1
00:00:18,070 --> 00:00:25,620
Okay given the element from the range space
and the operator T I want to find out x This
2
00:00:25,620 --> 00:00:32,840
is called as inverse problem okay and these
are the problems which we normally have to
3
00:00:32,840 --> 00:00:39,460
solve So the core of this particular course
is dealing with these kind of problems and
4
00:00:39,460 --> 00:00:54,410
a next class which is identification problems
So here what falls under this So
5
00:00:54,410 --> 00:01:03,399
solving Ax equal to b given A and b classical
problem which right hand side given operator
6
00:01:03,399 --> 00:01:06,200
you want to find out x right
7
00:01:06,200 --> 00:01:13,640
This is probably the problem which we will
solve most often in this course So we are
8
00:01:13,640 --> 00:01:32,229
given some differential equation
9
00:01:32,229 --> 00:01:36,930
so the other classical problem is ODE initial
value problem We are given a differential
10
00:01:36,930 --> 00:01:48,229
equation so this is the operator okay We are
given ft ft is equivalent to b or ft is equivalent
11
00:01:48,229 --> 00:01:56,710
to the vector in the range space I want to
find out solution xt which satisfies the condition
12
00:01:56,710 --> 00:02:02,369
that initial value equal to alpha and initial
rate equal to beta
13
00:02:02,369 --> 00:02:07,790
I have given two initial conditions I have
given the operator I have given the vector
14
00:02:07,790 --> 00:02:15,329
in the range space I want to find out solution
xt okay inverse problem Operator is known
15
00:02:15,329 --> 00:02:21,380
the range space vector is known okay initial
conditions are known I am going to find out
16
00:02:21,380 --> 00:02:31,100
xt inverse problem So likewise ODE boundary
value problem is an inverse problem or solving
17
00:02:31,100 --> 00:02:37,519
a partial differential equation that we encounter
in engineering mostly inverse problems
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00:02:37,519 --> 00:02:42,161
We are given the vector in the range space
we are given the operator we have to find
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00:02:42,161 --> 00:02:49,079
out x that satisfies the differential equation
boundary conditions and solution gives you
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00:02:49,079 --> 00:02:57,040
the vector in the range space okay So these
two problems are conceptually similar Ax equal
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00:02:57,040 --> 00:03:04,600
to b or this operator operating on xt giving
you this vector ft okay and then the solution
22
00:03:04,600 --> 00:03:06,820
should satisfy these two
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00:03:06,820 --> 00:03:14,120
So these kinds of problems are inverse problems
The third class of problems that you encounter
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00:03:14,120 --> 00:03:37,780
in the engineering mathematics is identification
problems
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00:03:37,780 --> 00:03:48,570
So you are given x and y and you are asked
to find out operator T okay The classic problem
26
00:03:48,570 --> 00:04:07,880
here is model parameter estimation Suppose
I want for some particular material you want
27
00:04:07,880 --> 00:04:10,940
to find out cp as a function of temperature
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00:04:10,940 --> 00:04:22,560
So you have this a plus bT plus cT square
I do not know a b c I have been given values
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00:04:22,560 --> 00:04:30,900
of cp I have been given values of temperature
okay so I have been given x I have been given
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00:04:30,900 --> 00:04:38,870
y y here is cp okay x here is temperature
what I want to find out is the correlation
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00:04:38,870 --> 00:04:44,820
is the operator I want to find out the operator
Finding operator in this case it is to finding
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00:04:44,820 --> 00:04:47,470
out a b and c okay
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00:04:47,470 --> 00:05:12,020
So given data parameter estimation problem
What are the other parameter estimation problem
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00:05:12,020 --> 00:05:20,120
you have seen in chemical engineering reaction
rate expressions You have measured the rate
35
00:05:20,120 --> 00:05:25,500
of change of concentration of particular spaces
and then you have a proposed expression okay
36
00:05:25,500 --> 00:05:32,220
You do not know the parameters okay You have
rate values you have concentration values
37
00:05:32,220 --> 00:05:33,650
you want to fit
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00:05:33,650 --> 00:05:40,320
Find out the parameters of the rate expression
or you know you are trying to fit some thermodynamic
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00:05:40,320 --> 00:05:46,690
correlation PVT correlation you have data
for PV and temperature and you have a proposed
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00:05:46,690 --> 00:05:51,920
model we do not know the parameters you can
fit estimate the parameters from data So you
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00:05:51,920 --> 00:05:59,470
are trying to find out the operator okay knowing
I mean if you look at y as a effect and x
42
00:05:59,470 --> 00:06:08,650
is a cause so operator T operates on x gives
y y is the effect
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00:06:08,650 --> 00:06:13,430
So you know cause and effect you want to find
out the operator Another example is estimation
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00:06:13,430 --> 00:06:18,680
of transfer functions in process control okay
You give a input perturbation you measured
