Computational Science by James E. Gentle

Foundations of Computational Science

Chapter 0. Preview/Review for Students in the Computational Sciences

There are a number of fundamental skills and a basic set of knowledge that must be in the command of all who work in the computational sciences. This book addresses major subsets of that body of knowledge, and some of that knowledge is prerequsite background for this book. The exercises in this text assume also a set of technical skills. Careful attention to the exercises will hone these skills.

The following outline indicates the necessary background (what this book is not about) and provides a preview of the material covered in the text (what this book is about). The outline also serves as a list of topics for review. The terms listed in the Subject Index provide a vocabulary review.

Knowledge and skills must evolve over time. Technical skills in particular must be kept current.

Technical Skills

Work in the computational sciences often involves hands-on computing. This book assumes a certain level of copmputational technical skills. Some of the technical skills required at this point in time will not be required in the future. They will be replaced by new skills.

General familiarity with both Unix and Microsoft Windows.
Ability to program in Fortran or C.
General familiarity with sources of scientific software.
Facility in the use of at least one of the following packages: Matlab, S-Plus, PV-Wave, Maple, and Mathematica.
General familiarity with appropriate structures for representing scientific and statistical data.
Ability to work in a software development team.
General familiarity with literature and software sources, and ability to locate specific sources as needed.

Communication Skills

In all areas of the sciences, communication is an important activity. Scientific advancces are of little value unless they can be described clearly.

Ability to organize and write grammatically correct descriptions of mathematical and scientific concepts.
Ability to design and produce appropriate visual displays of information.

General Mathematics Background

All research workers in the computational sciences need at least some mathematics background in each of the areas listed above. The required levels of knowledge in particular areas vary considerably with the specific area of the computational sciences.

Linear algebra.
Advanced calculus.
Differential equations.
Probability and statistics.

Except for some material in linear algebra, these background topics are not covered directly in this text. The requirement for expertise in differential equations and in probability and statistics, in particular, is quite different in different types of scientific investigation.

Knowledge of Computational Science

This text is about computational science and numerical analysis. Many of the topics required in common by all research workers in the computational sciences are addressed in this text. Following a course in the foundations of computational science, the student should understand the topics listed below, be able to describe any of them briefly, and more importantly, be able to apply the understanding in the context of scientific investigations.

Basics of numerical computations on digital computers (Chapter 1).
- Floating-point model, and its implications.
- Principles of algorithm analysis and design.
- Principles of software design.
- Use of advanced computer architectures.
Basic methods of random number generation and issues relating to the quality of the generators (Chapter 2).
Methods of simulating specific distributions (Chapter 2).
- Inverse CDF.
- Acceptance/rejection principles.
Monte Carlo methods (Chapter 2).
- Quadrature - multivariate integrals; improper integrals.
- Variance reduction.
Methods for more general simulation (Chapter 2} and other chapters).
General numerical methods for vectors and matrices (Chapter 3).
Methods for solving linear systems; matrix decompositions such as $QR$, Cholesky, $LU$, and SVD (Chapter 3).
Basic methods for eigensystems and singular values (Chapter 3).
Methods for solving nonlinear systems (Chapter 4).
Methods for optimization of differentiable functions: Newton, quasi-Newton, Gauss-Newton, stochastic approximation methods, and so on (Chapter 4).
Methods for optimization over dense domains, that do not use derivatives: Nelder-Mead, dud, and so on (Chapter 4).
Methods for combinatorial optimization: simulated annealing, genetic algorithms, and so on (Chapter 4).
Methods for optimization under constraints (Chapter 4).
Orthogonal systems; expansions; transforms, of both functions and data (Chapter 5).
Data analysis (Chapters 5 and 6).
Fitting models to data (Chapter 5).
- Interpolation.
- Approximation/estimation.
  - Minimizing residuals.
  - Maximizing likelihood.
Basic operations with multivariate data (Chapter 6).
- Data structures and storage formats.
- Ordering of multivariate data (graphs, minimal spanning trees, etc.).
- Finding patterns and identifying groups in data (clustering, classification).
- Dimension reduction (principal components analysis and related techniques).
Quadrature (Chapter 7).
Solution of ODEs (Chapter 7).
Solution of PDEs (Chapter 7).
Solution of SDEs (Chapter 7).
Fitting differential equations to data (Chapter 7).
Building and analyzing mathematical and computer models (Chapters 5, 6, and 7).

Applications

This text mentions a number of applications, but does not discuss any of them extensively. The student will be expected to complete a project that applies the skills and topics listed above to some area of the computational sciences.