Math 181-Fall 2001

Instructor: E. Arthur Robinson, Jr.

Lecture 1

What is Computational Mathematics

What is "Computational Mathematics?"

Answer 1: Computational Mathematics can be defined in terms of its relation to computational science,
but what is that?

The idea of Computational Science--as a discipline in its own right--has recently emerged, encouraged by some of the National Laboratories (Oak Ridge,Los Alamos, Sandia). The idea has also caught on in some academic circles.

The idea is that computational science has actually been practiced for a long time (perhaps hundreds of years), but that previously practitioners were not identified as "Computational Scientists".

The idea is that computational science is a mode of doing science based on computing and thus a computational scientist is a scientist who uses computing in her/his work in a fundamental way.

"In broad terms [Computational Science] is about using computers to analyze scientific problems."----CSEPElectronic Book, Oak Ridge National Laboratory

This point of view (which seems to use Physics as its model) holds that traditionally scientists have been either:

theoretical scientists, or

experimental scientists.

Now (i.e. with widely available computer technology), scientists can also be

computational scientists.

Here is an example that illustrates this concept:

J. Zachary (Introduction to Scientific Programming) considers the following:

Tycho Brahe (1546-1601) measured and recorded position of sun, moon and planets using instruments he invented and built (the telescope was not invented until after his death). He produced and published a great deal of data about planetary motion.

Johannes Kepler (1571-1630) used Brahe's data to find his 3 laws of planetary motion.

Kepler's laws:

planetary orbits are ellipses with sun at one focus.

The line between a planet and the sun sweeps equal areas in equal time.

The square of the orbital period is proportional to the cube of the average distance to the sun

Isaac Newton (1642-1727) c ame up with what we call Newton's laws

Newton's laws explain why Kepler's laws work.

Newton needed to invent calculus to describe his laws (in "Calculus III" classes we sometimes show how to derive Kepler's laws from Newton's laws).

Newton's laws are a very general mathematical model that can make many predictions beyond planetary motion (classical mechanics).

In general, the equations that arise from Newton's laws are extremely difficult to solve (even the case of a sun and two plannets--the 3 body problem--is intractible)

Zachary's point is that in this story:

Brahe is an experimental scientist

Newton is a theoretical scientist, and

Kepler is a computational scientist.

Kepler was able to take Brahe's data and by computing (the old fashioned way, with pencil and paper) find relationships within it. His laws essentially explained how to generate the data computationally, without trying to explain why. In a sence it reduced Brahe's data to the smaller amount of "data" needed to express (and implement) Kepler's laws. Without Kepler's laws it is doubtful we would have Newton's laws.

There are two ways to fit "Computational Mathematics" into this framework.

It is the computational part of the science of Mathematics.
It is the mathematical part of of Computational Science in general.

These ideas are really complementary since the sciences likely to have computational parts are likely to be at least somewhat mathematical in nature.

Some problems with this model:

The notion of experimental, computational, and theoretical science does not fit all sciences equally well. In particular, if Mathematics has an experimental branch, it is probably part of it's computational branch.

We don't want to take the term "science" so literally as to be exclusive.

Here are some "sciences" that have computational branches:

Physical science:

physics, astronomy, chemistry, earth sciences

Biological science:
genetics, neuroscience, molecular biology

Engineering science:
electrical, mechanical, computer, environmental

Mathematical science:
mathematics, statistics, operations research

Social science:
economics, finance, demography, political

What is computational mathematics?

Answer 2: What computational mathematics is actually about.

Here are some closely related areas that computational mathematics is composed of:

Computer Science.
Scientific Programming/Scientific Computing.
High performance computing.
Numerical Analysis.
Applied Mathematics.
Mathematical Modeling.
Probability and Statistics.

We consider each in turn:

Computer Science

Broad definition: The practice (science) of using a computer solve problems.
Important Skill: Knowing how to program well.
In practice (as a field): Concerned with: computer architecture, operating systems, programming languages, networking, data structures, algorithms, computational complexity, databases, user interfaces, graphics, computer vision, artificial intelligence, scientific programming, etc.

Related field: Computer engineering.

Comment: The phrase "Computational Science" (discussed above) is meant to suggest a distinction between it and "Computer Science". Computational Scientists are clearly computer scientists in the broad sense. Some of the specific skills of the discipline are also relevant. But Computational Scientists need skills that go beyond computer science.