45
00:06:18,680 --> 00:06:24,330
the output you tried to fit the transfer function
into the data okay all these are examples
46
00:06:24,330 --> 00:06:33,130
of identification problems okay So bulk of
our work in this courses going to be inverse
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00:06:33,130 --> 00:06:38,200
problems and then we will also look at identification
problems to a large extent
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00:06:38,200 --> 00:06:43,340
Direct problems are not going to be focused
I am not saying the direct problems are not
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00:06:43,340 --> 00:06:47,610
important but relatively easy to deal with
these two problems are more difficult and
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00:06:47,610 --> 00:06:54,930
we should get the better understanding of
these problems Now the main problem in most
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00:06:54,930 --> 00:06:59,930
of the cases not in every case in most of
the cases is that once you have formulated
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00:06:59,930 --> 00:07:07,180
a problem it may not be possible to construct
analytical solutions to the problem okay
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00:07:07,180 --> 00:07:14,060
Particularly if a operator is nonlinear okay
So I can say in general when operator T is
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00:07:14,060 --> 00:07:21,200
nonlinear you cannot construct analytical
solutions Well there are of course many exceptions
55
00:07:21,200 --> 00:07:27,020
but the cases where you cannot solve are far
more than the cases where you can solve so
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00:07:27,020 --> 00:07:36,460
in general you can say that when the operator
T is nonlinear
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00:07:36,460 --> 00:07:44,280
so lot of numerical analysis is all about
transforming a problem which is not analytically
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00:07:44,280 --> 00:07:50,550
computable to a form which is numerically
computable okay
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00:07:50,550 --> 00:08:01,970
So actually my original problem is y equal
to T of x okay and where you know y belongs
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00:08:01,970 --> 00:08:13,010
to space Y and x belongs to space X okay I
am not able to solve this original problem
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00:08:13,010 --> 00:08:37,940
so what I do is I approximate T using approximation
theory and then I get let us call it T cap
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00:08:37,940 --> 00:08:51,580
okay I get a T cap which actually works on
x tilde and gives me y tilde okay here y tilde
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00:08:51,580 --> 00:08:59,560
belongs to Yn and x tilde belongs to Xn
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00:08:59,560 --> 00:09:11,960
So these are finite dimensional spaces
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00:09:11,960 --> 00:09:20,010
typically and then we end up solving this
problem not this problem We hope that the
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00:09:20,010 --> 00:09:25,520
solution that you get suppose I solve a inverse
problem which is I wanted to solve the original
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00:09:25,520 --> 00:09:32,710
inverse problem I end up solving an approximate
inverse problem okay and I hope that the solution
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00:09:32,710 --> 00:09:41,250
x tilde is close to x okay
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00:09:41,250 --> 00:09:46,600
So this is generally the situation now how
do you get from here to here next about 10
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00:09:46,600 --> 00:09:50,930
to 12 lectures are going to be how do I go
from here to here So this looks abstract right
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00:09:50,930 --> 00:09:57,350
now but keep this in mind in background that
this is what we are going to do okay So we
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00:09:57,350 --> 00:10:02,770
might start with the partial differential
equation and end up with nonlinear algebraic
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00:10:02,770 --> 00:10:07,330
equations okay See this might be a partial
differential equation what do you end up here
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00:10:07,330 --> 00:10:13,060
might be linear algebraic equations or nonlinear
algebraic equations okay
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00:10:13,060 --> 00:10:19,000
So what you start with and what you end up
with can be completely different okay So it
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00:10:19,000 --> 00:10:23,150
is not that because I start with the differential
equation I will end up with the differential
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00:10:23,150 --> 00:10:28,550
equation okay Now when you go from here to
here there is no unique way of constructing
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00:10:28,550 --> 00:10:35,330
T cap there are multiple ways of constructing
T cap okay Same problem can be approximated
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00:10:35,330 --> 00:10:40,670
discretized in multiple possible ways that
is what we are going to see here okay
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00:10:40,670 --> 00:10:46,210
And each one of them has advantage and disadvantage
So there is nothing like though method to
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00:10:46,210 --> 00:10:54,363
