Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced

computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...

capabilities to understand and solve complex problems. It is an area of science that spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems. *

Algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

s ( numerical and non-numerical):

mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

computational model A computational model uses computer programs to simulate and study complex systems using an algorithmic or mechanistic approach and is widely used in a diverse range of fields spanning from physics, chemistry and biology to economics, psychology, ...

s, and

computer simulation Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...

s developed to solve

science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...

(e.g.,

biological Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary in ...

physical Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...

, and

social Social organisms, including human(s), live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchange is voluntary or not. Etymology The word "social" derives from ...

engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

, and

humanities Humanities are academic disciplines that study aspects of human society and culture. In the Renaissance, the term contrasted with divinity and referred to what is now called classics, the main area of secular study in universities at the t ...

problems *

Computer hardware Computer hardware includes the physical parts of a computer, such as the computer case, case, central processing unit (CPU), Random-access memory, random access memory (RAM), Computer monitor, monitor, Computer mouse, mouse, Computer keyboard, ...

that develops and optimizes the advanced system hardware,

firmware In computing, firmware is a specific class of computer software that provides the low-level control for a device's specific hardware. Firmware, such as the BIOS of a personal computer, may contain basic functions of a device, and may provide h ...

, networking, and

data management Data management comprises all disciplines related to handling data as a valuable resource. Concept The concept of data management arose in the 1980s as technology moved from sequential processing (first punched cards, then magnetic tape) to r ...

components needed to solve computationally demanding problems * The computing infrastructure that supports both the science and engineering problem solving and the developmental computer and

information science Information science (also known as information studies) is an academic field which is primarily concerned with analysis, collection, Categorization, classification, manipulation, storage, information retrieval, retrieval, movement, dissemin ...

In practical use, it is typically the application of

and other forms of

computation Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm). Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. An es ...

from

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

and

theoretical computer science Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumsc ...

to solve problems in various scientific disciplines. The field is different from theory and laboratory experiments, which are the traditional forms of science and

. The scientific computing approach is to gain understanding through the analysis of mathematical models implemented on

computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...

s. Scientists and engineers develop

computer programs A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program ...

and

application software Application may refer to: Mathematics and computing * Application software, computer software designed to help the user to perform specific tasks ** Application layer, an abstraction layer that specifies protocols and interface methods used in a c ...

that model systems being studied and run these programs with various sets of input parameters. The essence of computational science is the application of numerical algorithms and

computational mathematics Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006 ...

. In some cases, these models require massive amounts of calculations (usually

floating-point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...

) and are often executed on

supercomputer A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second ( FLOPS) instead of million instructions ...

s or

distributed computing A distributed system is a system whose components are located on different computer network, networked computers, which communicate and coordinate their actions by message passing, passing messages to one another from any system. Distributed com ...

platforms.

The computational scientist

The term

computational scientist A computational scientist is a person skilled in scientific computing. This person is usually a scientist, a statistician, an applied mathematician, or an engineer who applies high-performance computing and sometimes cloud computing in different w ...

is used to describe someone skilled in scientific computing. Such a person is usually a scientist, an engineer, or an applied mathematician who applies

high-performance computing High-performance computing (HPC) uses supercomputers and computer clusters to solve advanced computation problems. Overview HPC integrates systems administration (including network and security knowledge) and parallel programming into a mult ...

in different ways to advance the state-of-the-art in their respective applied disciplines in physics, chemistry, or engineering. Computational science is now commonly considered a third mode of

, complementing and adding to

experimentation An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when ...

observation Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. The ...

and

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...

(see image). Here, one defines a

system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...

as a potential source of data, an

experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...

as a process of extracting data from a system by exerting it through its inputs and a

model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

(''M'') for a system (''S'') and an experiment (''E'') as anything to which ''E'' can be applied in order to answer questions about ''S''. A computational scientist should be capable of: * recognizing complex problems * adequately conceptualizing the system containing these problems * designing a framework of algorithms suitable for studying this system: the simulation * choosing a suitable computing infrastructure (

parallel computing Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different fo ...

grid computing Grid computing is the use of widely distributed computer resources to reach a common goal. A computing grid can be thought of as a distributed system with non-interactive workloads that involve many files. Grid computing is distinguished from co ...