Scientific computing:

This is a modern term for the part of computer science that deals with implementing mathematical models and/or numerical algorithms on the computer. An older name for the same area is "scientific programming". In current usage, it is nearly synonomous with the kind of computing computational scientists do.

Cliche:
The favorite language for scientific computing is FORTRAN (rather than C).

High Performance Computing:

The practice of taking advantage of high performance computing hardware, including RISC proessors, memory caching schemes, parallel architectures, etc.
This includes specific programming techniques (eg, loop unrolling) and types of software (eg, optimizing compilers) to take advantages of the improved hardware.
High performance computing implicitly often refers to scientific applications, where execution speed is the limiting factor in what problems can be solved.

Numerical Analysis:

The branch of applied mathematics that deals with how to solve (or approximately solve) mathematical problems numerically.

Closely connected with Scientific Computing.

Includes the theory of errors in calculations and computer arithmetic (the latter is also part of computer science and computer engineering).

Applied mathematics:

Broad definition: Mathematics applied to "real world" problems.
As opposed to "pure mathematics" which is mathematics for its own sake (or perhaps mathematics that has not yet found an application).
In practice: Certain fields in mathematics are considered "applied": mostly differential equations (PDE's and ODE's) & numerical analysis.

Mathematical modeling:

Constructing mathematical models of "real-world" phenomena.
Models can sometimes be solved "analytically" but more often numerical solutions are necessary (e.g. Newton's laws). Numerical solutions are often implemented computationally.
Related term: simulation.

If Scientific Programming lies somewhere between Computer Science and Applied Mathematics

then

Mathematical modeling lies somewhere between Applied Mathematics and the scientific discipline being modeled.

Probability and Statistics

These disciplines are similar (in spirit) to applied mathematics, but for historical reasons have developed independently.
Many important mathematical models are statistical or stochastic in nature (i.e., involve elements of probability).

Skills needed to sucessfully apply computational mathematics to your field:

Scientific Programming/Scientific Computing (including good programming skills and familiarity with CAS environments).
Computer Science (a select set of practical skills, as well as in the broad sense)
High performance computing (knowledge of how to take advantage of advances in computer technology).
Numerical Analysis/Applied Mathematics (a working knowledge of principles and techniques).
Mathematical Modeling (including Probability and Statistics).

Why the sudden interest in computing?

Recent advances in computer technology have made a more viable tool than ever before.

Advance: Effect:

Higher CPU speed/RISC architecture/parallel processing.
Cheaper memory/faster cache.
(Moore's Law) Larger computations possible. Rougher programming may still work.

More economical faster mass data storage. Computer data gathering, massive databases.
(Human Genome, U.S. economy, National weather service)

Networking (local and Global). Data and software sharing, scientific collaboration, e-publication, virtual parallel architecture

Improvements in graphics hardware and software. Desktop publishing, visualization as a scientific tool.

Software/Language improvements. New models for scientific computing (CAS).
New approaches to programming, e.g. FORTRAN 90, Java, C++.
Numerical analysis using spreadsheet

Availability of laptop computers Portable, ever present computing.

What does the future hold?

The end of Moore's law? (or not?)^*
Quantum computing?
More portable and more powerful computers?
Computer consciousness/artificial life?

Moore's law: Every 18 months, the number of transistors that will fit on a silicon chip doubles.

-Gordon Moore, Intel co-founder, 1965 (original version said 24 months).

Implications: Faster and cheaper computers continually become available.

Higher CPU speed/RISC architecture/parallel processing. Cheaper memory/faster cache. (Moore's Law)	Larger computations possible. Rougher programming may still work.
More economical faster mass data storage.	Computer data gathering, massive databases. (Human Genome, U.S. economy, National weather service)
Networking (local and Global).	Data and software sharing, scientific collaboration, e-publication, virtual parallel architecture
Improvements in graphics hardware and software.	Desktop publishing, visualization as a scientific tool.
Software/Language improvements.	New models for scientific computing (CAS). New approaches to programming, e.g. FORTRAN 90, Java, C++. Numerical analysis using spreadsheet
Availability of laptop computers	Portable, ever present computing.