discretize okay and as you go along doing
numerical problems you will develop your own
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00:10:54,363 --> 00:10:59,710
preferences as to so I am going to talk about
not just one method I am going to talk about
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00:10:59,710 --> 00:11:03,830
multiple methods So you might wonder why I
am talking about multiple methods because
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00:11:03,830 --> 00:11:06,870
there is no one way to solve the problems
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00:11:06,870 --> 00:11:11,730
Sometimes some methods are simple but if they
do not work you need to go to more complex
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00:11:11,730 --> 00:11:20,600
methods and so on okay So when you attack
a problem you should have a repository of
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00:11:20,600 --> 00:11:25,540
tools or repository of you know approaches
to deal with a problem and then you can go
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00:11:25,540 --> 00:11:30,000
on you know simple method first if does not
work go to a more complex method if it does
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00:11:30,000 --> 00:11:31,649
not work go to a more complex method
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00:11:31,649 --> 00:11:36,860
What you mean by does not work the proof of
the pudding is x tilde close to x does not
91
00:11:36,860 --> 00:11:43,700
make sense Now you in the real situation you
never know what is x true x but since you
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00:11:43,700 --> 00:11:47,830
are an engineer okay if you look at the solution
you can make out whether this makes physical
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00:11:47,830 --> 00:11:54,170
sense or not okay Whether the solution makes
sense as a engineer as a scientist as a physicist
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00:11:54,170 --> 00:11:59,040
you can make a judgment and then decide whether
your method is giving resemble results or
95
00:11:59,040 --> 00:12:00,420
not okay
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00:12:00,420 --> 00:12:07,770
So there is a lot of subjective element here
which requires development of expertise okay
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00:12:07,770 --> 00:12:13,529
So even though we are dealing with applied
math which everything cannot be automated
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00:12:13,529 --> 00:12:20,980
and that is why we are in business okay There
is still scope for improvement you know for
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00:12:20,980 --> 00:12:30,630
interpretations for doing it differently getting
better solutions and so on so now let us begin
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00:12:30,630 --> 00:12:36,080
okay So what is the basic trick that is
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00:12:36,080 --> 00:12:41,089
So if you ask me to distill out one basic
idea which is used to do this transformations
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00:12:41,089 --> 00:12:49,529
from T to T hat okay Cutting across all the
methods for boundary value problems or partial
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00:12:49,529 --> 00:12:56,110
differential equations for all kinds of things
what is one trick that is used Well if you
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00:12:56,110 --> 00:13:03,540
ask me to summit up I will say that approximate
a function by polynomial that is the trick
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00:13:03,540 --> 00:13:05,270
that is the underlying trick
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00:13:05,270 --> 00:13:10,610
You know if I cut across many of the methods
and what I am going to show in next few lectures
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is that how this one trick is used to you
know deal with variety of problems okay Starting
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00:13:16,680 --> 00:13:22,111
from nonlinear algebraic equations to partial
differential equations to boundary value problem
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00:13:22,111 --> 00:13:39,589
so all kinds of terms we just use one trick
in multiple ways So basic idea is approximate
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00:13:39,589 --> 00:13:45,959
Now the questions is well is it just an observation
or is the basis why should I approximate a
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00:13:45,959 --> 00:13:50,340
function by a polynomial function why not
cosine functions You know why not exponential
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00:13:50,340 --> 00:13:55,850
functions Why polynomials What is so get about
them Well of course they are convenient when
113
00:13:55,850 --> 00:14:00,600
you do calculations but not just that there
is something deeper into why polynomials are
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00:14:00,600 --> 00:14:10,680
used for approximating functions and then
developing different methods for solving the
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00:14:10,680 --> 00:14:11,970
problems
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00:14:11,970 --> 00:14:21,650
So fundamental concept here is the concept
of a dense set see approximations is something
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00:14:21,650 --> 00:14:29,140
that we very very often use in mathematics
for example pi you know when you start using
118
00:14:29,140 --> 00:14:40,529
pi you start using that 22 by 7 right But
pi is not equal to 22 by 7 it is an okay approximation
119
00:14:40,529 --> 00:14:49,060
