
for optimization numerical optimization so
these are optimization based formulations
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00:38:06,990 --> 00:38:18,940
and ordinary differential equations initial
value problems arise when you start looking
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00:38:18,940 --> 00:38:25,310
at control at dynamic simulation of a chemical
plant
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00:38:25,310 --> 00:38:31,660
or when you want to do hazop analysis using
dynamics dynamic simulators so these problems
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00:38:31,660 --> 00:38:37,070
in abstract terms are nothing but solving
coupled ordinary differential equations subject
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00:38:37,070 --> 00:38:45,600
to given initial conditions or given input
scenarios then you may end up with not just
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00:38:45,600 --> 00:38:51,300
differential equations you may end up with
algebraic differential equations well common
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00:38:51,300 --> 00:38:57,580
example is distillation columns where you
have a phase equilibrium giving rise to algebraic
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00:38:57,580 --> 00:39:01,220
equations which could be highly nonlinear
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00:39:01,220 --> 00:39:07,410
you have differential equations coming from
dynamics on the trace temperature dynamics
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00:39:07,410 --> 00:39:12,561
composition dynamics material balance on the
trace if you want to simulate the dynamic
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00:39:12,561 --> 00:39:17,530
behaviour not just do the design then you
get differential algebraic equations coupled
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00:39:17,530 --> 00:39:22,470
equations and these equations are notoriously
difficult to solve than the differential equations
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00:39:22,470 --> 00:39:25,130
alone or algebraic equations alone
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00:39:25,130 --> 00:39:32,390
so these are the situations where you know
these differential algebraic equations arise
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00:39:32,390 --> 00:39:39,260
when you have phenomena which are operating
at different time scales so some phenomena
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00:39:39,260 --> 00:39:47,230
are fast some phenomena are slow and in such
situations the slow phenomena you retain them
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00:39:47,230 --> 00:39:54,010
as differential equations the fast phenomena
you can neglect the derivatives and you know
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00:39:54,010 --> 00:39:58,150
approximate those equations associated with
those equations as algebraic equations and
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00:39:58,150 --> 00:40:02,000
that gives rise to differential algebraic
equations
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00:40:02,000 --> 00:40:08,790
if you want to do detail analysis of let us
say some reactor plug flow reactor or a packed
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00:40:08,790 --> 00:40:17,700
bed column then you do not have option but
to use partial differential equations whereas
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00:40:17,700 --> 00:40:25,030
when you are doing a very gross analysis you
know taking it just as a unit in a plant and
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00:40:25,030 --> 00:40:29,880
doing energy material balance you can probably
neglect those variations but if you want to
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00:40:29,880 --> 00:40:37,200
study one unit operation in detail you often
have to use you know distributed parameter
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00:40:37,200 --> 00:40:40,170
models that is partial differential equations
363
00:40:40,170 --> 00:40:46,070
so these partial differential equations arise
when you are looking at packed bed columns
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00:40:46,070 --> 00:40:53,020
plug flow reactors and so on so in the beginning
it is good to make these associations to understand
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00:40:53,020 --> 00:40:58,550
where these abstract equation forms arise
but as i said my lectures are going to start
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00:40:58,550 --> 00:41:08,870
from module 2 because these are meant for
somewhat advanced users in the final year
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00:41:08,870 --> 00:41:14,740
of a chemical engineering undergraduate program
or maybe first year of a graduate program
368
00:41:14,740 --> 00:41:16,520
of chemical engineering
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00:41:16,520 --> 00:41:24,310
well here we begin with fundamentals of vector
spaces so what are the learning objective
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00:41:24,310 --> 00:41:31,340
so first thing is i would like to understand
2 fundamental operations vector addition and
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00:41:31,340 --> 00:41:38,830
vector and scalar multiplication and see how
these operations hold in any vector space
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00:41:38,830 --> 00:41:44,910
what i mean by any vector space i am going
to define sets which are called as vector
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00:41:44,910 --> 00:41:54,560
spaces where these 2 operations holds and
these sets are going to be other than the
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00:41:54,560 --> 00:41:56,990
familiar 3 dimensional vector spaces
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00:41:56,990 --> 00:42:06,140
for example i would introduce set of continuous
functions over some domain say 0 to 1 or i
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00:42:06,140 --> 00:42:12,550
might introduce i meant start talking about
a set of continuous functions over 0 to infinity
377
00:42:12,550 --> 00:42:19,580
these kind of functions these kind of sets
arise when we are solving differential equations
378
00:42:19,580 --> 00:42:26,150
partial differential equations and if you
have understanding basic understanding or
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00:42:26,150 --> 00:42:34,790
of the geometric understanding of these underlying
spaces then it is much easier to develop the
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00:42:34,790 --> 00:42:38,500
solutions for these kind of equations
381
00:42:38,500 --> 00:42:47,310
so we are going to look at these abstract
notions of vector space and generalize vector
382
00:42:47,310 --> 00:42:58,670
spaces like function spaces so a vector in
this vector space is a function for example
383
00:42:58,670 --> 00:43:08,020
you know set of all continuous functions over
0 to 2pi okay and say sin x is a vector in
384
00:43:08,020 --> 00:43:17,110
this set or cos x or cos 2x is a vector in
this set of continuous functions well another
385
00:43:17,110 --> 00:43:24,690
vector could be just a line a plus bt defined
over 0 to 2pi and so on or some polynomial
386
00:43:24,690 --> 00:43:27,270
defined over 0 to 2pi
387
00:43:27,270 --> 00:43:33,380
so these sets are generalized sets not just
3 dimensional vector spaces that you are familiar
388
00:43:33,380 --> 00:43:40,780
with and what you will study in this particular
module as how these sets to qualify to be
