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Coarse Space (numerical Analysis)
: ''This article deals with a component of numerical methods. For coarse space in topology, see coarse structure.'' In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given larger system of equations. A coarse problem is basically a version of the same problem at a lower resolution, retaining its essential characteristics, but with fewer variables. The purpose of the coarse problem is to propagate information throughout the whole problem globally. In multigrid methods for partial differential equations, the coarse problem is typically obtained as a discretization of the same equation on a coarser grid (usually, in finite difference methods) or by a Galerkin approximation on a subspace, called a coarse space. In finite element methods, the Galerkin approximation is typically used, with the coarse space generated by larger elements on the same domain. Typically, the coarse problem corresponds to a grid that is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Coarse Structure
In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties. Properly, a coarse structure is not the large-scale analog o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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BDDC
In numerical analysis, BDDC (balancing domain decomposition by constraints) is a domain decomposition method for solving large symmetric, positive definite systems of linear equations that arise from the finite element method. BDDC is used as a preconditioner to the conjugate gradient method. A specific version of BDDC is characterized by the choice of coarse degrees of freedom, which can be values at the corners of the subdomains, or averages over the edges or the faces of the interface between the subdomains. One application of the BDDC preconditioner then combines the solution of local problems on each subdomains with the solution of a global coarse problem with the coarse degrees of freedom as the unknowns. The local problems on different subdomains are completely independent of each other, so the method is suitable for parallel computing. With a proper choice of the coarse degrees of freedom (corners in 2D, corners plus edges or corners plus faces in 3D) and with regular sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Michel Bercovier
Michel Bercovier (Hebrew: מישל ברקוביאר; born: 10 September 1941) is a French-Israeli Professor (Emeritus) of Scientific Computing and Computer Aided Design (CAD) in The Rachel and Selim Benin School of Computer Science and Engineering at the Hebrew University of Jerusalem. Bercovier is also the head of the School of Computer Science at the Hadassah Academic College, Jerusalem. Early life and education Michel Bercovier was born in Lyon, France. He received his B.Sc in Mathematics from Paris University in 1964. He was from 1964 to 1965 vice president of Union of French Jewish Students and co-principal editor of its magazine ''Kadima''. During the years 1965-67 he served in the French Army. He earned his D. Es Sc. in 1976 at the Faculté des Sciences de Rouen. Bercovier authored the thesis Régularisation duale des problèmes variationnels mixtes (Dual regularization of mixed variational problems), under the supervision of Jacques-Louis Lions. He belongs to the secon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Olof B
Olov (or Olof) is a Swedish form of Olav/ Olaf, meaning "ancestor's descendant". A common short form of the name is ''Olle''. The name may refer to: *Per-Olov Ahrén (1926–2004), Swedish clergyman, bishop of Lund from 1980 to 1992 * Per-Olov Brasar (born 1950), retired professional ice hockey forward * Olov Englund (born 1983), Swedish bandy player * Per Olov Enquist (1934–2020), one of Sweden's internationally best known authors * Olle Hagnell (1924–2011), Swedish psychiatrist *Karl Olov Hedberg (1923–2007), botanist, taxonomist, author, professor at Uppsala University *Olle Hellbom (1925–1982), Swedish film director *Per Olov Jansson (1920–2019), Finnish photographer * Olof Johansson (born 1937), Swedish politician *Per-Olov Kindgren (born 1956), Swedish musician, composer, guitarist and music teacher *Olov Lambatunga, Archbishop of Uppsala, Sweden, 1198–1206 * Sven-Olov Lawesson (1926–1988), Swedish chemist known for his popularization of Lawesson's reagent with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Jan Mandel
Jan Mandel is a Czech- American mathematician. He received his PhD from the Faculty of Mathematics and Physics, Charles University in Prague and was a Senior Research Scientist there. Since 1986, he is professor of Mathematics at the University of Colorado Denver. Since 2013, he is senior scientist at the Institute of Computer Science of the Czech Academy of Sciences. He has worked in the field of multigrid methods and domain decomposition methods. He developed the balancing domain decomposition method and, with coauthors, published the convergence proofs of the FETI, FETI-DP, and BDDC methods, and the proof of the equivalence of the FETI-DP and the BDDC methods. He has been involved in the field of dynamic data driven application systems and data assimilation with applications in wildfire and epidemic An epidemic (from Greek ἐπί ''epi'' "upon or above" and δῆμος ''demos'' "people") is the rapid spread of disease to a large number of patients among a given popula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Scalable
Scalability is the property of a system to handle a growing amount of work by adding resources to the system. In an economic context, a scalable business model implies that a company can increase sales given increased resources. For example, a package delivery system is scalable because more packages can be delivered by adding more delivery vehicles. However, if all packages had to first pass through a single warehouse for sorting, the system would not be as scalable, because one warehouse can handle only a limited number of packages. In computing, scalability is a characteristic of computers, networks, algorithms, networking protocols, programs and applications. An example is a search engine, which must support increasing numbers of users, and the number of topics it indexes. Webscale is a computer architectural approach that brings the capabilities of large-scale cloud computing companies into enterprise data centers. In mathematics, scalability mostly refers to closure u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Elliptic Partial Differential Equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. A PDE written in this form is elliptic if :B^2-AC, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^2. To transform our PDE into the desired canonical f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Markov Chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematical Economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Iterative Aggregation Method
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms. Mathematics In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algebraic Multigrid Method
In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners. The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a ''global'' correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem. The coarse problem, while cheaper to solve, is similar to the fine grid problem in that it also has short- and long-wavelength erro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |