Computational science

Encyclopedia

**Computational science**is the field of study concerned with constructing mathematical models and quantitative analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

techniques and using computers to analyze and solve scientific problems. In practical use, it is typically the application of computer simulation

Computer simulation

A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

and other forms of computation

Computation

Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...

to problems in various scientific disciplines.

The field is distinct from computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

(the study of computation

Computation

Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...

, computers

Computer

A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

and information processing

Information processing

Information processing is the change of information in any manner detectable by an observer. As such, it is a process which describes everything which happens in the universe, from the falling of a rock to the printing of a text file from a digital computer system...

). 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 computer

Computer

A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

s.

Scientists and engineers develop computer programs, application software

Application software

Application software, also known as an application or an "app", is computer software designed to help the user to perform specific tasks. Examples include enterprise software, accounting software, office suites, graphics software and media players. Many application programs deal principally with...

, 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 supercomputer

Supercomputer

A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s or distributed computing

Distributed computing

Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...

platforms.

Numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical 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

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 theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

to model networks, especially those connecting individuals, organizations, and websites.

### Computational optimization

- 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 analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
- Application of Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

as convergent and asymptotic series - ComputingComputingComputing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...

derivatives by Automatic differentiationAutomatic differentiationIn mathematics and computer algebra, automatic differentiation , sometimes alternatively called algorithmic differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program...

(AD) - ComputingComputingComputing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...

derivatives by finite differences - Graph theoretic suites
- High order difference approximations via Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

and Richardson extrapolationRichardson extrapolationIn numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century. In the words of Birkhoff and Rota, "..... - Methods of integration on a uniform mesh: rectangle rule (also called
*midpoint rule*), trapezoid rule, Simpson's ruleSimpson's ruleIn numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:... - Runge Kutta method for solving ordinary differential equations
- Monte Carlo methodMonte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

s - Molecular dynamicsMolecular dynamicsMolecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
- Numerical linear algebraNumerical linear algebraNumerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, Telecommunication, computational...
- ComputingComputingComputing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...

the LULU decompositionIn linear algebra, LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear...

factors by Gaussian eliminationGaussian eliminationIn linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix... - Cholesky factorizationsCholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...
- Discrete Fourier transformDiscrete Fourier transformIn mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...

and applications. - Newton's methodNewton's methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley'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)

R (programming language)

R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....

, MATLAB

MATLAB

MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

, Mathematica

Mathematica

Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...

, SciLab

Scilab

Scilab is an open source, cross-platform numerical computational package and a high-level, numerically oriented programming language. Itcan be used for signal processing, statistical analysis, image enhancement, fluid dynamics simulations, numerical optimization, and modeling and simulation of...

, GNU Octave

GNU Octave

GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...

, COMSOL Multiphysics

COMSOL Multiphysics

COMSOL Multiphysics is a finite element analysis, solver and Simulation software / FEA Software package for various physics and engineering applications, especially coupled phenomena, or multiphysics. COMSOL Multiphysics also offers an extensive interface to MATLAB and its toolboxes for a large...

, Python (programming language)

Python (programming language)

Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...

with SciPy

SciPy

SciPy is an open source library of algorithms and mathematical tools for the Python programming language.SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and...

, and PDL

Perl Data Language

PDL is a set of array programming extensions to the Perl programming language.PDL is an extension to Perl v5, intended for scientific and other data intensive programming tasks...

. The more computationally intensive aspects of scientific computing will often utilize some variation of C

C (programming language)

C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....

or Fortran

Fortran

Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...

and optimized algebra libraries such as BLAS

Blas

Blas is mainly a Spanish given name and surname, related to Blaise. It may refer to-Places:*Piz Blas, mountain in Switzerland*San Blas , many places - see separate article, also**Cape San Blas Light, lighthouse...

or LAPACK

LAPACK

-External links:* : a modern replacement for PLAPACK and ScaLAPACK* on Netlib.org* * * : a modern replacement for LAPACK that is MultiGPU ready* on Sourceforge.net* * optimized LAPACK for Solaris OS on SPARC/x86/x64 and Linux* * *...

.

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

Computational scientist

A computational scientist is a person 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...

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

Science

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, complementing and adding to experimentation/observation

Observation

Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...

and theory

Theory

The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...

. The essence of computational science is numerical algorithm

and/or computational mathematics

Computational mathematics

Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. Computation in the research is prominent. Computational mathematics emerged as a distinct part of applied...

. 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.

## Education

Scientific computation is most often studied through an applied mathematicsApplied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

or computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

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

Bachelor's degree

A bachelor's degree is usually an academic degree awarded for an undergraduate course or major that generally lasts for three or four years, but can range anywhere from two to six years depending on the region of the world...

and master's programs in computational science. Some schools also offer the Ph.D. in computational science, computational engineering

Computational engineering

Computational science and engineering is a relatively new discipline of engineering. It is typically offered as a masters or doctorate program at several institutions...

, computational science and engineering, or scientific computation.

There are also programs in areas such as computational physics

Computational physics

Computational physics is the study and implementation of numerical algorithms to solve problems in physics for which a quantitative theory already exists...

, computational chemistry

Computational chemistry

Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...

, etc.

## See also

- Comparison of computer algebra systems
- List of molecular modeling software
- List of numerical analysis software
- List of statistical packages
- Simulated realitySimulated realitySimulated reality is the proposition that reality could be simulated—perhaps by computer simulation—to a degree indistinguishable from "true" reality. It could contain conscious minds which may or may not be fully aware that they are living inside a simulation....