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, ...
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 problems or to perform a computation. Algorithms are used as specifications for performing ...
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, ...
s,
computational models, and
computer simulations developed to solve
science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...
(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 ...
,
physical, 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 th ...
problems
*
Computer hardware 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 ...
,
networking
Network, networking and networked may refer to:
Science and technology
* Network theory, the study of graphs as a representation of relations between discrete objects
* Network science, an academic field that studies complex networks
Mathematics
...
, and
data management 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, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information. ...
In practical use, it is typically the application of
computer simulation and other forms of
computation from
numerical analysis 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
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 ...
. The scientific computing approach is to gain understanding through the analysis of mathematical models implemented on
computers. 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 progra ...
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 ...
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) and are often executed on
supercomputers or
distributed computing platforms.
The computational scientist

The term
computational scientist 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 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
science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...
, complementing and adding to
experimentation/
observation 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 ...
(see image). Here, one defines a
system 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 cause-and-effect by demonstrating what outcome occurs wh ...
as a process of extracting data from a system by exerting it through its inputs and a
model (''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/
grid computing/
supercomputers)
* 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 security, develop friendly relations among nations, achieve international cooperation, and be a centre for harmonizi ...
estimates, 68% of the world's population will be urban. 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
In financial markets, 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 computational models. 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 algorithms. 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. Examples of these techniques are high-throughput sequencing, high-throughput quantitative PCR, intra-cellular imaging, in-situ hybridization
''In situ'' hybridization (ISH) is a type of hybridization that uses a labeled complementary DNA, RNA or modified nucleic acids strand (i.e., probe) to localize a specific DNA or RNA sequence in a portion or section of tissue (''in situ'') or ...
of gene expression, three-dimensional imaging techniques like Light Sheet Fluorescence Microscopy
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahe ...
, 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 needs to develop new methods to discover meaningful patterns in these large data sets. Model-based reconstruction of gene networks
A gene (or genetic) regulatory network (GRN) is a collection of molecular regulators that interact with each other and with other substances in the cell to govern the gene expression levels of mRNA and proteins which, in turn, determine the fu ...
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 and embryogenesis. The sub-processes like gene regulation, organic molecules interacting with the mineral deposition process, cellular processes, 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 chemic ...
, 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 scales (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, non-equilibrium dynamics, and explicit simulations, computational systems theory tries to uncover the true nature of complex adaptive systems.
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 high-performance computing, 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 practical disciplines (includin ...
, 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 enginee ...
, 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, including symbolic computation in fields such as statistics, equation solving, algebra, calculus, geometry, linear algebra, tensor analysis (multilinear algebra), optimization
* Numerical analysis,[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.][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, ...
derivatives by finite differences
** Application of Taylor series as convergent and asymptotic series
** 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, ...
derivatives by Automatic differentiation (AD)
** Finite element method for solving PDEs[Brenner, 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 Taylor series 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, ...
** Methods of integration
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
on a uniform mesh: rectangle rule (also called ''midpoint rule''), trapezoid rule, Simpson's rule
** Runge–Kutta methods for solving ordinary differential equations
** Newton's method
* Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...
* 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 deter ...
s
* Numerical linear algebra, including decompositions and eigenvalue algorithms
* Linear programming
* Branch and cut
* Branch and bound
* 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 th ...
, Car–Parrinello molecular dynamics
* Space mapping
* Time stepping methods for dynamical systems
Historically and today, Fortran remains popular for most applications of scientific computing. Other programming languages and computer algebra systems commonly used for the more mathematical aspects of scientific computing applications include GNU Octave, Haskell,[ Julia,][ Maple,][ Mathematica, ]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, implementa ...
,[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 i ...
.[Barnes, B., & Fulford, G. R. (2011). Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB. Chapman and Hall/CRC.] Python (with third-party SciPy library), Perl
Perl is a family of two High-level programming language, high-level, General-purpose programming language, general-purpose, Interpreter (computing), interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it ...
(with third-party PDL PDL is an initialism for:
Politics
*Democratic Liberal Party (Romania), Democratic Liberal Party (''Partidul Democrat Liberal''), a former political party in Romania
*Labour Democratic Party (''Partito Democratico del Lavoro''), a former politi ...
library), R, Scilab, and TK Solver. The more computationally intensive aspects of scientific computing will often use some variation of C or Fortran and optimized algebra libraries such as BLAS or LAPACK. In addition, parallel computing 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), divided over many CPU nodes networked together (such as with MPI
MPI or Mpi may refer to:
Science and technology Biology and medicine
* Magnetic particle imaging, an emerging non-invasive tomographic technique
* Myocardial perfusion imaging, a nuclear medicine procedure that illustrates the function of the hear ...
), or is run on one or more GPUs (typically using either CUDA or OpenCL).
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
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 ...
, 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 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, co ...
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, master's, 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 bein ...
and the Vrije Universiteit 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 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 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, 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 mo ...
, earth systems, and global changes, 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 physics, space sciences, and computational statistics.
The School of Computational and Integrative Sciences, Jawaharlal Nehru University (erstwhile School of Information Technology) also offers a vibrant master's science program for computational science with two specialties: Computational Biology and Complex Systems.
Related fields
* Bioinformatics
* Car–Parrinello molecular dynamics
* Cheminformatics
* Chemometrics
* Computational archaeology
* Computational astrophysics
* Computational biology
* 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 mo ...
* Computational materials science
* 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 mod ...
* Computational electromagnetics
* 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 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 t ...
* Computational forensics
* Computational geophysics
* Computational history
* Computational informatics
* Computational intelligence
* Computational law
* 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 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 ...
* Computational particle physics
* Computational physics
* 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 s ...
* Computational statistics
* Computational sustainability
* Computer algebra
* Computer simulation
* Financial modeling
* Geographic information system (GIS)
* High-performance computing
* 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
* Neuroinformatics
* Numerical linear algebra
* Numerical weather prediction
* Pattern recognition
* Scientific visualization
* Simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...
See also
* Computational science and engineering
* Modeling and simulation
* Comparison of computer algebra systems
* Differentiable programming
* List of molecular modeling software
* List of numerical analysis software
* List of statistical packages
* Timeline of scientific computing
* Simulated reality
* Extensions for Scientific Computation (XSC)
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 (2010)
* A.K. Hartmann
Practical Guide to Computer Simulations
World Scientific (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
* 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 Academic publishing, 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, ...
(2008)
External links
Journal of Computational Science
The Journal of Open Research Software
The National Center for Computational Science at Oak Ridge National Laboratory
{{DEFAULTSORT:Computational Science
Applied mathematics
Computational fields of study