of pi for doing you know rough calculations
not exact calculations When you start using
120
00:14:49,060 --> 00:14:56,089
for example E we never use the true value
E right
121
00:14:56,089 --> 00:15:03,600
We use an approximation a finite truncated
approximation of E and do calculations right
122
00:15:03,600 --> 00:15:10,800
So what allows you to do that What allows
you to do that is that you can approximate
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00:15:10,800 --> 00:15:20,380
a real number using a rational number okay
this property of rational number there is
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00:15:20,380 --> 00:15:23,860
something special about rational numbers You
can approximate any real number as close as
125
00:15:23,860 --> 00:15:27,420
you want by a rational number okay
126
00:15:27,420 --> 00:15:33,839
And this particular property is expected by
us when we do computations and the best example
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00:15:33,839 --> 00:15:38,769
I said is pi being replaced by 22 by 7 or
whatever There are different different rational
128
00:15:38,769 --> 00:15:47,220
approximations of pi you remember something
else some 141 by or there is some other approximations
129
00:15:47,220 --> 00:15:52,290
also which are not so popular in the school
books But what we use in school book is 22
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00:15:52,290 --> 00:15:53,610
by 7 right
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00:15:53,610 --> 00:16:03,180
We are always so why we can do this or I told
you that when you are doing computing all
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00:16:03,180 --> 00:16:11,470
the number are finite procedure right No number
in the computer so there will be missing numbers
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00:16:11,470 --> 00:16:17,120
if there is a finite procedure okay Not all
numbers can be represented particularly it
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00:16:17,120 --> 00:16:24,180
is not possible to represent you know all
real numbers pi may not be truly representable
135
00:16:24,180 --> 00:16:26,310
you can represent using some truncations
136
00:16:26,310 --> 00:16:32,550
Because when you do finite precision okay
if I write some expansion for pi I can write
137
00:16:32,550 --> 00:16:38,440
that integer divided by some 10 to the power
something and that will be a rational number
138
00:16:38,440 --> 00:16:49,079
right So it is not the correct value for pi
it is a rational approximation for pi okay
139
00:16:49,079 --> 00:17:50,769
So I am able to do this because of this denseness
property so what is a dense set
140
00:17:50,769 --> 00:17:52,419
So let us go over the definition
141
00:17:52,419 --> 00:17:59,809
A set D is said to be dense in a normed space
so first of all you have to worked in a normed
142
00:17:59,809 --> 00:18:10,509
space meaning norm okay without that we cannot
work both with these numbers or vectors So
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00:18:10,509 --> 00:18:18,539
for any element if I give you any element
x in x I should be able to find and if I give
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00:18:18,539 --> 00:18:26,779
you an epsilon I should be able to find an
element d belonging to dense set D which is
145
00:18:26,779 --> 00:18:32,759
close to x how close there is an epsilon
146
00:18:32,759 --> 00:18:41,630
So you know it is like saying if my x is pi
and if I specify epsilon you should be able
147
00:18:41,630 --> 00:18:47,020
to come up with a rational number which is
close to pi such that the difference is less
148
00:18:47,020 --> 00:18:53,620
than You know she might my epsilon is 10 to
the power minus 3 okay I will come up with
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00:18:53,620 --> 00:18:59,980
1d which is one rational approximation of
pi pi minus d is less than 10 to power minus
150
00:18:59,980 --> 00:19:03,639
3 and he comes up when says no no no I do
not accept 10 to the power minus 3
151
00:19:03,639 --> 00:19:11,309
I want 10 to the power minus 9 is it possible
It is possible to find given pi it is possible
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00:19:11,309 --> 00:19:18,289
to find rational approximation such that pi
minus that number is less than 10 to the power
153
00:19:18,289 --> 00:19:23,191
minus 9 and somebody does not accept 10 to
the power minus 9 and you know he says 10
154
00:19:23,191 --> 00:19:33,419
to the power minus 17 fine I can find a rational
number which is pi minus that rational number
155
00:19:33,419 --> 00:19:41,100
will be less than 10 to the power minus 17
So any epsilon that is very important okay
156
00:19:41,100 --> 00:19:47,609
So what does it mean that on real line these
rational numbers are everywhere you know I
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00:19:47,609 --> 00:19:52,429
can use them as approximation of something
else which I am not able to represent which
158
00:19:52,429 --> 00:19:59,860
is very nice okay So I can use rational numbers
as an approximation of a real number and that
159
00:19:59,860 --> 00:20:06,590
is why I can work in a computer okay So when
I am working in n dimensional space okay
160
00:20:06,590 --> 00:20:16,549
In Rn even though I may not be able to represent
you know all elements of the vector Rn because
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00:20:16,549 --> 00:20:24,580
a real number may not be exactly representable
I can replaced by its rational approximation
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00:20:24,580 --> 00:20:29,749
okay See suppose I have a vector which is
like this Suppose I have vector which is e
163
00:20:29,749 --> 00:20:42,409
minus pi pi square root 7 In a computer can
I really work with e minus pi I cannot right
164
00:20:42,409 --> 00:20:47,799