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00:43:40,780 --> 00:43:47,650
called as vector spaces and how the geometric
ideas that hold in 3 dimensions can be extended
390
00:43:47,650 --> 00:43:54,040
to these higher dimensional spaces so we will
go on to generalize the concepts such as subspace
391
00:43:54,040 --> 00:44:00,451
such as linear dependence such as span of
vectors what is the basis in a vector space
392
00:44:00,451 --> 00:44:01,500
and so on
393
00:44:01,500 --> 00:44:08,810
and we will examine examples of different
sets that qualify to be vector spaces or that
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00:44:08,810 --> 00:44:17,310
qualified to be subspace of a vector space
and so on so this is this is beginning of
395
00:44:17,310 --> 00:44:26,580
the geometric generalization this grand geometric
generalization was carried out probably 60
396
00:44:26,580 --> 00:44:36,530
70 80100 years back in the domain of mathematics
and if you have some idea about the generalizations
397
00:44:36,530 --> 00:44:41,960
then it becomes very very easy to understand
underlying foundations of numeric analysis
398
00:44:41,960 --> 00:44:46,250
so that is why first few lectures are going
to be devoted to understanding these generalized
399
00:44:46,250 --> 00:44:52,180
sets well when we work in a 3 dimensional
vector space what are the things that you
400
00:44:52,180 --> 00:45:00,540
actually need well we need when you work with
vectors we need to know about length of the
401
00:45:00,540 --> 00:45:08,170
vector okay so when you move on to generalize
vector spaces we define something called norm
402
00:45:08,170 --> 00:45:15,950
of a vector which could be viewed as a generalization
of concept of length of a vector
403
00:45:15,950 --> 00:45:22,240
so we are going to distil out essential properties
that define length in 3 dimensions and generalize
404
00:45:22,240 --> 00:45:28,420
them to this concept of norms is there a unique
way of defining a norm what we will find out
405
00:45:28,420 --> 00:45:35,280
is that a norm can be defined in multiple
ways okay the way we define the so called
406
00:45:35,280 --> 00:45:46,280
length in 3 dimensions is one way of defining
norm it is a special case now it is good to
407
00:45:46,280 --> 00:45:51,780
do visualizations in 3 dimensions or 2 dimensions
one can do visualizations
408
00:45:51,780 --> 00:45:56,870
and maybe if you understand visualizations
in 2 and 3 dimensions then you might be at
409
00:45:56,870 --> 00:46:04,910
least able to do some imagination and or extend
your imagination to see what is happening
410
00:46:04,910 --> 00:46:09,020
in a higher dimensional space or a function
space so that is what we are going to look
411
00:46:09,020 --> 00:46:20,890
at in this part well when you are dealing
with numerical analysis a thing that you have
412
00:46:20,890 --> 00:46:27,420
to you invariably encountered is convergence
of a numerical scheme
413
00:46:27,420 --> 00:46:35,680
so we have to understand whether a particular
we start with a guess solution and we construct
414
00:46:35,680 --> 00:46:41,720
a new solution from an initial solution so
whether this sequence of vectors that you
415
00:46:41,720 --> 00:46:49,230
get in the process of generating approximate
solutions is it converging to some point is
416
00:46:49,230 --> 00:46:55,840
it going to somewhere is it going somewhere
in the same space we need to examine this
417
00:46:55,840 --> 00:47:01,870
thoroughly when it comes to understanding
numerical behaviour of solutions
418
00:47:01,870 --> 00:47:08,400
so in abstract terms we are going to look
at sequences of vectors okay and we also have
419
00:47:08,400 --> 00:47:13,660
to talk about conversions in fact when you
talk of conversions of vectors we have to
420
00:47:13,660 --> 00:47:18,450
talk of nearness of 2 vectors and if you want
to talk of nearness of 2 vectors you have
421
00:47:18,450 --> 00:47:22,920
to find out distance between the 2 vectors
and this is where the concept of norm becomes
422
00:47:22,920 --> 00:47:31,070
very very vital so the ideas that we use in
3 dimensions need to be generalized to higher
423
00:47:31,070 --> 00:47:32,070
dimensions
424
00:47:32,070 --> 00:47:39,940
well we will look at very briefly at the concept
of a normed space that means a space on which
425
00:47:39,940 --> 00:47:46,500
a norm is defined so and we will also have
you know we will also understand very briefly
426
00:47:46,500 --> 00:47:54,580
what are called as banach spaces or the complete
normed spaces well these spaces you may not
427
00:47:54,580 --> 00:47:58,680
encounter later the concept of banach space
may not be required throughout the course
428
00:47:58,680 --> 00:48:07,320
but it is good to have understanding of this
idea when we start looking atwhen you start
429
00:48:07,320 --> 00:48:11,560
generalizing the concept of a vector space
430
00:48:11,560 --> 00:48:20,050
the most important concept that we use in
3 dimensions when we do geometry in 3 dimensions
431
00:48:20,050 --> 00:48:30,860
is orthogonally okay we like to work with
orthogonal sets we like to work with you know
432
00:48:30,860 --> 00:48:36,920
coordinate definitions which are orthogonal
to each other x y z or you know coordinates
433
00:48:36,920 --> 00:48:43,500
that we normally take or you know is a orthogonal
coordinate system so orthogonality that is
434
00:48:43,500 --> 00:48:49,790
so useful in 3 dimensions is also useful when
it goes to other spaces like function spaces
435
00:48:49,790 --> 00:48:50,940
okay
436
00:48:50,940 --> 00:48:57,280
so we need to generalize the concept of orthogonality
to higher dimensional spaces to other spaces
437
00:48:57,280 --> 00:49:02,380
and this is done through what is called as
inner products so we are going to define a
438
00:49:02,380 --> 00:49:09,780
special class of spaces vector spaces called
as inner product spaces okay that is a vector
439
00:49:09,780 --> 00:49:18,520
space a set of objects on which an inner product
is defined well when it comes to a 3 dimensional
440
00:49:18,520 --> 00:49:22,420
space all of you would be familiar with a
dot product okay
441
00:49:22,420 --> 00:49:27,510
so when i am generalizing the when i am generalizing
the idea of vector space from 3 dimensions
442
00:49:27,510 --> 00:49:34,390
to some other you know some other sets which
are more general sets i also would like to
443
00:49:34,390 --> 00:49:41,370
have ideas which are similar to a dot product
okay so this inner product is going to give
444
00:49:41,370 --> 00:49:46,620
me something similar to a dot product in fact
what we will see is that dot product is one
445
00:49:46,620 --> 00:49:48,420
way of defining an inner product
446
00:49:48,420 --> 00:49:55,760
so inner product is a generalization is a
grand generalization which will help us to
447
00:49:55,760 --> 00:50:03,230
generalize the concepts of orthogonality orthogonal
vectors in general spaces function spaces
448
00:50:03,230 --> 00:50:14,050
and so on so we are going to look at inner
product spaces inner product spaces are generalization
449
00:50:14,050 --> 00:50:23,990
of 3 dimensional vector spaces with dot product
defined in them so we are very very familiar
450
00:50:23,990 --> 00:50:29,830
with dot product we use dot product to define
angle between 2 vectors in 3 dimensions
451
00:50:29,830 --> 00:50:37,580
and when we move on to more general spaces
set of functions set of the polynomials we
452
00:50:37,580 --> 00:50:44,421
need this concept we need something like dot
product which is going to be this inner product
453
00:50:44,421 --> 00:50:53,730
in these spaces so we are going to look at
variety of inner product spaces so there are
454
00:50:53,730 --> 00:50:58,010
different ways of defining inner product that