s) * hereby, maximizing the computational power of the simulation * assessing to what level the output of the simulation resembles the systems: the model is validated * adjusting the conceptualization of the system accordingly * repeat the cycle until a suitable level of validation is obtained: the computational scientist trusts that the simulation generates adequately realistic results for the system under the studied conditions Substantial effort in computational sciences has been devoted to developing algorithms, efficient implementation in programming languages, and validating computational results. A collection of problems and solutions in computational science can be found in Steeb, Hardy, Hardy, and Stoop (2004). Philosophers of science addressed the question to what degree computational science qualifies as science, among them Humphreys and Gelfert. They address the general question of epistemology: how does gain insight from such computational science approaches? Tolk uses these insights to show the epistemological constraints of computer-based simulation research. As computational science uses mathematical models representing the underlying theory in executable form, in essence, they apply modeling (theory building) and simulation (implementation and execution). While simulation and computational science are our most sophisticated way to express our knowledge and understanding, they also come with all constraints and limits already known for computational solutions.

Applications of computational science

Problem domains for computational science/scientific computing include:

Predictive computational science

Predictive computational science is a scientific discipline concerned with the formulation, calibration, numerical solution, and validation of mathematical models designed to predict specific aspects of physical events, given initial and boundary conditions, and a set of characterizing parameters and associated uncertainties. In typical cases, the predictive statement is formulated in terms of probabilities. For example, given a mechanical component and a periodic loading condition, "the probability is (say) 90% that the number of cycles at failure (Nf) will be in the interval N1

Urban complex systems

In 2018, over half the world's population lives in cities. By 2050, the

United Nations The United Nations (UN) is an intergovernmental organization whose stated purposes are to maintain international peace and international security, security, develop friendly relations among nations, achieve international cooperation, and be ...

estimates, 68% of the world's population will be

urban Urban means "related to a city". In that sense, the term may refer to: * Urban area, geographical area distinct from rural areas * Urban culture, the culture of towns and cities Urban may also refer to: General * Urban (name), a list of people ...

. This urban growth is focused in the urban populations of developing countries where city dwellers will more than double, increasing from 2.5 billion in 2009 to almost 5.2 billion in 2050. Cities are massively complex systems created by humans, made up of humans, and governed by humans. Trying to predict, understand and somehow shape the development of cities in the future requires complex thinking and computational models and simulations to help mitigate challenges and possible disasters. The focus of research in urban complex systems is, through modeling and simulation, to build a greater understanding of city dynamics and help prepare for the coming

urbanization Urbanization (or urbanisation) refers to the population shift from rural to urban areas, the corresponding decrease in the proportion of people living in rural areas, and the ways in which societies adapt to this change. It is predominantly t ...

Computational finance

financial market A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial markets ...

s, huge volumes of interdependent assets are traded by a large number of interacting market participants in different locations and time zones. Their behavior is of unprecedented complexity and the characterization and measurement of the risk inherent to this highly diverse set of instruments is typically based on complicated

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

s. Solving these models exactly in closed form, even at a single instrument level, is typically not possible, and therefore we have to look for efficient

numerical algorithm Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

s. This has become even more urgent and complex recently, as the credit crisis has clearly demonstrated the role of cascading effects going from single instruments through portfolios of single institutions to even the interconnected trading network. Understanding this requires a multi-scale and holistic approach where interdependent risk factors such as market, credit, and liquidity risk are modeled simultaneously and at different interconnected scales.

Computational biology

Exciting new developments in

biotechnology Biotechnology is the integration of natural sciences and engineering sciences in order to achieve the application of organisms, cells, parts thereof and molecular analogues for products and services. The term ''biotechnology'' was first used b ...

are now revolutionizing biology and

biomedical research Medical research (or biomedical research), also known as experimental medicine, encompasses a wide array of research, extending from "basic research" (also called ''bench science'' or ''bench research''), – involving fundamental scientif ...

. Examples of these techniques are

high-throughput sequencing DNA sequencing is the process of determining the nucleic acid sequence – the order of nucleotides in DNA. It includes any method or technology that is used to determine the order of the four bases: adenine, guanine, cytosine, and thymine. The ...

, high-throughput

quantitative PCR A real-time polymerase chain reaction (real-time PCR, or qPCR) is a laboratory technique of molecular biology based on the polymerase chain reaction (PCR). It monitors the amplification of a targeted DNA molecule during the PCR (i.e., in real ...