I actually replace this by some approximations
165
00:20:47,799 --> 00:20:53,810
So some rational approximation of e some rational
approximation of this some rational approximation
166
00:20:53,810 --> 00:21:03,059
of this and so on and why we can do this because
set of no no understand the philosophy why
167
00:21:03,059 --> 00:21:09,879
we can do this is because set of rational
numbers is a dense okay Rational numbers are
168
00:21:09,879 --> 00:21:15,911
everywhere you can just if you want to represent
a real number pick a very close rational number
169
00:21:15,911 --> 00:21:20,190
you know you will be having a good approximation
okay
170
00:21:20,190 --> 00:21:33,739
Now I want a similar result to this in set
of continuous functions okay What I am going
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00:21:33,739 --> 00:21:39,059
to work with now we have seen that you know
when you deal with partial differential equations
172
00:21:39,059 --> 00:21:41,850
when you deal with boundary value problems
when you deal with ordinary differential equations
173
00:21:41,850 --> 00:21:48,039
we will dealing with set of continuous functions
okay So this is nice here that you know I
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00:21:48,039 --> 00:21:53,289
can use rational approximations
175
00:21:53,289 --> 00:22:04,090
You know some q1 q2 q3 q4 so this is a rational
approximation of this and in my computer I
176
00:22:04,090 --> 00:22:10,830
can do calculations with this in the same
way I want analogy in set of continuous functions
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00:22:10,830 --> 00:22:17,840
This is a similar idea conceptually a similar
idea is that set of polynomials is dense in
178
00:22:17,840 --> 00:22:28,169
set of continuous functions
In the same sense the set of rational number
179
00:22:28,169 --> 00:22:32,859
is dense in set of real numbers okay
180
00:22:32,859 --> 00:22:38,130
Polynomials you know you can approximate anything
by a polynomial any continuous function by
181
00:22:38,130 --> 00:22:44,820
a polynomial so these are the Lindemann theorem
given by a German mathematician Weierstrass
182
00:22:44,820 --> 00:23:13,059
I think somewhere in 1850s or 1860s and this
is a celebrated theorem by called as
183
00:23:13,059 --> 00:23:21,389
so this is a well known result well this particular
result is what is called as an existence result
184
00:23:21,389 --> 00:23:22,389
okay
185
00:23:22,389 --> 00:23:28,169
I will tell you what I mean by existence result
It does not tell you how do construct a polynomial
186
00:23:28,169 --> 00:23:34,749
approximation It assures that given a continuous
function there exist a polynomial which is
187
00:23:34,749 --> 00:23:39,980
arbitrarily close to the continuous function
Now what is arbitrarily closeness You need
188
00:23:39,980 --> 00:24:08,020
norm okay what is arbitrarily closeness You
need concept of norm So now what do we consider
189
00:24:08,020 --> 00:24:09,020
here
190
00:24:09,020 --> 00:24:14,169
We consider the set of continuous functions
over an interval a b together with infinite
191
00:24:14,169 --> 00:24:23,730
norm okay So this is the space this is the
norm defined on it this is the norm linear
192
00:24:23,730 --> 00:24:38,289
space okay Now Weierstrass theorem tells us
that well I will move on to here to complete
193
00:24:38,289 --> 00:24:50,149
this theorem statement
so if I give you any epsilon greater than
194
00:24:50,149 --> 00:24:57,419
0 any degree of accuracy epsilon will specify
how accurate you want the approximation okay
195
00:24:57,419 --> 00:25:04,970
And if I pick up any continuous function ft
from c a b set of continuous functions over
196
00:25:04,970 --> 00:25:28,019
a b then there exists a polynomial
very very important result
197
00:25:28,019 --> 00:25:35,259
okay So what does it say given any epsilon
you give me the accuracy that you want How
198
00:25:35,259 --> 00:25:41,799
close an approximation you want okay You can
specify that epsilon an give me any function
199
00:25:41,799 --> 00:25:44,849
ft which is the continuous function okay
200
00:25:44,849 --> 00:25:52,869
Then there exists a polynomial approximation
I am going to call this as Pnt and we will
201
00:25:52,869 --> 00:26:06,950
be let us say order of the polynomial okay
Such that ft minus Pnt is less than epsilon
202
00:26:06,950 --> 00:26:19,259
okay This is clear So what is this norm This
norm is absolute norm We are finding out difference
203
00:26:19,259 --> 00:26:28,840
between maximum of the absolute value see
if I give you a function let us say sine t
204
00:26:28,840 --> 00:26:36,099
okay this theorem tells me there is nth order
polynomial such that sine t minus the polynomial
205
00:26:36,099 --> 00:26:38,239
absolute of this okay
206
00:26:38,239 --> 00:26:48,379
Maximum is the interval will not exceed epsilon
Use specify epsilon I will construct a Pn
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okay You give me an epsilon how do you construct
a Pn Is not what is told by this theorem It
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just says that there exists okay how do you
construct that approximation Well that is
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a different story It only assures that there
exist a polynomial which is arbitrarily closed
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How do we find out that particular polynomial
is not given by this theorem but it tells
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you that there exists a polynomial so which
means when I am approximating a transformation
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I could use this basic idea could transform
a differential equation or transform you know
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boundary value problem or partial differential
equation into some simplified form We will