not only one way in 3 dimensional space you
455
00:50:58,010 --> 00:51:00,210
only know one way of defining an inner product
456
00:51:00,210 --> 00:51:04,640
but there are other ways of defining inner
products and we will look at those different
457
00:51:04,640 --> 00:51:13,990
methods of defining inner products well one
of the fundamental equation that we use in
458
00:51:13,990 --> 00:51:23,360
3 dimensions is that cos theta angle between
any 2 vectors is dot product of two unit vectors
459
00:51:23,360 --> 00:51:31,000
in two directions so if i have a vector a
and b i find out we need vectors along a i
460
00:51:31,000 --> 00:51:36,110
find a unit vector along b and then i take
a dot product which gives me cos theta between
461
00:51:36,110 --> 00:51:37,490
a and b
462
00:51:37,490 --> 00:51:46,000
a generalization of this particular concept
in inner product spaces is nothing but the
463
00:51:46,000 --> 00:51:52,220
so called cauchy schwarz inequality the name
cauchy schwarz inequality might sound very
464
00:51:52,220 --> 00:51:59,010
intimidating but this is a very fundamental
result in inner product spaces and it will
465
00:51:59,010 --> 00:52:04,790
help us to define angle between 2 vectors
so here a vector as i said is going to be
466
00:52:04,790 --> 00:52:10,590
a function and then we need to talk about
orthogonal functions okay
467
00:52:10,590 --> 00:52:17,750
so so see you might have come across some
statements in your undergraduate education
468
00:52:17,750 --> 00:52:24,460
saying that sin theta sin 2 theta sin 3 theta
these are orthogonal to each other why they
469
00:52:24,460 --> 00:52:29,810
are orthogonal okay so if you understand the
concept of inner products and inner product
470
00:52:29,810 --> 00:52:36,250
spaces this will no longer be a mystery okay
so generalization of concept of angle between
471
00:52:36,250 --> 00:52:41,340
any 2 vectors is achieved through inner product
472
00:52:41,340 --> 00:52:46,410
and then the cauchy schwarz inequality is
a fundamental inequality which is nothing
473
00:52:46,410 --> 00:52:56,750
but generalization of the fact that cos theta
is dot product of 2 vectors through unit vectors
474
00:52:56,750 --> 00:53:02,751
in 3 dimensions okay so we are going to study
this cauchy schwarz inequality then we will
475
00:53:02,751 --> 00:53:09,120
look at variety of orthogonal or orthonormal
sets that are very often used in numeric analysis
476
00:53:09,120 --> 00:53:15,380
for example legendre polynomial or lagrange
polynomial now these names we encountered
477
00:53:15,380 --> 00:53:17,170
in maths courses
478
00:53:17,170 --> 00:53:26,960
and often we do not know why they are they
are called orthogonal sets or why they are
479
00:53:26,960 --> 00:53:33,730
called as orthogonal polynomials if you start
from fundamentals of vector spaces you will
480
00:53:33,730 --> 00:53:44,740
get in depth understanding as why these sets
are called as orthogonal sets well it is not
481
00:53:44,740 --> 00:53:49,980
always that you have a set of vectors which
are orthogonal okay but if you have a non
482
00:53:49,980 --> 00:53:56,520
orthogonal set of vectors then one can systematically
construct a set of vectors that is orthogonal
483
00:53:56,520 --> 00:54:03,740
for example in 3 dimensions you may have come
across this method called gram schmidt orthogonalization
484
00:54:03,740 --> 00:54:10,560
okay which is you start with 3 vectors which
are not orthogonal and starting from these
485
00:54:10,560 --> 00:54:16,430
vectors one can systematically construct 3
new vectors which are orthogonal to each other
486
00:54:16,430 --> 00:54:17,430
okay
487
00:54:17,430 --> 00:54:23,290
so constructing an orthogonal set from non
orthogonal set this process is called gram
488
00:54:23,290 --> 00:54:27,971
schmidt orthogonalization and this we are
going to study in a general inner product
489
00:54:27,971 --> 00:54:36,450
space it is very useful to get again insight
into how different orthogonal sets are developed
490
00:54:36,450 --> 00:54:42,700
and then we will look at examples of generating
orthogonal sets starting from non orthogonal
491
00:54:42,700 --> 00:54:51,680
sets so from inner product spaces we then
move on to the third module
492
00:54:51,680 --> 00:54:58,170
now this is going to be a very very important
module in this course i would say heart of
493
00:54:58,170 --> 00:55:05,730
this course how do you discretize the problem
using approximation theory so as i told you
494
00:55:05,730 --> 00:55:15,790
in the beginning it is often not possible
to solve a given problem in its original form
495
00:55:15,790 --> 00:55:22,090
most of the times the problem that you have
is not a linear which means it could consist
496
00:55:22,090 --> 00:55:29,010
of nonlinear algebraic equations nonlinear
differential equations nonlinear partial differential
497
00:55:29,010 --> 00:55:30,590
equations
498
00:55:30,590 --> 00:55:35,500
well when where you have linear differential
equations linear partial differential equations
499
00:55:35,500 --> 00:55:42,980
you can many times construct solutions analytically
at least for some idealized situations this
500
00:55:42,980 --> 00:55:49,520
becomes very very difficult even if there
are slight nonlinearities and it may not be
501
00:55:49,520 --> 00:55:56,080
possible to have analytical solutions this
means you have to construct numerical solutions
502
00:55:56,080 --> 00:56:04,150
to construct numerical solutions we have to
first transform into standard forms
503
00:56:04,150 --> 00:56:10,140
see this is because we do not have tools to
solve all kinds of problems we can only tackle
504
00:56:10,140 --> 00:56:18,600
certain types of equation forms so first step
is to convert a different problem into a problem
505
00:56:18,600 --> 00:56:28,869
which can be tackled using standard tools
okay and then we attack the problem to construct
506
00:56:28,869 --> 00:56:37,800
the solution okay so by hook or crook by some
means by using multiple ideas together from
507
00:56:37,800 --> 00:56:43,020
approximation theory we actually transform
the problem to a computable form
508
00:56:43,020 --> 00:56:48,210
is there a unique way of doing this obviously
not a given problem can be transformed into
509
00:56:48,210 --> 00:56:54,150
a computable form by variety of means and
if you have to choose between different means
510
00:56:54,150 --> 00:56:58,680
of transformations you have to have in depth
understanding of how these transformations
511
00:56:58,680 --> 00:57:05,390
are done why do you choose one over the other
whether i should use a taylor series approximation
512
00:57:05,390 --> 00:57:07,800
or whether i should use interpolation
513
00:57:07,800 --> 00:57:13,250
unless you know the foundations it is difficult
to make these choices so it is good to have
514
00:57:13,250 --> 00:57:24,570
a basis of foundation of you know approximation
theory this step of model transformation is
515
00:57:24,570 --> 00:57:30,740
often referred to as problem discretization
and in this module in this set of lectures
516
00:57:30,740 --> 00:57:37,920
we are going to look at popular approaches
that are available in the literature for approximations
517
00:57:37,920 --> 00:57:44,790
or approximate a given problem to computable
forms
518
00:57:44,790 --> 00:57:53,680
so first thing that i want to do here before
i begin this transformation is to show that
519
00:57:53,680 --> 00:58:03,340
actually different problems that you encounter
in numeric analysis they are only seemingly
520
00:58:03,340 --> 00:58:13,690
different once you start viewing these problems
from the viewpoint of vector spaces generalized
521
00:58:13,690 --> 00:58:21,369
vector spaces they do not really appear different
problems one can come up with a grand generalization
522
00:58:21,369 --> 00:58:26,790