, intra-cellular imaging, in-situ hybridization of gene expression, three-dimensional imaging techniques like Light Sheet Fluorescence Microscopy, and Optical Projection (micro)-Computer Tomography. Given the massive amounts of complicated data that is generated by these techniques, their meaningful interpretation, and even their storage, form major challenges calling for new approaches. Going beyond current bioinformatics approaches,

computational biology Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has fo ...

needs to develop new methods to discover meaningful patterns in these large data sets. Model-based reconstruction of gene networks can be used to organize the gene expression data in a systematic way and to guide future data collection. A major challenge here is to understand how gene regulation is controlling fundamental biological processes like

biomineralization Biomineralization, also written biomineralisation, is the process by which living organisms produce minerals, often to harden or stiffen existing tissues. Such tissues are called mineralized tissues. It is an extremely widespread phenomenon; ...

and

embryogenesis An embryo is an initial stage of development of a multicellular organism. In organisms that reproduce sexually, embryonic development is the part of the life cycle that begins just after fertilization of the female egg cell by the male sperm ...

. The sub-processes like

gene regulation Regulation of gene expression, or gene regulation, includes a wide range of mechanisms that are used by cells to increase or decrease the production of specific gene products (protein or RNA). Sophisticated programs of gene expression are wi ...

, organic molecules interacting with the mineral deposition process,

cellular processes The cell is the basic structural and functional unit of life forms. Every cell consists of a cytoplasm enclosed within a membrane, and contains many biomolecules such as proteins, DNA and RNA, as well as many small molecules of nutrients and ...

physiology Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...

, and other processes at the tissue and environmental levels are linked. Rather than being directed by a central control mechanism, biomineralization and embryogenesis can be viewed as an emergent behavior resulting from a complex system in which several sub-processes on very different temporal and

spatial scale Spatial scale is a specific application of the term scale for describing or categorizing (e.g. into orders of magnitude) the size of a space (hence ''spatial''), or the extent of it at which a phenomenon or process occurs. For instance, in phy ...

s (ranging from nanometer and nanoseconds to meters and years) are connected into a multi-scale system. One of the few available options to understand such systems is by developing a multi-scale model of the system.

Complex systems theory

Using

information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

, non-equilibrium dynamics, and explicit simulations, computational systems theory tries to uncover the true nature of

complex adaptive system A complex adaptive system is a system that is ''complex'' in that it is a dynamic network of interactions, but the behavior of the ensemble may not be predictable according to the behavior of the components. It is ''adaptive'' in that the individ ...

Computational science in engineering

Computational science and engineering (CSE) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with

, to solve complex physical problems arising in engineering analysis and design (computational engineering) as well as natural phenomena (computational science). CSE has been described as the "third mode of discovery" (next to theory and experimentation). In many fields, computer simulation is integral and therefore essential to business and research. Computer simulation provides the capability to enter fields that are either inaccessible to traditional experimentation or where carrying out traditional empirical inquiries is prohibitively expensive. CSE should neither be confused with pure

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, nor with

computer engineering Computer engineering (CoE or CpE) is a branch of electrical engineering and computer science that integrates several fields of computer science and electronic engineering required to develop computer hardware and software. Computer engineers ...

, although a wide domain in the former is used in CSE (e.g., certain algorithms, data structures, parallel programming, high-performance computing), and some problems in the latter can be modeled and solved with CSE methods (as an application area).

Methods and algorithms

Algorithms and mathematical methods used in computational science are varied. Commonly applied methods include: *

Computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...

, including symbolic computation in fields such as statistics, equation solving, algebra, calculus, geometry, linear algebra, tensor analysis (multilinear algebra), optimization *

Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

,Stoer, J., & Bulirsch, R. (2013). Introduction to numerical analysis. Springer Science & Business Media.Conte, S. D., & De Boor, C. (2017). Elementary numerical analysis: an algorithmic approach.

Society for Industrial and Applied Mathematics Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific socie ...

.Greenspan, D. (2018). Numerical Analysis. CRC Press.Linz, P. (2019). Theoretical numerical analysis. Courier Dover Publications. including

Computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...

derivatives by

finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...

s ** Application of

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

as convergent and asymptotic series **

derivatives by

Automatic differentiation In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techniques to evaluate the derivative of a function s ...

