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I'd like to now give you an overview of
the course, and what is expected over the

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ten week period of this course.
There are 27 lectures.

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And in those lectures, we cover a very
broad range of techniques and mathematical

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ideas in scientific computing.
In fact, the goal of the course is to

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really highlight all the major techniques
used today.

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And so that you get yourself up to a level
where you can attack problems that are

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complex and know the kind of right methods
to be applied.

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Now before we start I wanna give you an
idea of what the expectations are for you

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in terms of background.
Certainly linear algebra is key in all the

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methodology used, in all the techniques
developed.

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So a strong background in linear algebra
is really imperative.

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Along with this some familiarity, or
potentially a good background in

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differential equations.
Certainly brushing up on differential

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equations is a very good idea, as many of
the systems we will study and try to

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characterize, or compute solutions for,
involve differential equations.

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Now, we're not so interested in analytic
solution techniques, so much as just

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understanding some of the basic properties
of differential equations.

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And third, you should have some background
in programming.

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Now this can come in a variety of ways,
mostly the idea is, do you understand some

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basic structures?
In particular, for loops, if statements, I

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think of those as the building blocks for
making any kind of very complex code.

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Now once you have that background in
place, we're gonna be using two, major

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tools in developing almost everything we
do for the course.

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First, a computer, of course that's kind
of expected, scientific computing.

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So, it's expected that you have access to
a computer or a magic abacus which you can

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do a transform on.
But, I think those are very difficult to

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find.
So you want to have a good computer,

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fairly fast.
It doesn't need to be brand new, but

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certainly within the last couple of years
as computing power has really gone up.

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The second thing is the course will really
focus in using high level language for

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programming.
In particular, we'll focus on using

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MATLAB.
Matlab is basically the leading product on

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the market for high level language in
computing.

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Now MATLAB is available for students of
Coursera.

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For around $100 you can download this,
-load this online or find it in a

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bookstore.
There are also versions of MATLAB or other

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sources that are similar to MATLAB that
can be downloaded.

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One of them is Octave for instance.
I'll put resources on the web page for you

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to follow through.
So, there are fee versions available, but

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they are basically knockoffs or mockups of
what MATLAB is.

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If you're very serious about learning the
material, MATLAB is actually a very

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wonderful investment, extremely powerful,
allowing us to do very complex things,

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very efficiently and in some sense that's
the mantra of the course.

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Take a very complex system.
Something that, maybe ten years ago, 50

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years ago was very difficult to solve, and
with something like MATLAB, you can bring

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in this very high level functionality.
You can bring in code that is written by

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some of the world's best people, and you
understand the very basics of these codes

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and how to combine them together to make
for yourself a program that can simulate a

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very complex system.
The course is breaking down into four

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parts.
In the first part of the course, what

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we're gonna cover is basically
differential equations in boundary value

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problems.
So, when we think about differential

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equations, oftentimes we've been
introduced to them from the point of view

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of an initial value problem solving.
So you give the initial state of the

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system, some prescribed set of dynamics,
and what is the future state of that

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system?
We'll develop this techniques primarily

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because they purvey the rest of the
course.

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Everything we do has something to do with
how to time-step into the future, mostly

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because most complex systems is all about
predicting the future state of the system.

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Once we have the initial value problem
solved, we'll move on to boundary value

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problems.
Boundary value problems typically

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prescribe some kind of domain in which you
need a solution to exist.

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And you would like to figure out how to
correctly prescribe the boundary

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conditions that are also self consistent
with the behavior in between those

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boundaries.
Armed with those two methods, we'll move

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on to, the heart of the course, which is,
partial differential equations.

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Partial differential equations, are
differential equations in more than one

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variable.
So for instance, time and space.

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And typically this is what we're
interested in solving.

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You take a problem.
For instance, you wanna predict, the

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weather.
Well, here, in the weather, pattern, you

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wanna see how the weather, changes, both
in time, and in space.

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So it is important for you to understand
that you have both time and space

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variables and how they interact, is the
heart of what's the science of computing

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method is trying to solve for.
We'll start out with finite difference

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methods, which are some of the more easy
to understand techniques out there, and

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very common.
They can handle sufficiently complicated

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boundary conditions, and are fairly
straightforward to implement.

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They give you fairly reasonable accuracy.
And everything's based upon Taylor

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Expansions.
So once you know how to Taylor Expand, you

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can develop a whole host of finite
different schemes.

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In addition to the finite difference,
we'll move next to spectral methods.

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Spectral methods are ideas that transform
your variables into what's called the

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Fourier domain, or spectral domain.
The reason these methods are attractive is

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they're highly accurate, and it's very
fast.

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One of the most famous routines, top ten
algorithms of the last century, was the

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fast Fourier transform.
The fact that an algorithm has the name

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fast in it means that it's extra fast,
because all algorithms on a computer are

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fairly fast.
So, the fast Fourier transform is

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something we would like to take advantage
of as much as possible.

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It is one of the leading techniques for
solving certain kind of problems.

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So we use these spectral ideas and
spectral technologies and build up ideas

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for solving partial difference equations
within the context of spectral methods as

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well.
Finally after understanding finite

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difference in spectral techniques we'll
move on to finite element methods.

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Now finite element methods can handle
arbitrary boundaries.

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Complex domains that you wanna solve for.
Very difficult, interior meth-, domains

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also.
This is typically beyond the scope of what

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you would do with finite difference or
spectral methods.

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However, we'll cover some of the basic
ideas behind finite element technique.

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Now, we will not spend time coding the
finite element technique, because it's

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beyond the scope of this course, but,
we'll give you the main ideas and show you

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how to use some commercial software.
To solve these [inaudible], these finite

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element based problems, where you have
very complex geometry, or very complex

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interior geometries to your problem.
It's one of the leading methods when you

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have such complex geometries but also,
typically tends to fall under commercial

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software development.
Those are the majors themes and ideas that

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we will cover.
The idea is to mostly focus in on building

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code, getting code to work.
That is where the assessment and the value

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of the course is.
We are after using the techniques,

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implementing the techniques, not just
knowing about how they work in theory, but

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then applying them in practice and solving
complex problems directly on your laptop.

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So by the end of the course you too will
be able to solve very complicated

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problems, and get yourself to a point
where you can pretty much do research

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level problems.
You will have the skills and techniques to

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do so.
Not only that, you'll know about the major

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methods.
You'll what they entail, why you should

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use one method versus another, and
basically have a good characterization of

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the strengths and weaknesses of all the
key, the key ingredients and key elements

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of these techniques so that, when you come
to your problem or, a problem you are

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trying to solve.
You can come in, use these techniques,

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make an evaluation of the right technique
to apply, solve the problem in very fast

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and simple way through use of a high level
code like MATLAB, and produce a solution.

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That's the basic architecture of the
course.

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Application, application, application.
I want you to know how to code by the time

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this course is over.
I want you to know how to evaluate what

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techniques to be applying, and the
strength and weaknesses of different

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techniques.
I want you to take your level of

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computational fluency a significant step
upward in your professional development.