that there is a one single problem well in
a particular vector space this problem will
523
00:58:26,790 --> 00:58:29,030
be called as set of algebraic equations
524
00:58:29,030 --> 00:58:33,710
in the particular vector space in another
kind of vector space a similar problem will
525
00:58:33,710 --> 00:58:40,010
be called as solving differential equations
initial value problems in some other vector
526
00:58:40,010 --> 00:58:47,770
space this problem will be called as a problem
which is partial differential equation okay
527
00:58:47,770 --> 00:58:59,260
so if you understand this grand generalization
very briefly then it helps us to develop discretization
528
00:58:59,260 --> 00:59:02,730
in a better way into the computable forms
529
00:59:02,730 --> 00:59:12,310
so basic problem you can show is that is nothing
but operator operating on a vector giving
530
00:59:12,310 --> 00:59:21,550
another vector and there are 3 problems associated
with this fundamental equation is either given
531
00:59:21,550 --> 00:59:29,470
the operator and a vector find the solution
so given operator say t operating on a vector
532
00:59:29,470 --> 00:59:41,410
x find y the second problem that you encounter
is given operator at t and y find x that means
533
00:59:41,410 --> 00:59:48,810
i know the solution i know the effect i want
to find out the cause so operator at t when
534
00:59:48,810 --> 00:59:51,369
it operates on x gives me y
535
00:59:51,369 --> 00:59:57,200
i know y i know t i want to find out x these
are called as inverse problems the first problem
536
00:59:57,200 --> 01:00:04,040
where you look at or you are given operator
and you are given x you find out y is called
537
01:00:04,040 --> 01:00:11,180
as direct problems our course is mostly going
to be dealing with inverse problems that is
538
01:00:11,180 --> 01:00:17,950
given an operator operating on a vector and
you are given y the effect then you want to
539
01:00:17,950 --> 01:00:28,260
find out cause that is x then you know we
will look at specific tools that are used
540
01:00:28,260 --> 01:00:30,040
in problem approximation
541
01:00:30,040 --> 01:00:37,960
what it turns out is that the backbone of
approximation is approximating a given function
542
01:00:37,960 --> 01:00:45,210
using set of polynomials okay it is the fundamental
theorem in approximation theory called as
543
01:00:45,210 --> 01:00:51,930
weierstrass approximation theorem and this
lays the foundation of all the problem discretization
544
01:00:51,930 --> 01:00:57,980
methods that are used in numerical analysis
okay so this particular theorem states that
545
01:00:57,980 --> 01:01:04,120
any continuous function over a finite domain
can be approximated with arbitrary degree
546
01:01:04,120 --> 01:01:09,190
of accuracy using a set of polynomials okay
547
01:01:09,190 --> 01:01:15,200
so it does not tell you which polynomial to
use it just tells you the existence of such
548
01:01:15,200 --> 01:01:20,720
a polynomial approximation well it is up to
us to construct the polynomial approximations
549
01:01:20,720 --> 01:01:28,750
but the study of weiestrass theorem very briefly
we will give you the foundation of how this
550
01:01:28,750 --> 01:01:35,070
whole business is done of approximating or
how transforming a problem original problem
551
01:01:35,070 --> 01:01:37,390
into a computable form
552
01:01:37,390 --> 01:01:42,970
so we will just very briefly look at the weiestrass
approximation theorem and then we will one
553
01:01:42,970 --> 01:01:49,860
by one start looking at commonly used polynomial
approximations okay so which is the most commonly
554
01:01:49,860 --> 01:01:54,520
used polynomial approximation as i said the
most commonly used polynomial approximation
555
01:01:54,520 --> 01:02:04,940
is taylor series approximation so this is
used in variety of numerical tools for example
556
01:02:04,940 --> 01:02:10,619
for solving or developing this method called
method of finite difference
557
01:02:10,619 --> 01:02:19,970
method of finite difference is used for discretization
of ordinary differential equations boundary
558
01:02:19,970 --> 01:02:31,180
value problems ode bvp they get transformed
into set of algebraic equations this method
559
01:02:31,180 --> 01:02:40,600
is also used for transforming partial differential
equations into set of algebraic equations
560
01:02:40,600 --> 01:02:52,140
okay so we will also study this method in
a different context for example you probably
561
01:02:52,140 --> 01:02:59,609
are familiar with newtons method or sometimes
called as newton raphson method for solving
562
01:02:59,609 --> 01:03:01,010
nonlinear algebraic equations
563
01:03:01,010 --> 01:03:11,540
and again this method originates from taylor
series approximation that is approximating
564
01:03:11,540 --> 01:03:16,920
a nonlinear differential equation or nonlinear
set of equations locally using taylor series
565
01:03:16,920 --> 01:03:25,210
and then converting into a set of a sequence
of linear algebraic equation problems so we
566
01:03:25,210 --> 01:03:30,290
will look at taylor series approximation as
a fundamental tool and how it is applied to
567
01:03:30,290 --> 01:03:31,320
do problem transformations
568
01:03:31,320 --> 01:03:36,690
a variety of problem transformations transforming
a partial differential equation transforming
569
01:03:36,690 --> 01:03:44,930
boundary value problem transforming set of
nonlinear algebraic equations then we continue
570
01:03:44,930 --> 01:03:51,150
our journey into other type of approximation
the second most important or not second most
571
01:03:51,150 --> 01:03:58,760
important equally important approximation
is polynomial interpolations so in the beginning
572
01:03:58,760 --> 01:04:02,751
we will have a brief understanding of lagrange
interpolation
573
01:04:02,751 --> 01:04:12,040
well it is a large vast area and then we cannot
do justice to every aspect of interpolation
574
01:04:12,040 --> 01:04:18,140
i am just going to give you a brief introduction
to some important concepts so we will begin
575
01:04:18,140 --> 01:04:22,230
with lagrange interpolation we will move on
to piecewise polynomial approximations or
576
01:04:22,230 --> 01:04:29,480
interpolations or not approximations piecewise
polynomial interpolation and then we will
577
01:04:29,480 --> 01:04:35,140
also look at not just polynomial interpolation
we will also look at function interpolations
578
01:04:35,140 --> 01:04:36,140
okay
579
01:04:36,140 --> 01:04:43,680
so linearly independent functions are used
to construct interpolating functions and then
580
01:04:43,680 --> 01:04:50,119
we will look at problem discretization using
this approach so i am going to again look
581
01:04:50,119 --> 01:04:54,240
at a boundary value problem ordinary differential
equation boundary value problem and discretize
582
01:04:54,240 --> 01:05:01,140
it using interpolation polynomials or i am
going to discretize a boundary value partial
583
01:05:01,140 --> 01:05:04,970
differential equation using interpolation
polynomial
584
01:05:04,970 --> 01:05:13,040
so this is my next task that is study how
interpolation plays a role in problem discretization
585
01:05:13,040 --> 01:05:19,890
in particular we are going to look at this
method of orthogonal collocations which is
586
01:05:19,890 --> 01:05:26,160
a very powerful method used in solving variety
of chemical engineering problems and then
587
01:05:26,160 --> 01:05:36,800
have a brief probably look at orthogonal collocations
on finite elements so the third important
588
01:05:36,800 --> 01:05:43,500
tool or third important approach that is used
for problem discretization is least squares
589
01:05:43,500 --> 01:05:52,090
so we are going to study various ways of approximating
problems using method of least squares first
590