(AD) **

Finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...

for solving PDEsBrenner, S., & Scott, R. (2007). The mathematical theory of finite element methods (Vol. 15). Springer Science & Business Media.Oden, J. T., & Reddy, J. N. (2012). An introduction to the mathematical theory of finite elements. Courier Corporation. ** High order difference approximations via

and

Richardson extrapolation In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, we ...

** Methods of integration on a uniform

mesh A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands. Types * A plastic mesh may be extruded, oriented, ex ...

rectangle rule In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or ...

(also called ''midpoint rule''),

trapezoid rule In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral. \int_a^b f(x) \, dx. The trapezoidal rule works b ...

Simpson's rule In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, ...

Runge–Kutta methods In numerical analysis, the Runge–Kutta methods ( ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. The ...

for solving ordinary differential equations **

Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...

Discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...

s *

Numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. ...

, including decompositions and

eigenvalue algorithm In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors Given an square ...

s *

Linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...

Branch and cut Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. Branch and cut involves running a branc ...

Branch and bound Branch and bound (BB, B&B, or BnB) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate soluti ...

Molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...

Car–Parrinello molecular dynamics Car–Parrinello molecular dynamics or CPMD refers to either a method used in molecular dynamics (also known as the Car–Parrinello method) or the computational chemistry software package used to implement this method. The CPMD method is one of th ...

Space mapping The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The kno ...

Time stepping Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also ...

methods for dynamical systems Historically and today, Fortran remains popular for most applications of scientific computing. Other

programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...

s and computer algebra systems commonly used for the more mathematical aspects of scientific computing applications include

GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a langu ...

Haskell Haskell () is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research and industrial applications, Haskell has pioneered a number of programming lan ...

Julia Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...

Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...

Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...

MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...

,Gander, W., & Hrebicek, J. (Eds.). (2011). Solving problems in scientific computing using Maple and Matlab®.

Springer Science & Business Media Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

.Barnes, B., & Fulford, G. R. (2011). Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB. Chapman and Hall/CRC.

Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (pro ...

(with third-party

SciPy SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing. SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal ...

library),

Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offici ...

(with third-party PDL library), R,

Scilab Scilab is a free and open-source, cross-platform numerical computational package and a high-level, numerically oriented programming language. It can be used for signal processing, statistical analysis, image enhancement, fluid dynamics simulat ...

, and

TK Solver TK Solver (originally TK!Solver) is a mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc. History Invented by Milos Konopasek in the late ...

. The more computationally intensive aspects of scientific computing will often use some variation of C or Fortran and optimized algebra libraries such as

BLAS Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix ...

LAPACK LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also ...

. In addition,

is heavily used in scientific computing to find solutions of large problems in a reasonable amount of time. In this framework, the problem is either divided over many cores on a single CPU node (such as with

OpenMP OpenMP (Open Multi-Processing) is an application programming interface (API) that supports multi-platform shared-memory multiprocessing programming in C, C++, and Fortran, on many platforms, instruction-set architectures and operating syste ...

), divided over many CPU nodes networked together (such as with MPI), or is run on one or more

GPUs A graphics processing unit (GPU) is a specialized electronic circuit designed to manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display device. GPUs are used in embedded systems, mobil ...

(typically using either

CUDA CUDA (or Compute Unified Device Architecture) is a parallel computing platform and application programming interface (API) that allows software to use certain types of graphics processing units (GPUs) for general purpose processing, an approach ca ...

OpenCL OpenCL (Open Computing Language) is a framework for writing programs that execute across heterogeneous platforms consisting of central processing units (CPUs), graphics processing units (GPUs), digital signal processors (DSPs), field-progra ...

). Computational science application programs often model real-world changing conditions, such as weather, airflow 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 differential equations that describe how the system operates, and then repeat the process to calculate the next state.

Conferences and journals

In 2001, the ''International Conference on Computational Science (ICCS)'' was first organized. Since then, it has been organized yearly. ICCS is an A-rank conference in CORE classification. The ''Journal of Computational Science'' published its first issue in May 2010. The ''Journal of Open Research Software'' was launched in 2012''.'' The

ReScience C ''ReScience C'' is a journal created in 2015 by Nicolas Rougier and Konrad Hinsen with the aim of publishing researchers' attempts to replicate computations made by other authors, using independently written, free and open-source software (FOSS), ...

initiative, which is dedicated to replicating computational results, was started on

GitHub GitHub, Inc. () is an Internet hosting service for software development and version control using Git. It provides the distributed version control of Git plus access control, bug tracking, software feature requests, task management, continuous ...

in 2015.

