Discussion from UC-HiPACC on Vimeo. Also See: Bolshoi Simulation: WMAP Explorer
“It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do? So I have often made the hypotheses that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities”.
Computational science (or scientific computing) is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems. In practical use, it is typically the application of computer simulation and other forms of computation to problems in various scientific disciplines.
The field is distinct from computer science (the study of computation, computers and information processing). It is also different from theory and experiment which are the traditional forms of science and engineering. The scientific computing approach is to gain understanding, mainly through the analysis of mathematical models implemented on computers.
Scientists and engineers develop computer programs, application software, that model systems being studied and run these programs with various sets of input parameters. Typically, these models require massive amounts of calculations (usually floating-point) and are often executed on supercomputers or distributed computing platforms.
Numerical analysis is an important underpinning for techniques used in computational science.
Applications of computational science
Problem domains for computational science/scientific computing include:
Numerical simulations have different objectives depending on the nature of the task being simulated:
- Reconstruct and understand known events (e.g., earthquake, tsunamis and other natural disasters).
- Predict future or unobserved situations (e.g., weather, sub-atomic particle behaviour).
Model fitting and data analysis
- Appropriately tune models or solve equations to reflect observations, subject to model constraints (e.g. oil exploration geophysics, computational linguistics).
- Use graph theory to model networks, especially those connecting individuals, organizations, and websites.
- Optimize known scenarios (e.g., technical and manufacturing processes, front-end engineering).
Methods and algorithms
Algorithms and mathematical methods used in computational science are varied. Commonly applied methods include:
- Numerical analysis
- Application of Taylor series as convergent and asymptotic series
- Computing derivatives by Automatic differentiation (AD)
- Computing derivatives by finite differences
- Graph theoretic suites
- High order difference approximations via Taylor series and Richardson extrapolation
- Methods of integration on a uniform mesh: rectangle rule (also called midpoint rule), trapezoid rule, Simpson’s rule
- Runge Kutta method for solving ordinary differential equations
- Monte Carlo methods
- Molecular dynamics
- Numerical linear algebra
- Computing the LU factors by Gaussian elimination
- Cholesky factorizations
- Discrete Fourier transform and applications.
- Newton’s method
- Time stepping methods for dynamical systems
Programming languages commonly used for the more mathematical aspects of scientific computing applications include R (programming language), MATLAB, Mathematica, SciLab, GNU Octave, COMSOL Multiphysics, Python (programming language) with SciPy, and PDL. The more computationally intensive aspects of scientific computing will often utilize some variation of C or Fortran and optimized algebra libraries such as BLAS or LAPACK.
Computational science application programs often model real-world changing conditions, such as weather, air flow around a plane, automobile body distortions in a crash, the motion of stars in a galaxy, an explosive device, etc. Such programs might create a ‘logical mesh’ in computer memory where each item corresponds to an area in space and contains information about that space relevant to the model. For example in weather models, each item might be a square kilometer; with land elevation, current wind direction, humidity, temperature, pressure, etc. The program would calculate the likely next state based on the current state, in simulated time steps, solving equations that describe how the system operates; and then repeat the process to calculate the next state.
The term computational scientist is used to describe someone skilled in scientific computing. This person is usually a scientist, an engineer or an applied mathematician who applies high-performance computers in different ways to advance the state-of-the-art in their respective applied disciplines in physics, chemistry or engineering. Scientific computing has increasingly also impacted on other areas including economics, biology and medicine.
Computational science is now commonly considered a third mode of science, complementing and adding to experimentation/observation and theory. The essence of computational science is numerical algorithm and/or computational mathematics. In fact, substantial effort in computational sciences has been devoted to the development of algorithms, the efficient implementation in programming languages, and validation of computational results. A collection of problems and solutions in computational science can be found in Steeb, Hardy, Hardy and Stoop, 2004.
Scientific computation is most often studied through an applied mathematics or computer science program, or within a standard mathematics, sciences, or engineering program. At some institutions a specialization in scientific computation can be earned as a “minor” within another program (which may be at varying levels). However, there are increasingly many bachelor’s and master’s programs in computational science. Some schools also offer the Ph.D. in computational science, computational engineering, computational science and engineering, or scientific computation.
- Computational biology
- Computational chemistry
- Computational economics
- Computational electromagnetics
- Computational engineering
- Computational finance
- Computational fluid dynamics
- Computational forensics
- Computational geophysics
- Computational linguistics
- Computational mathematics
- Computational mechanics
- Computational neuroscience
- Computational particle physics
- Computational physics
- Computational statistics
- Computer algebra
- Environmental simulation
- Financial modeling
- Geographic information system (GIS)
- High performance computing
- Machine learning
- Network analysis
- Numerical linear algebra
- Numerical weather prediction
- Pattern recognition
- Comparison of computer algebra systems
- List of molecular modeling software
- List of numerical analysis software
- List of statistical packages
- Simulated reality
- ^ National Center for Computational Science
- ^ Mathematica 6 Scientific Computing World, May 2007
- ^ Siam.org
- ^ Nonweiler T. R., 1986. Computational Mathematics: An Introduction to Numerical Approximation, John Wiley and Sons
- ^ Yang X. S., 2008. Introduction to Computational Mathematics, World Scientific Publishing
- ^ Steeb W.-H., Hardy Y., Hardy A. and Stoop R., 2004. Problems and Solutions in Scientific Computing with C++ and Java Simulations, World Scientific Publishing. ISBN 981-256-112-9