01:05:52,090 --> 01:06:01,750
we will develop analytical solution of linear
least square problem okay look at its geometric
591
01:06:01,750 --> 01:06:10,160
interpolations this will give us insight that
is very very valuable that can be you know
592
01:06:10,160 --> 01:06:19,570
extended when we understand approximations
in higher dimensional spaces and then we will
593
01:06:19,570 --> 01:06:28,440
actually extend this idea to general spaces
or general hilbert spaces
594
01:06:28,440 --> 01:06:34,650
so the fundamental to this least square approximation
is the idea of projections now projections
595
01:06:34,650 --> 01:06:42,330
we normally study in engineering engineering
drawing or we study projections even starting
596
01:06:42,330 --> 01:06:50,180
at a school where you were to project find
a nearest point in a plane from a given point
597
01:06:50,180 --> 01:06:55,230
outside the plane so projections are very
very important and how do these idea projections
598
01:06:55,230 --> 01:07:00,340
is used in the problem approximation is what
we want to study next
599
01:07:00,340 --> 01:07:07,200
so we will also have a brief peek at function
approximation based models and the formulation
600
01:07:07,200 --> 01:07:20,240
of the parameter estimation problem and in
this before we move on to the main the remaining
601
01:07:20,240 --> 01:07:28,230
part that is understanding the tools we will
also look at least square problems for linear
602
01:07:28,230 --> 01:07:33,130
in parameters models least square formulations
for nonlinear in parameters models so in particular
603
01:07:33,130 --> 01:07:36,190
we are going to look at a method called gauss
newton method
604
01:07:36,190 --> 01:07:43,441
so this gauss newton method is a combination
of least squares and taylor series so we look
605
01:07:43,441 --> 01:07:47,350
at this taylor series approximation and least
square approximation so we are going to look
606
01:07:47,350 --> 01:07:57,760
at this method and then finally move to problem
transformations which we have been already
607
01:07:57,760 --> 01:08:02,070
looking at that is how do you transform a
boundary value problem or how do you discretize
608
01:08:02,070 --> 01:08:07,840
a partial differential equation using method
of least squares
609
01:08:07,840 --> 01:08:17,790
so these methods are known as method of minimum
residual methods so a popular method in this
610
01:08:17,790 --> 01:08:27,230
class is gelarkin method and we will have
wewill study this method actually the discretization
611
01:08:27,230 --> 01:08:34,109
of ordinary differential equations boundary
value problems or partial differential equations
612
01:08:34,109 --> 01:08:40,489
using least square approach leads to the so
called finite element methods we will not
613
01:08:40,489 --> 01:08:45,869
go in depth into this but we will have a very
brief introduction to what element finite
614
01:08:45,869 --> 01:08:52,099
element method using and how it is related
to least square approximation
615
01:08:52,099 --> 01:09:03,960
so with this we will come to an end to of
our module which talks about problem transformations
616
01:09:03,960 --> 01:09:11,940
so this will almost we come to half of the
course now what remains to be done is attack
617
01:09:11,940 --> 01:09:18,099
the problems which are transformed before
that we will very briefly look at what are
618
01:09:18,099 --> 01:09:25,719
the errors that come up in problem transformations
and what are the approximation errors and
619
01:09:25,719 --> 01:09:31,589
what it is bearing on the solutions numerical
solutions
620
01:09:31,589 --> 01:09:36,880
so after having done this after having transformed
the problem now we begin our journey into
621
01:09:36,880 --> 01:09:47,499
tools okay the first tool that we are going
to look at is solving linear algebraic equations
622
01:09:47,499 --> 01:09:52,390
and here well you might wonder we have been
solving linear algebraic equations since school
623
01:09:52,390 --> 01:09:57,409
days what is so new about it what am i going
to learn about it maybe you are already familiar
624
01:09:57,409 --> 01:09:59,440
with gaussian elimination
625
01:09:59,440 --> 01:10:10,079
and then in gaussian elimination you may have
studied even some advanced things like when
626
01:10:10,079 --> 01:10:17,889
you know how to do pivoting and so on but
there is much more to linear equation solving
627
01:10:17,889 --> 01:10:23,280
than just gaussian elimination there are many
other methods there are iterative methods
628
01:10:23,280 --> 01:10:33,131
for solving linear algebraic equations and
we are going to have look at them even optimization
629
01:10:33,131 --> 01:10:37,499
methods based methods or numerical optimization
based methods are used to solve linear algebraic
630
01:10:37,499 --> 01:10:41,400
equations and we will be studying those equations
631
01:10:41,400 --> 01:10:46,530
but apart from studying these numerical schemes
i am going to discuss one very important thing
632
01:10:46,530 --> 01:10:55,499
here that is matrix conditioning matrix conditioning
talks about how well posed or how ill posed
633
01:10:55,499 --> 01:11:01,870
a given problem is a given set of linear equations
are and then that gives your insight into
634
01:11:01,870 --> 01:11:10,630
behaviour of the numerical solution it may
happen that you have ill posed problem and
635
01:11:10,630 --> 01:11:14,889
then the solution that you compute numerically
is not quite reliable
636
01:11:14,889 --> 01:11:21,639
you should be able to differentiate between
ill posed problem and not reliable solution
637
01:11:21,639 --> 01:11:29,510
and well posed problem but mistake that you
have made in computing the solution okay so
638
01:11:29,510 --> 01:11:34,929
this is possible using the concept of condition
number or matrix conditioning and we are going
639
01:11:34,929 --> 01:11:44,260
to have a look at these the concept of conditional
numbers as a part of this module so we will
640
01:11:44,260 --> 01:11:48,889
begin with the study of conditions for existence
of solutions for linear algebraic equations
641
01:11:48,889 --> 01:11:54,010
we move on to the geometric interpretation
of the solutions very very important so i
642
01:11:54,010 --> 01:12:01,949
look at the problem through 2 pictures a row
picture and a column picture we will look
643
01:12:01,949 --> 01:12:09,460
at the solution from a 2 different viewpoints
geometric viewpoints we will interpret the
644
01:12:09,460 --> 01:12:17,100
what is the meaning of a singular matrix geometrically
and here essentially in the beginning we will
645
01:12:17,100 --> 01:12:24,090
just have a some understanding of 4 fundamental
subspaces associated with the matrix row space
646
01:12:24,090 --> 01:12:29,599
column space null space and left null space
647
01:12:29,599 --> 01:12:36,059
so up to now we were not talking about any
numerical scheme where our solution scheme
648
01:12:36,059 --> 01:12:40,909
we were talking about problem transformation
and just now i started about solving linear
649
01:12:40,909 --> 01:12:45,289
algebraic equations but even in the beginning
i am talking about geometric ideas and now
650
01:12:45,289 --> 01:12:51,139
we will move into numerical schemes okay the
first time in this course will be encountering
651
01:12:51,139 --> 01:12:52,170
actual numerical schemes
652
01:12:52,170 --> 01:13:01,960
so first of course i am going to look at gaussian
elimination very briefly and lu decomposition
653
01:13:01,960 --> 01:13:12,289
and will spend some time on the number of
computations that are required in carrying
654
01:13:12,289 --> 01:13:18,870
out a gaussian elimination process and see
whether you know there are methods that can
655
01:13:18,870 --> 01:13:24,479
even improve that can even reduce the number
of computations
656