Education

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 A bachelor's degree (from Middle Latin ''baccalaureus'') or baccalaureate (from Modern Latin ''baccalaureatus'') is an undergraduate academic degree awarded by colleges and universities upon completion of a course of study lasting three to six ye ...

master's A master's degree (from Latin ) is an academic degree awarded by universities or colleges upon completion of a course of study demonstrating mastery or a high-order overview of a specific field of study or area of professional practice.

, and doctoral programs in computational science. The joint degree program master program computational science at the

University of Amsterdam The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university located in Amsterdam, Netherlands. The UvA is one of two large, publicly funded research universities in the city, the other being ...

and the

Vrije Universiteit The Vrije Universiteit Amsterdam (abbreviated as ''VU Amsterdam'' or simply ''VU'' when in context) is a public research university in Amsterdam, Netherlands, being founded in 1880. The VU Amsterdam is one of two large, publicly funded research ...

in computational science was first offered in 2004. In this program, students: * learn to build computational models from real-life observations; * develop skills in turning these models into computational structures and in performing large-scale simulations; * learn theories that will give a firm basis for the analysis of complex systems; * learn to analyze the results of simulations in a virtual laboratory using advanced numerical algorithms.

ETH Zurich (colloquially) , former_name = eidgenössische polytechnische Schule , image = ETHZ.JPG , image_size = , established = , type = Public , budget = CHF 1.896 billion (2021) , rector = Günther Dissertori , president = Joël Mesot , ac ...

offers a bachelor's and master's degree in Computational Science and Engineering. The degree equips students with the ability to understand scientific problem and apply numerical methods to solve such problems. The directions of specializations include Physics, Chemistry, Biology and other Scientific and Engineering disciplines.

George Mason University George Mason University (George Mason, Mason, or GMU) is a public research university in Fairfax County, Virginia with an independent City of Fairfax, Virginia postal address in the Washington, D.C. Metropolitan Area. The university was origin ...

was one of the early pioneers first offering a multidisciplinary doctorate Ph.D. program in Computational Sciences and Informatics in 1992 that focused on a number of specialty areas, including

bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...

computational chemistry Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...

, earth systems, and global changes,

computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...

, space sciences, and

computational statistics Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computi ...

. The School of Computational and Integrative Sciences,

Jawaharlal Nehru University Jawaharlal Nehru University (JNU) is a public major research university located in New Delhi, India. It was established in 1969 and named after Jawaharlal Nehru, India's first Prime Minister. The university is known for leading faculties and r ...

(erstwhile School of Information Technology) also offers a vibrant master's science program for computational science with two specialties:

Computational Biology Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has fo ...

and

Complex Systems A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication s ...

Related fields

Bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...

Cheminformatics Cheminformatics (also known as chemoinformatics) refers to use of physical chemistry theory with computer and information science techniques—so called "''in silico''" techniques—in application to a range of descriptive and prescriptive problem ...

Chemometrics Chemometrics is the science of extracting information from chemical systems by data-driven means. Chemometrics is inherently interdisciplinary, using methods frequently employed in core data-analytic disciplines such as multivariate statistics, a ...

Computational archaeology Computational archaeology describes computer-based analytical methods for the study of long-term human behaviour and behavioural evolution. As with other sub-disciplines that have prefixed 'computational' to their name (e.g., computational biol ...

Computational astrophysics Computational astrophysics refers to the methods and computing tools developed and used in astrophysics research. Like computational chemistry or computational physics, it is both a specific branch of theoretical astrophysics and an interdiscipli ...

Computational biology Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has fo ...

Computational chemistry Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...

Computational materials science ''Computational Materials Science'' is a monthly peer-reviewed scientific journal published by Elsevier. It was established in October 1992. The editor-in-chief is Susan Sinnott. The journal covers computational modeling and practical research fo ...

Computational economics Computational Economics is an interdisciplinary research discipline that involves computer science, economics, and management science.''Computational Economics''."About This Journal"an"Aims and Scope" This subject encompasses computational model ...

Computational electromagnetics Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computer ...

Computational engineering Computational science and engineering (CSE) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with high-performance computing, to solve complex physical problems ...

