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Now this lecture is going to be very brief
introduction to mathematical model.
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Here we aim to explain what goes on in mathematical
modelling and why we actually do it, and then
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finally we end with examples to highlight
the importance of math modelling in the real
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world.
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We start by asking, what really is mathematical
modelling?
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We offer rather a simple minor definition
that is representation in mathematical terms
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of the behaviour of a real world or of real
devices and real processes.
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We now take rather elementary view of science.
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So what we do is we take the world and break
up into two books, one is real world and other
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is conceptual world.
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We go ahead and divide into two worlds.
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In the real world, is where we observe phenomena
and in the conceptual world we conduct experiments
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and device models both of which leading to
predictions.
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Central idea is try and work out, what are
all is going out in real world?
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That is the central idea, so we highlight
the real world.
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Now has this is really central with the modelling
process.
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We go ahead and highlight it.
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In the modelling process, it is rather important,
go from real world to conceptual world and
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then back from conceptual world to the real
world.
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Now within the conceptual world, insights
from experiments should inform models and
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inference from the model should guide experiments.
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Now let us consider the math modelling and
engineering.
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Now what we find is that engineers are typically
interested in designing devices and the systems.
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And to that end design turns out to be rather
important and key features of engineering.
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Let us now look at the basic process of modelling.
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We always start with real world, problem in
construct a model, which is then subjected
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to analysis and finally it all feeds into
design.
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So we got to and fro between real world and
final design in getting to the model.
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We need to simplify the real world the objective
of the analysis is to allow us to make inferences.
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The iterative process between the real world
and final design is important.
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As it is the only way that models can be improved
and validated, few comments are enviable.
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Now initially it is very difficult to know,
what really to include in the model.
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So, if you happen to be a research student
and struggling in formulating a good initial
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model rest assured the lots of others who
will be in the very same boat.
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But, if the model is chosen very carefully,
we can then understand the real world rather
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quickly.
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Now the reason, we device models is because
we want guidelines about how system should
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be designed, built or operated.
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However, the model is only really useful,
if we can translate the modelling conclusions
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and the inference into real world predictions.
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Now let us look at the process the modelling
in some more detail.
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We would actually start with the real world
at one end and with mathematical model at
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the other end.
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And of course, while we connect with real
world to a model, we would need to make some
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simplification to the real world along the
way.
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And of course, the model has to be connected
all the way back into the real world.
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We will start with mathematical analysis followed
by computational/numerical analysis.
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Conduct simple simulations and then move on
to more detailed simulations.
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Then try and conduct some very simple test
bed experiments before moving on to the real-world
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experiments and finally implementation in
the real world.
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It is really useful to keep going between
the different levels of detail.
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That way we can understand which parts of
the system are important and perhaps which
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parts can be left out.
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The above diagram was adopted from performance
modelling notes of Hemant B Kaushik
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Now we get on to the question, on why we actually
do mathematical modelling.
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Or in other words, what really is mathematical
modelling look for?
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Well, one key reason is predicting system
behaviour.
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Now we would like to know whether we can still
observe some real world behaviour.
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When we create a simplified model of the system
even after leaving out certain real world
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problems, we like to know were, where we would
predict the system behaviour, even when it
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becomes too large to simulate.
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Modelling turns out to be the quick way to
get insight into emergent behaviour.
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Another important reason is design of new
products.
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For example, most systems have lot of parameters.
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Can we understand which of these parameters
really matter in the system? Modelling can
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also suggests where problems in fact are most
likely to occur.
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We can then actually conform this with simulations
and or experiments.
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In fact, we can also avoid or fix these problems
with the help of modelling.
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We can also understand various trade-offs
that may impact design options.
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We can also understand performance limits
before the product is actually built.
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And finally, we can understand weather system
or the product is achieving what we really
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wanted to achieve.
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With all the stock of modelling we should
also be asking, is there such a thing as a
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correct model?
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The short answer is not really.
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Now we have to be crystal clear on the purpose
of a model.
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Different models are useful for different
purposes and different purposes in fact, need
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different models.
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They can be different types of models.
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Some nearly help to describe, observe behaviour.
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Others help to explain, why the behaviour
occurred in the first place?
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We have models that would allow us to predict
future behaviour.
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And of course, we could also have models which
are used to persuade other people view point
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of view.
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I would not have I said that going between
real world and the model can actually be quite
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hard.
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So we now offer some pointers on how to cope
with this part of the modelling process.
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First always construct simple models from
simple assumptions.
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This will lead to models that are mathematically
tractable.
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And it does not really matter if initially
the model does not fully represents the real
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world.
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The mathematical analysis will in fact, help
develop some insight.
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And one should always check that insight with
simulations.
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After this part, we should refine the models
with enhanced assumptions.
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Of course, this will make models more realistic.
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And one should repeat the analysis, the computations,
simulations and the experiments till we have
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satisfied the purpose of the model.
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Now for some words of caution always be aware
of the limitations of a model.
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The limitations are often based on the underline
assumptions that you make, and always remember
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that models are only abstractions of the real
world and if the behaviour that is predicted
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by the model does not reflect, what we see
or measure in the real world and the time
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for you to revisit the model.
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Now, let us talk about some real life examples,
live in this lecture.
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The first example will be beautiful Millennium
bridge, which is in London.
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It is the bridge for people to cross the river
Thames.
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That is the picture of the bridge, so nice
beautiful bridge, which is opened on 10th
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of June 2000.
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And then something rather dramatic happened
the bridge closed within two days of it opening.
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Essentially what happened was that unexpected
lateral vibrations forced the bridge to closed.
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Now, what really happened was that the natural
sway of motion of people walking caused small
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lateral oscillations in the bridge.
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This in turn caused all the people on the
bridge to sway in step.
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And which in turn increased the amplitude
of the bridge oscillations.
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Now these vibrational modes had in fact not
been anticipated during the design of the
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bridge.
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After extensive analysis, following solution
was proposed, retrofitting of 37 fluid viscous
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dampers to control horizontal movement and
52 mass dampers to control vertical movement.
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The bridge opened again on the 22 of February
2002.
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Almost two years later, all has been well
since with the bridge.
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But Londoners still rather affectionately
call it the Wobbly bridge.
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The bridge was also featured in the very famous
movie “Harry Potter and the Half-blood prince”
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in 2009.
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Now in the movie the bridge actually famously
collapses after an attack by the notorious
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Death Eaters.
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Now let us look at another real life example.
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This is an example that we use almost everyday
of our life it is PageRank.
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The PageRank was the first algorithm that
was used by Google search to rank websites.
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Now how it works is roughly as follows.
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The algorithm counts the number and quality
of the links to a page, to get some idea of
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how important the website actually is.
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The underline assumption is that the more
important websites are likely to receive more
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links from other websites.
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Now in the time frame about let us highlight
underline principle, the science of each circle
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is proportional to the total size of the other
circles, which are pointing to it.
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So essentially if the circles represent websites,
then larger the circle, the larger the PageRank
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that is stands out to be absolutely excellent
example of modelling.
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The models was constructed, analysis was done,
prototype developed and taken on to the real
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world.
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Based on this work Google was founded in 1998
and life has not really been same since.
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So hopefully this video has managed do to
so at least some insight into, what mathematical
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modelling is why one should be really doing
it at the first place and how it can truly
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have tremendous impact in .