01:13:24,479 --> 01:13:31,780
so the main focus in this part is going to
be introduction to the iterative methods but
657
01:13:31,780 --> 01:13:37,059
before that we will look at some special methods
for solving linear algebraic equations and
658
01:13:37,059 --> 01:13:43,449
these are going to be called methods for sparse
linear systems so many problems have very
659
01:13:43,449 --> 01:13:51,260
nice structure sparse systems are one in which
lot of elements are zeros and there are only
660
01:13:51,260 --> 01:13:54,690
few nonzero elements in a big matrix
661
01:13:54,690 --> 01:13:59,699
in solving problems which are large scale
let us say you are doing simulation of a section
662
01:13:59,699 --> 01:14:06,170
of a plant you may have thousands of equations
and when you actually start solving them let
663
01:14:06,170 --> 01:14:10,909
us say by newtons method you linearize them
when you linearize them you get linear set
664
01:14:10,909 --> 01:14:15,949
of equations which are say 1000 plus 1000
or 10 000 plus 10 000 but this matrix which
665
01:14:15,949 --> 01:14:21,889
is 10 000 plus 10 000 may not be fully populated
it will have many many zeros and it is possible
666
01:14:21,889 --> 01:14:28,949
to take advantage of this structure and then
come up with special schemes
667
01:14:28,949 --> 01:14:33,309
so these are called as schemes for sparse
linear systems and we are going to look at
668
01:14:33,309 --> 01:14:40,420
just few of them it is an iceberg and we can
only touch the tip of the iceberg so i am
669
01:14:40,420 --> 01:14:46,760
going to look at block diagonal matrices i
am going to present the thomas algorithm for
670
01:14:46,760 --> 01:14:54,559
tridiagonal matrices and block tridiagonal
matrices we will look at triangular matrices
671
01:14:54,559 --> 01:15:00,630
and block triangular matrices but as i said
this is only a brief introduction and we are
672
01:15:00,630 --> 01:15:05,440
going to move on to the iterative schemes
673
01:15:05,440 --> 01:15:09,300
the main thing here is to familiarize you
with the notion of sparse matrices and then
674
01:15:09,300 --> 01:15:16,100
maybe when you encounter them you will remember
to use them in your application the study
675
01:15:16,100 --> 01:15:21,909
of iterative solutions of or study of solving
linear algebraic equations using iterative
676
01:15:21,909 --> 01:15:25,942
solution scheme is the next component that
we will look at so there are variability of
677
01:15:25,942 --> 01:15:32,260
iterative schemes you start with the guess
solution and then you iteratively refine the
678
01:15:32,260 --> 01:15:37,130
solution and finally you approach the true
solution this is the iterative approach
679
01:15:37,130 --> 01:15:44,639
and this we are going to study different methods
very popular methods in this category are
680
01:15:44,639 --> 01:15:51,360
jacobi method or gauss siedel method or the
relaxation method so we will study these methods
681
01:15:51,360 --> 01:15:56,429
their algorithms but more importantly we will
study the convergence analysis of these iterative
682
01:15:56,429 --> 01:16:00,239
schemes i am going to spend quite a bit of
time in understanding the convergence of these
683
01:16:00,239 --> 01:16:01,239
schemes
684
01:16:01,239 --> 01:16:08,389
the question is if i start with a particular
guess what is the guarantee that the solution
685
01:16:08,389 --> 01:16:13,729
iterative scheme will converge to the solution
of solving linear algebraic equations so that
686
01:16:13,729 --> 01:16:25,090
will be you know that will be analysed systematically
using concept of eigen values and we will
687
01:16:25,090 --> 01:16:32,170
see the rules of eigen values in speed of
conversions or the conversions itself and
688
01:16:32,170 --> 01:16:35,929
then we will look at some special form of
matrices that enhance convergence
689
01:16:35,929 --> 01:16:43,780
we then move on to optimization based schemes
for solving linear algebraic equations okay
690
01:16:43,780 --> 01:16:51,070
so here i am going to use a numerical optimization
tools such as gradient search method or conjugate
691
01:16:51,070 --> 01:16:58,199
gradient method to solve set of linear algebraic
equations that is solving ax equal to b is
692
01:16:58,199 --> 01:17:03,341
going to be done using optimization it turns
out that in many situations this can be a
693
01:17:03,341 --> 01:17:09,119
very fast tools particularly when you are
solving large set of equations
694
01:17:09,119 --> 01:17:16,840
and in the end of this module i am going to
understand i am going to present the concept
695
01:17:16,840 --> 01:17:23,289
of matrix conditioning or condition number
of a matrix and its relationship with behaviour
696
01:17:23,289 --> 01:17:29,929
of numerical solutions of linear algebraic
equations so we will end with a deeper understanding
697
01:17:29,929 --> 01:17:36,860
into how good or how bad a numerical solution
is and we associate that with the conditioning
698
01:17:36,860 --> 01:17:39,290
of the matrix
699
01:17:39,290 --> 01:17:44,590
we then move on to the next tool the next
tool that i am going to study is going to
700
01:17:44,590 --> 01:17:56,519
be solving nonlinear algebraic equation so
in this toolbox well nonlinear equations are
701
01:17:56,519 --> 01:18:01,119
more often encountered than the real equations
most of the real engineering problems or real
702
01:18:01,119 --> 01:18:07,999
engineering models consist of nonlinear coupled
equations you do not have them in single variables
703
01:18:07,999 --> 01:18:15,409
you have multiple variables which are coupled
which give rise to coupled nonlinear algebraic
704
01:18:15,409 --> 01:18:16,409
equations
705
01:18:16,409 --> 01:18:21,979
if you are modelling section of a plant and
understanding the steady state behaviour of
706
01:18:21,979 --> 01:18:27,840
energy and material balance it might be thousands
of coupled nonlinear algebraic equations that
707
01:18:27,840 --> 01:18:33,010
need to be solved simultaneously that is very
very important in this method in this particular
708
01:18:33,010 --> 01:18:39,150
module we will look at variety of iterative
methods that are used for solving nonlinear
709
01:18:39,150 --> 01:18:41,989
algebraic equations okay
710
01:18:41,989 --> 01:18:46,300
in the end we will also have a brief introduction
to the convergence analysis of these methods
711
01:18:46,300 --> 01:18:53,440
based on a famous principle in function analysis
called as contraction mapping principle so
712
01:18:53,440 --> 01:19:01,239
again this is just a brief introduction to
let you to tell you that what goes in in understanding
713
01:19:01,239 --> 01:19:07,960
the convergence analysis of this scheme so
we will begin with the method of successive
714
01:19:07,960 --> 01:19:14,139
substitutions this is one of the very preliminary
methods which is used
715
01:19:14,139 --> 01:19:19,719
these are derivative free methods so there
are a variety of derivative free methods like
716
01:19:19,719 --> 01:19:24,849
jacobi iterations or gaussian iterations or
relaxation iterations we will study these
717
01:19:24,849 --> 01:19:34,610
methods and then from this we will move on
to derivative based iterative methods the
718
01:19:34,610 --> 01:19:41,230
well known derivative based internet methods
are newtons method so we will first look at
719
01:19:41,230 --> 01:19:48,130
univaraite newton type methods where you find
out the local derivatives either exactly or
720
01:19:48,130 --> 01:19:50,360
approximately
721
01:19:50,360 --> 01:19:57,130
then we will formulate a multivariate secant
method which is an approximate derivative
722
01:19:57,130 --> 01:20:04,980
based method or popularly known as wegstein
iterations then we will move on to the wellknown
723
01:20:04,980 --> 01:20:12,469