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do it much more in detail
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But I will just give you a very very simple
example okay So do you see parallels between
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here and here We are talking about finding
out a rational number which is arbitrarily
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close to a real number and using that rational
number for calculations this is not the real
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number okay Same thing same idea we are going
to do here okay The true solution would be
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a continuous function okay
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I am going to approximate that continuous
function by a polynomial function why that
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will help me to solve the problem and you
know by transforming the operator by solve
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the problem in a different way which is easier
than the original problem Well one simple
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this kind of things will hit on later Let
us look at a very very simple demonstration
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See if I have dx by dt equal to some f of
x okay
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And I have given you initial condition corresponds
to say x not okay Now what is xt let us say
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x star t is the true solution
okay Is this a continuous function it has
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to be a continuous function It has to be in
fact differentiable function not just continuous
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It has to be a differentiable function so
it is a continuous function Any continuous
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function can be approximated by a polynomial
function
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I propose a polynomial solution which is say
Pnt or I will call it xt which is a not plus
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a1t plus a2t square Let me propose a polynomial
solution Now this is a polynomial approximation
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This is not a true solution but I can substitute
it here I can substitute here and I can say
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that well I won the approximation solution
such that so what is dx by dt a1 plus 2a2t
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right and then I can substitute this here
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So for any t I want this equation to hold
that is a1 plus 2a2t equal to f of a not plus
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a1t plus a2t square right I am doing something
which is very lieu we do it much more sophisticated
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mind there afterwards I just want to carry
some point here Look this is a differentiated
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equation I started with I approximated using
a polynomial form with unknown coefficients
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I do not know a not a1 a2
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True solution is x star With what boldness
I can do this I know at a continuous function
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can be approximated by a polynomial function
okay I substituted this what happen what look
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like originally a differential equation now
looks like an algebraic equation with unknowns
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a1 a2 a not The problem is transform from
a differential equation to an algebraic equation
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okay
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So this idea of using a polynomial approximation
of a continuous function will be used to transform
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problems which are originally so original
operator t was a differential operator t prime
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or t cap what you are getting here looks like
an algebraic so I was talking about you know
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starting with an original problem transforming
the problem and solving the transform problem
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So we might actually computationally this
is easier to track than this We might solve
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00:32:05,009 --> 00:32:10,859
this as compare to this What we get by this
approach is the approximate solution not the
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00:32:10,859 --> 00:32:17,339
true solution remember that okay This is the
approximate solution but if we can accept
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22 by 7 in place of pi we can accept this
approximate solution and as long as it is
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close and you know your physics is you know
preserved in some sense qualitatively you
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00:32:32,090 --> 00:32:35,440
do not bother too much about the difference
between the two okay
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00:32:35,440 --> 00:32:44,909
So this is how it is going to help us in transforming
the problems So what is used so Weierstrass
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approximation theorem as such we never revisit
again but it is the foundation everywhere
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you know Weierstrass theorem comes at in a
