I'd like to now give you an overview of
the course, and what is expected over the
ten week period of this course.
There are 27 lectures.
And in those lectures, we cover a very
broad range of techniques and mathematical
ideas in scientific computing.
In fact, the goal of the course is to
really highlight all the major techniques
used today.
And so that you get yourself up to a level
where you can attack problems that are
complex and know the kind of right methods
to be applied.
Now before we start I wanna give you an
idea of what the expectations are for you
in terms of background.
Certainly linear algebra is key in all the
methodology used, in all the techniques
developed.
So a strong background in linear algebra
is really imperative.
Along with this some familiarity, or
potentially a good background in
differential equations.
Certainly brushing up on differential
equations is a very good idea, as many of
the systems we will study and try to
characterize, or compute solutions for,
involve differential equations.
Now, we're not so interested in analytic
solution techniques, so much as just
understanding some of the basic properties
of differential equations.
And third, you should have some background
in programming.
Now this can come in a variety of ways,
mostly the idea is, do you understand some
basic structures?
In particular, for loops, if statements, I
think of those as the building blocks for
making any kind of very complex code.
Now once you have that background in
place, we're gonna be using two, major
tools in developing almost everything we
do for the course.
First, a computer, of course that's kind
of expected, scientific computing.
So, it's expected that you have access to
a computer or a magic abacus which you can
do a transform on.
But, I think those are very difficult to
find.
So you want to have a good computer,
fairly fast.
It doesn't need to be brand new, but
certainly within the last couple of years
as computing power has really gone up.
The second thing is the course will really
focus in using high level language for
programming.
In particular, we'll focus on using
MATLAB.
Matlab is basically the leading product on
the market for high level language in
computing.
Now MATLAB is available for students of
Coursera.
For around $100 you can download this,
-load this online or find it in a
bookstore.
There are also versions of MATLAB or other
sources that are similar to MATLAB that
can be downloaded.
One of them is Octave for instance.
I'll put resources on the web page for you
to follow through.
So, there are fee versions available, but
they are basically knockoffs or mockups of
what MATLAB is.
If you're very serious about learning the
material, MATLAB is actually a very
wonderful investment, extremely powerful,
allowing us to do very complex things,
very efficiently and in some sense that's
the mantra of the course.
Take a very complex system.
Something that, maybe ten years ago, 50
years ago was very difficult to solve, and
with something like MATLAB, you can bring
in this very high level functionality.
You can bring in code that is written by
some of the world's best people, and you
understand the very basics of these codes
and how to combine them together to make
for yourself a program that can simulate a
very complex system.
The course is breaking down into four
parts.
In the first part of the course, what
we're gonna cover is basically
differential equations in boundary value
problems.
So, when we think about differential
equations, oftentimes we've been
introduced to them from the point of view
of an initial value problem solving.
So you give the initial state of the
system, some prescribed set of dynamics,
and what is the future state of that
system?
We'll develop this techniques primarily
because they purvey the rest of the
course.
Everything we do has something to do with
how to time-step into the future, mostly
because most complex systems is all about
predicting the future state of the system.
Once we have the initial value problem
solved, we'll move on to boundary value
problems.
Boundary value problems typically
prescribe some kind of domain in which you
need a solution to exist.
And you would like to figure out how to
correctly prescribe the boundary
conditions that are also self consistent
with the behavior in between those
boundaries.
Armed with those two methods, we'll move
on to, the heart of the course, which is,
partial differential equations.
Partial differential equations, are
differential equations in more than one
variable.
So for instance, time and space.
And typically this is what we're
interested in solving.
You take a problem.
For instance, you wanna predict, the
weather.
Well, here, in the weather, pattern, you
wanna see how the weather, changes, both
in time, and in space.
So it is important for you to understand
that you have both time and space
variables and how they interact, is the
heart of what's the science of computing
method is trying to solve for.
We'll start out with finite difference
methods, which are some of the more easy
to understand techniques out there, and
very common.
They can handle sufficiently complicated
boundary conditions, and are fairly
straightforward to implement.
They give you fairly reasonable accuracy.
And everything's based upon Taylor
Expansions.
So once you know how to Taylor Expand, you
can develop a whole host of finite
different schemes.
In addition to the finite difference,
we'll move next to spectral methods.
Spectral methods are ideas that transform
your variables into what's called the
Fourier domain, or spectral domain.
The reason these methods are attractive is
they're highly accurate, and it's very
fast.
One of the most famous routines, top ten
algorithms of the last century, was the
fast Fourier transform.
The fact that an algorithm has the name
fast in it means that it's extra fast,
because all algorithms on a computer are
fairly fast.
So, the fast Fourier transform is
something we would like to take advantage
of as much as possible.
It is one of the leading techniques for
solving certain kind of problems.
So we use these spectral ideas and
spectral technologies and build up ideas
for solving partial difference equations
within the context of spectral methods as
well.
Finally after understanding finite
difference in spectral techniques we'll
move on to finite element methods.
Now finite element methods can handle
arbitrary boundaries.
Complex domains that you wanna solve for.
Very difficult, interior meth-, domains
also.
This is typically beyond the scope of what
you would do with finite difference or
spectral methods.
However, we'll cover some of the basic
ideas behind finite element technique.
Now, we will not spend time coding the
finite element technique, because it's
beyond the scope of this course, but,
we'll give you the main ideas and show you
how to use some commercial software.
To solve these [inaudible], these finite
element based problems, where you have
very complex geometry, or very complex
interior geometries to your problem.
It's one of the leading methods when you
have such complex geometries but also,
typically tends to fall under commercial
software development.
Those are the majors themes and ideas that
we will cover.
The idea is to mostly focus in on building
code, getting code to work.
That is where the assessment and the value
of the course is.
We are after using the techniques,
implementing the techniques, not just
knowing about how they work in theory, but
then applying them in practice and solving
complex problems directly on your laptop.
So by the end of the course you too will
be able to solve very complicated
problems, and get yourself to a point
where you can pretty much do research
level problems.
You will have the skills and techniques to
do so.
Not only that, you'll know about the major
methods.
You'll what they entail, why you should
use one method versus another, and
basically have a good characterization of
the strengths and weaknesses of all the
key, the key ingredients and key elements
of these techniques so that, when you come
to your problem or, a problem you are
trying to solve.
You can come in, use these techniques,
make an evaluation of the right technique
to apply, solve the problem in very fast
and simple way through use of a high level
code like MATLAB, and produce a solution.
That's the basic architecture of the
course.
Application, application, application.
I want you to know how to code by the time
this course is over.
I want you to know how to evaluate what
techniques to be applying, and the
strength and weaknesses of different
techniques.
I want you to take your level of
computational fluency a significant step
upward in your professional development.