newtons method and look at its variations
like damped newton method you can try to improve
724
01:20:12,469 --> 01:20:21,469
the convergence behaviour or we will develop
numerically more friendly versions of newtons
725
01:20:21,469 --> 01:20:29,940
method which are you know called as quasi
newton methods or with rank 1 updates of the
726
01:20:29,940 --> 01:20:31,099
jacobian matrix
727
01:20:31,099 --> 01:20:35,630
the problem with newton method is that you
have to compute derivative matrix jacobian
728
01:20:35,630 --> 01:20:41,110
matrix if there are any questions and n variables
every iteration you have to compute an n cross
729
01:20:41,110 --> 01:20:48,010
n matrix and this can be numerically quite
complex if you have thousands of equations
730
01:20:48,010 --> 01:20:55,909
this quasi newton method allow you to do approximate
update of the jacobian so they construct a
731
01:20:55,909 --> 01:21:02,369
new jacobian using the old jacobian and this
way they save computations so we are going
732
01:21:02,369 --> 01:21:05,110
to have a brief introduction to this quasi
newton methods
733
01:21:05,110 --> 01:21:15,459
then we move on to solving nonlinear algebraic
equations using optimization optimization
734
01:21:15,459 --> 01:21:22,150
numerical optimization is a powerful tool
which is used for solving non linear problems
735
01:21:22,150 --> 01:21:29,079
nonlinear algebraic equations one of the popular
method in this class is conjugate gradient
736
01:21:29,079 --> 01:21:34,080
method so we will have a brief look at conjugate
gradient method this is a gradient based method
737
01:21:34,080 --> 01:21:39,739
there is a hessian or second order derivative
based method which are called in this category
738
01:21:39,739 --> 01:21:40,909
they are called as newtons method
739
01:21:40,909 --> 01:21:47,179
we also have quasi newton method which are
again simplifications of newtons method or
740
01:21:47,179 --> 01:21:51,619
hessian based methods so we will have a brief
peek or brief introduction to quasi newton
741
01:21:51,619 --> 01:21:58,469
methods and finally we will look at a method
called leverberg marquardt method which is
742
01:21:58,469 --> 01:22:03,170
combination of the gradient method and newtons
method so you use gradient when it is helpful
743
01:22:03,170 --> 01:22:08,869
to use gradient you use hessian when you it
is helpful to use hessian so it is a merger
744
01:22:08,869 --> 01:22:10,329
of the 2 methods
745
01:22:10,329 --> 01:22:16,719
and we will just understand this towards the
end we will just briefly understand the concept
746
01:22:16,719 --> 01:22:21,739
of condition number of set of nonlinear equations
you cannot have one condition number you can
747
01:22:21,739 --> 01:22:28,780
define a local concept of condition number
here which is conceptually similar qualitatively
748
01:22:28,780 --> 01:22:35,510
similar to what we have done for linear algebraic
equations so before we wind up this particular
749
01:22:35,510 --> 01:22:42,989
module we look at 2 important aspects
750
01:22:42,989 --> 01:22:49,449
one was existence of solution of nonlinear
algebraic equations and its relation to convergence
751
01:22:49,449 --> 01:22:55,060
of iterative methods okay in the when we started
studying linear algebraic equations we began
752
01:22:55,060 --> 01:23:00,369
with the conditions for existence of solutions
we never talked about this when we started
753
01:23:00,369 --> 01:23:04,969
solving nonlinear algebraic equations here
i want to give a brief introduction to the
754
01:23:04,969 --> 01:23:07,499
conditions of existence of solutions
755
01:23:07,499 --> 01:23:15,300
and what is its relation to convergence of
iterative methods we look at contraction mapping
756
01:23:15,300 --> 01:23:22,999
principle or contraction mapping theorem we
will apply to understand convergence of method
757
01:23:22,999 --> 01:23:29,900
of successive substitutions we will also see
how contraction mapping principle can be used
758
01:23:29,900 --> 01:23:38,159
to analyse newtons method or newton raphson
method and with this we have come to or we
759
01:23:38,159 --> 01:23:44,239
will come to an end of module 5 which is on
solving non linear algebraic equations so
760
01:23:44,239 --> 01:23:50,380
we move on to the last tool that will be discussed
in this course that is solving ordinary differential
761
01:23:50,380 --> 01:23:52,929
equations initial value problems
762
01:23:52,929 --> 01:23:58,699
so this is another fundamental tool which
can be used to attack or to solve the transform
763
01:23:58,699 --> 01:24:07,781
problem so what are the learning objectives
here as it is evident from problem transformation
764
01:24:07,781 --> 01:24:13,979
module that many situations when you transform
a problem you get ordinary differential equations
765
01:24:13,979 --> 01:24:20,409
initial value problems so this is one of the
fundamental model type or equation type which
766
01:24:20,409 --> 01:24:21,729
needs to be dealt with
767
01:24:21,729 --> 01:24:31,059
and we have to arrive at or we have to develop
special methods to solve this class of problems
768
01:24:31,059 --> 01:24:41,479
so in the beginning we will very briefly introduce
the conditions for existence and uniqueness
769
01:24:41,479 --> 01:24:46,190
of solutions of ordinary differential equation
initial value problem this is a very very
770
01:24:46,190 --> 01:24:52,500
brief introduction and then we immediately
move to study of analytical solutions of linear
771
01:24:52,500 --> 01:24:55,849
ordinary differential equations in multiple
variables
772
01:24:55,849 --> 01:25:02,389
well you might wonder why am i doing this
analytical solution in a course which is meant
773
01:25:02,389 --> 01:25:10,750
to be for constructing numerical solutions
well this analytical solution part gives in
774
01:25:10,750 --> 01:25:17,780
depth understanding how local solutions behave
also this is going to help us when we understand
775
01:25:17,780 --> 01:25:23,369
or when we analyse convergence behaviour of
numerical schemes for solving ode ivp so as
776
01:25:23,369 --> 01:25:29,940
a background to develop numerical schemes
i am going to solve analytically linear ordinary
777
01:25:29,940 --> 01:25:33,030
differential equations given initial conditions
778
01:25:33,030 --> 01:25:39,070
so i will start with a scalar equation move
on to vector equations and then what is critical
779
01:25:39,070 --> 01:25:45,530
here is that i want to relate so what are
these kind of equations i am going to look
780
01:25:45,530 --> 01:25:53,280
at dx by dt equal to ax where a is a matrix
and then i want to understand relationship
781
01:25:53,280 --> 01:26:00,130
between the eigen values of matrix a and analytical
solution of this differential equation dx
782
01:26:00,130 --> 01:26:11,010
by dt equal to ax then actually you can get
the eigen values of this matrix i can qualitatively
783
01:26:11,010 --> 01:26:16,739
tell how the solution is going to behave asymptotically
as time goes to infinity
784
01:26:16,739 --> 01:26:21,510
so just looking at the eigen values we can
analyse the behaviour of the solutions and
785
01:26:21,510 --> 01:26:29,119
this elegant part we are going to study briefly
and then what is the relationship of linear
786
01:26:29,119 --> 01:26:33,780
equations and local linearization through
taylor series approximation is what we are
787
01:26:33,780 --> 01:26:42,310
going to look at here at end of this sub module
we now move to the proper numerical methods
788
01:26:42,310 --> 01:26:44,690
for solving ode ivp
789
01:26:44,690 --> 01:26:50,900
so before that we need to understand some
basic concepts like marching in time how do
790
01:26:50,900 --> 01:26:55,230
you develop a solution you want to solve a
problem you want to integrate a differential
791
01:26:55,230 --> 01:27:01,800