hidden form it is everywhere because we approximate
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continuous solutions using polynomials so
somebody asked what is the basis why polynomials
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Because polynomials are dense why should I
be so much worried about a dense set You know
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dense set is something which can pick an element
dense set and it can be as close as possible
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00:33:16,759 --> 00:33:24,809
to the original you know element in the original
set okay So dense set is a special set so
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just like set of rational numbers is a special
set in the real numbers polynomials set of
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polynomials is a special dense set instead
of continuous functions okay
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So this is a foundation this is result but
it does not tell you how to construct a polynomial
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approximation Now we are going to use three
different tricks for constructing polynomial
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approximations So there are three different
ways by which we are going to construct a
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00:34:03,229 --> 00:34:08,260
polynomial approximation First is the Taylor
series approximation you are familiar with
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00:34:08,260 --> 00:34:11,629
Taylor series expansions we will just revisit
them briefly in the next lecture
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00:34:11,629 --> 00:34:21,260
Then we move on to polynomial interpolation
polynomials and the third is the least square
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00:34:21,260 --> 00:34:30,370
approximation So if you understand these three
basic concepts most of the problem transformations
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00:34:30,370 --> 00:34:37,600
will be clear to you How the problem is transformed
to a computable form okay Then comes how to
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solve the transform problem okay so that is
the next part so till mid sem will be actually
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working on now this
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Will systematically look at different problems
particularly boundary value problems partial
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00:34:52,830 --> 00:34:58,270
differential equations nonlinear algebraic
equations and all kinds of things were we
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used these three ideas and then transform
the problem to a computable form So next the
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12 lectures are about problem formulation
okay
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You have formulated the problem from physics
and you got some problem which is coming from
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00:35:14,700 --> 00:35:21,830
your courses and transport reaction engineering
whatever heat transformer strength of materials
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whatever your specialization so those original
problem is coming from there okay I want to
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00:35:28,101 --> 00:35:34,370
computer a numerical solution for these problems
so I use all these tricks to transform the
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00:35:34,370 --> 00:35:40,050
problem to a computer bill form and then I
solve that computer bill form okay
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Construct a solution which is approximate
numerical solution to the problem okay So
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in the next class we will start with Taylor
series approximations okay I will very quickly
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00:35:53,770 --> 00:36:00,230
review Taylor series approximation what is
the basis behind Taylor series approximation
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00:36:00,230 --> 00:36:05,860
you are aware of only one variable Taylor
series we will move on to multivariable Taylor
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00:36:05,860 --> 00:36:11,690
series okay Polynomial functions in n variables
okay
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00:36:11,690 --> 00:36:15,910
And then we will look at for example one of
the application of the Taylor series would
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00:36:15,910 --> 00:36:21,730
be Newton Raphson method okay Then we will
move on to show that this Taylor series approximation
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00:36:21,730 --> 00:36:28,320
actually gives rise to the finite difference
method of solving boundary value problems
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00:36:28,320 --> 00:36:34,370
finite different method of solving partial
differential equations okay and so on or the
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polynomial interpolations
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So we will develop in the class method of
orthogonal collocations and see how an orthogonal
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00:36:40,730 --> 00:36:48,630
collocation arises from polynomial interpolations
and so on So all these three different approaches
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give rise to different ways of problem discretization
and that will be the center theme for next
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00:36:53,770 --> 00:36:54,319
few lectures