Computational finance Computational finance is a branch of applied computer science that deals with problems of practical interest in finance.Rüdiger U. Seydel, '' tp://nozdr.ru/biblio/kolxo3/F/FN/Seydel%20R.U.%20Tools%20for%20Computational%20Finance%20(4ed.,%20Spring ...

Computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...

Computational forensics Computational criminology is an interdisciplinary field which uses computing science methods to formally define criminology concepts, improve our understanding of complex phenomena, and generate solutions for related problems. Methods Computing sci ...

Computational geophysics Computational geophysics is the field of study that uses any type of numerical computations to generate and analyze models of complex geophysical systems. It can be considered an extension, or sub-field, of both computational physics and geophysics. ...

* Computational history * Computational informatics *

Computational intelligence The expression computational intelligence (CI) usually refers to the ability of a computer to learn a specific task from data or experimental observation. Even though it is commonly considered a synonym of soft computing, there is still no c ...

Computational law Computational Law is the branch of legal informatics concerned with the automation of legal reasoning. What distinguishes Computational Law systems from other instances of legal technology is their autonomy, i.e. the ability to answer legal quest ...

Computational linguistics Computational linguistics is an Interdisciplinarity, interdisciplinary field concerned with the computational modelling of natural language, as well as the study of appropriate computational approaches to linguistic questions. In general, comput ...

Computational mathematics Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006 ...

Computational mechanics Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "third w ...

Computational neuroscience Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to u ...

Computational particle physics Computational particle physics refers to the methods and computing tools developed in and used by particle physics research. Like computational chemistry or computational biology, it is, for particle physics both a specific branch and an inter ...

Computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...

Computational sociology Computational sociology is a branch of sociology that uses computationally intensive methods to analyze and model social phenomena. Using computer simulations, artificial intelligence, complex statistical methods, and analytic approaches like soc ...

Computational statistics Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computi ...

* Computational sustainability *

Computer simulation Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...

Financial modeling Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio o ...

Geographic information system A geographic information system (GIS) is a type of database containing Geographic data and information, geographic data (that is, descriptions of phenomena for which location is relevant), combined with Geographic information system software, sof ...

(GIS) *

High-performance computing High-performance computing (HPC) uses supercomputers and computer clusters to solve advanced computation problems. Overview HPC integrates systems administration (including network and security knowledge) and parallel programming into a mult ...

Machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...

Network analysis Network analysis can refer to: * Network theory, the analysis of relations through mathematical graphs ** Social network analysis, network theory applied to social relations * Network analysis (electrical circuits) See also *Network planning and ...

Neuroinformatics Neuroinformatics is the field that combines informatics and neuroscience. Neuroinformatics is related with neuroscience data and information processing by artificial neural networks. There are three main directions where neuroinformatics has to be ...

Numerical weather prediction Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in th ...

Pattern recognition Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphi ...

Scientific visualization Scientific visualization ( also spelled scientific visualisation) is an interdisciplinary branch of science concerned with the visualization of scientific phenomena.Michael Friendly (2008)"Milestones in the history of thematic cartography, stat ...

Simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...

References

Additional sources

* E. Gallopoulos and A. Sameh, "CSE: Content and Product". IEEE Computational Science and Engineering Magazine, 4(2):39–43 (1997) * G. Hager and G. Wellein, Introduction to High Performance Computing for Scientists and Engineers,

Chapman and Hall Chapman & Hall is an imprint owned by CRC Press, originally founded as a British publishing house in London in the first half of the 19th century by Edward Chapman and William Hall. Chapman & Hall were publishers for Charles Dickens (from 1840 ...

(2010) * A.K. Hartmann
Practical Guide to Computer Simulations

World Scientific World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various f ...

(2009) * Journa
Computational Methods in Science and Technology
(open access),

Polish Academy of Sciences The Polish Academy of Sciences ( pl, Polska Akademia Nauk, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of ...

* Journa
Computational Science and Discovery

Institute of Physics The Institute of Physics (IOP) is a UK-based learned society and professional body that works to advance physics education, research and application. It was founded in 1874 and has a worldwide membership of over 20,000. The IOP is the Physica ...

* R.H. Landau, C.C. Bordeianu, and M. Jose Paez
A Survey of Computational Physics: Introductory Computational Science

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial su ...

(2008)

External links

Journal of Computational ScienceThe Journal of Open Research SoftwareThe National Center for Computational Science at Oak Ridge National Laboratory
{{DEFAULTSORT:Computational Science Applied mathematics Computational fields of study