equations from some time 0 to time infinity
you actually do it in small steps this is
792
01:27:01,800 --> 01:27:09,810
marching in time so we will talk about this
if you look at the methods for solving numerical
793
01:27:09,810 --> 01:27:13,179
methods for solving ode initial value problems
there are 2 classes
794
01:27:13,179 --> 01:27:19,960
one is explicit methods other are implicit
methods so we will just have understanding
795
01:27:19,960 --> 01:27:25,599
of what is an implicit method what is an explicit
method and then we move on to study and important
796
01:27:25,599 --> 01:27:31,690
class of methods which is based on taylor
series approximation popularly these methods
797
01:27:31,690 --> 01:27:36,780
are known as runge kutta methods they actually
arise from taylor series approximation and
798
01:27:36,780 --> 01:27:41,790
this is where i relate it to the approximation
theory part that we have done earlier
799
01:27:41,790 --> 01:27:49,079
so we will actually derive here runge kutta
methods starting from basics initially for
800
01:27:49,079 --> 01:27:57,530
a scalar case and then move on to the multivariate
case we then move on to the next important
801
01:27:57,530 --> 01:28:04,179
method which is based on polynomial interpolation
so again you will see that the ideas of ideas
802
01:28:04,179 --> 01:28:10,590
of approximation theory are playing a role
when you are actually solving ordinary differential
803
01:28:10,590 --> 01:28:11,920
equation initial value problem
804
01:28:11,920 --> 01:28:19,010
so those ideas are so fundamental they just
are everywhere in numerical analysis so we
805
01:28:19,010 --> 01:28:25,110
are going to study methods called as multi
step methods okay or popularly known as predictor
806
01:28:25,110 --> 01:28:29,659
corrector methods okay we will develop we
will derive these algorithms starting from
807
01:28:29,659 --> 01:28:36,599
scratch starting from interpolation polynomials
and first for the scalar case and see how
808
01:28:36,599 --> 01:28:39,850
they can be generalized to multivariate case
809
01:28:39,850 --> 01:28:43,949
and then move on to solving initial value
problems ordinary differential equation initial
810
01:28:43,949 --> 01:28:54,670
value problems using orthogonal collocations
well after that we actually have a brief look
811
01:28:54,670 --> 01:29:01,420
at convergence analysis of numerical schemes
for solving initial value problems ode initial
812
01:29:01,420 --> 01:29:08,539
value problems and what is its relationship
with selection of integration step size when
813
01:29:08,539 --> 01:29:13,229
you are integrating nonlinear differential
equations one of the key things is how do
814
01:29:13,229 --> 01:29:15,769
you select integration step size okay
815
01:29:15,769 --> 01:29:22,760
to greater understanding into this we have
to have some understanding of you know convergence
816
01:29:22,760 --> 01:29:29,690
analysis so we will analyse of course linear
ordinary differential equation initial value
817
01:29:29,690 --> 01:29:36,639
problems use and we will apply approximate
solutions to these linear problems we already
818
01:29:36,639 --> 01:29:40,920
know their exact solutions and then we can
compare exact solution with approximate solution
819
01:29:40,920 --> 01:29:46,059
and get insights that is the reason i introduced
analytical solution of linear ode ivp in the
820
01:29:46,059 --> 01:29:48,110
beginning
821
01:29:48,110 --> 01:29:55,050
then we will see how this can be extended
to nonlinear ode ivps we will look at few
822
01:29:55,050 --> 01:30:01,909
concepts which are important in solving these
equations like stiff ordinary differential
823
01:30:01,909 --> 01:30:09,599
equations so stiffness of odes is what we
look at and then finally we look at what are
824
01:30:09,599 --> 01:30:16,320
called as variable step size implementation
of these ode ivp schemes with accuracy monitoring
825
01:30:16,320 --> 01:30:22,630
so these are all involved concepts of course
most of the tools that we use today most of
826
01:30:22,630 --> 01:30:28,179
the programs are available will have these
built in tools you should know when to use
827
01:30:28,179 --> 01:30:33,360
which one to use why to use particular choice
828
01:30:33,360 --> 01:30:36,900
if you have a stiff differential equation
you should use a particular tool if you have
829
01:30:36,900 --> 01:30:44,159
you know variables which are which have too
much difference in their timescales you should
830
01:30:44,159 --> 01:30:51,610
use variable steps as implementation and so
on so these things become very very important
831
01:30:51,610 --> 01:31:00,090
when it comes to in the end i am going to
talk about solving differential algebraic
832
01:31:00,090 --> 01:31:04,579
equations we have studied differential equations
we have studied algebraic equations nonlinear
833
01:31:04,579 --> 01:31:09,730
algebraic equations just a brief look at how
do you solve differential algebraic equations
834
01:31:09,730 --> 01:31:14,130
if they are encountered together
835
01:31:14,130 --> 01:31:19,570
then we will look at a special method for
solving ordinary differential equations boundary
836
01:31:19,570 --> 01:31:27,179
value problems called method of or a shooting
method so actually you use an initial value
837
01:31:27,179 --> 01:31:31,340
problem solver to solve the boundary value
problem okay so how this is done to look at
838
01:31:31,340 --> 01:31:42,989
this method and then again we look at conversion
analysis of solvers for ode ivp so this brings
839
01:31:42,989 --> 01:31:49,930
us to an end of this 6 modules introduction
to these 6 modules
840
01:31:49,930 --> 01:31:55,619
so if i want to sum up what is what is you
know overall learning objective in this course
841
01:31:55,619 --> 01:32:02,739
in this well first is you should know how
to transform a mathematical problem at hand
842
01:32:02,739 --> 01:32:09,309
into a computable form using of course principles
of approximations here that is the almost
843
01:32:09,309 --> 01:32:17,309
half the course is devoted to that then understand
basic properties of different tools particularly
844
01:32:17,309 --> 01:32:22,889
3 different tools solving linear algebraic
equations solving non linear algebraic equations
845
01:32:22,889 --> 01:32:28,639
and solving ordinary differential equations
subject to or given initial values or initial
846
01:32:28,639 --> 01:32:32,340
conditions odi ivps
847
01:32:32,340 --> 01:32:37,269
understand different methods of different
numerical schemes for solving these standard
848
01:32:37,269 --> 01:32:42,639
class of problems and understand their limitations
so that if you understand their limitations
849
01:32:42,639 --> 01:32:46,539
if you understand their strengths if you understand
how they are developed you have been much
850
01:32:46,539 --> 01:32:52,650
better position to employ them use them to
concoct a recipe finally what i wanted to
851
01:32:52,650 --> 01:32:58,479
learn or to understand is that a numerical
scheme is actually like a recipe and we are
852
01:32:58,479 --> 01:33:05,079
going to be a cook who will actually be able
to cook a recipe cook a recipe for a given
853
01:33:05,079 --> 01:33:06,150
problem
854
01:33:06,150 --> 01:33:11,429
so you have these fundamental tools you have
some fundamental tools coming from approximation
855
01:33:11,429 --> 01:33:17,110
theory use a combination of them first combination
of tools from the approximation theory to
856
01:33:17,110 --> 01:33:22,920
transform the problem then you solve the transform
problem using standard toolkits that you have
857
01:33:22,920 --> 01:33:29,320
okay so this journey is going to be fairly
long it is about 48 lectures and we begin
858
01:33:29,320 --> 01:33:32,630
our journey from the next lecture thank you