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Domain Decomposition
In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG. In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abstra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ddm Original Logo
DDM may refer to: Computing * Data Diffusion Machine, a virtual shared memory computer architecture from the 1990s * Digital diagnostics monitoring function in SFP transceivers * Distributed Data Management Architecture, an open, published architecture for creating, managing and accessing data on a remote computer. * Dynamic Data Masking, a form of data masking * Dynamic Device Mapping, an advanced technology for USB KVM switches Science and technology * Derrick Drilling Machine, or Top drive * Difference in the Depth of Modulation, an amplitude modulation method used in the Instrument Landing System * Differential dynamic microscopy, an optical technique * Direct Digital Manufacturing * Doctor of Dental Medicine, an academic degree for dentistry * Domain decomposition methods * Drift Diffusion Model, a method used in psychological choice testing * Maltosides (n-Dodecyl β-D-Maltopyranoside), a detergent used when purifying membrane proteins * Dyson Digital Motor, a two-pole s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Alternating Method
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Schwarz may refer to: * Schwarz, Germany, a municipality in Mecklenburg-Vorpommern, Germany * Schwarz (surname), a surname (and list of people with the surname) * Schwarz (musician), American DJ and producer * ''Schwarz'' (Böhse Onkelz album), released simultaneously with ''Weiß'', 1993 * ''Schwarz'' (Conrad Schnitzler album), a reissue of the 1971 Kluster album ''Eruption'' * Schwarz (cards), in some card games, a Schneider (low point score) in which no tricks are taken * Schwarz Gruppe, a multinational retail group * Schwarz Pharma, a German drug company See also * * * Schwartz (other) * Schwarzhorn (other) * Swartz (other) Swartz may refer to: ;Places * Swartz, Louisiana *Swartz Creek (other) *Swartz Bay, British Columbia on the north end of the Saanich Peninsula on Vancouver Island ** Swartz Bay Ferry Terminal *Swartz Nunataks, in Antarctica ;People *Swart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mortar Method
In numerical analysis, mortar methods are discretization methods for partial differential equations, which use separate finite element discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution.Y. Maday, C. Mavriplis, and A. T. Patera, ''Nonconforming mortar element methods: application to spectral discretizations'', in Domain decomposition methods (Los Angeles, CA, 1988), SIAM, Philadelphia, PA, 1989, pp. 392--418. B. I. Wohlmuth, ''A mortar finite element method using dual spaces for the Lagrange multiplier'', SIAM J. Numer. Anal., 38 (2000), pp. 989--1012. Mortar discretizations lend themselves naturally to the solution by iterative domain decomposition methods such as FETI and balancing domain decomposition In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FETI-DP
The FETI-DP method is a domain decomposition methodC. Farhat, M. Lesoinne, P. LeTallec, K. Pierson, and D. Rixen, ''FETI-DP: a dual-primal unified FETI method. I. A faster alternative to the two-level FETI method'', Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1523--1544. that enforces equality of the solution at subdomain interfaces by Lagrange multipliers except at subdomain corners, which remain primal variables. The first mathematical analysis of the method was provided by Mandel and Tezaur.J. Mandel and R. Tezaur, ''On the convergence of a dual-primal substructuring method'', Numerische Mathematik, 88 (2001), pp. 543--558. The method was further improved by enforcing the equality of averages across the edges or faces on subdomain interfacesC. Farhat, M. Lesoinne, and K. Pierson, ''A scalable dual-primal domain decomposition method'', Numer. Linear Algebra Appl., 7 (2000), pp. 687--714. Preconditioning techniques for large sparse matrix problems in industrial applications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function. The method can be summarized as follows: in order to find the maximum or minimum of a function f(x) subjected to the equality constraint g(x) = 0, form the Lagrangian function :\mathcal(x, \lambda) = f(x) + \lambda g(x) and find the stationary points of \mathcal considered as a function of x and the Lagrange mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 subdomai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balancing Domain Decomposition
In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method.J. Mandel, ''Balancing domain decomposition'', Comm. Numer. Methods Engrg., 9 (1993), pp. 233–241. In each iteration, it combines the solution of local problems on non-overlapping subdomains with a coarse problem created from the subdomain nullspaces. BDD requires only solution of subdomain problems rather than access to the matrices of those problems, so it is applicable to situations where only the solution operators are available, such as in oil reservoir simulation by mixed finite elements.L. C. Cowsar, J. Mandel, and M. F. Wheeler, ''Balancing domain decomposition for mixed finite elements'', Math. Comp., 64 (1995), pp. 989–1015. In its original formulation, BDD performs well only for 2nd order problems, such elasticity in 2D and 3D. For 4th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Additive Schwarz Method
Abstract may refer to: * ''Abstract'' (album), 1962 album by Joe Harriott * Abstract of title a summary of the documents affecting title to parcel of land * Abstract (law), a summary of a legal document * Abstract (summary), in academic publishing * Abstract art, artistic works that do not attempt to represent reality or concrete subjects * '' Abstract: The Art of Design'', 2017 Netflix documentary series * Abstract music, music that is non-representational * Abstract object in philosophy * Abstract structure in mathematics * Abstract type in computer science * The property of an abstraction * Q-Tip (musician) Kamaal Ibn John Fareed (born Jonathan William Davis, April 10, 1970), better known by his stage name Q-Tip, is an American rapper, record producer, singer, and DJ. Nicknamed The Abstract, he is noted for his innovative jazz-influenced style of ..., also known as "The Abstract" * Abstract and concrete See also * Abstraction (other) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Schwarz Method
In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. Overview Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down. :(Model problem) The heat distribution in a square metal plate such that the left edge is kept at 1 degree, and the other edges are kept at 0 degree, after letting it sit for a long period of time satisfies the following boundary value problem: ::''f''''xx''(''x'',''y'') + ''f''''yy''(''x'',''y'') = 0 ::''f''(0,''y'') = 1; ''f''(''x'',0) = ''f''(''x'',1) = ''f''(1,''y'') = 0 :where ''f'' is the unknown function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LOBPCG
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem :A x= \lambda B x, for a given pair (A, B) of complex Hermitian or real symmetric matrices, where the matrix B is also assumed positive-definite. Background Kantorovich in 1948 proposed calculating the smallest eigenvalue \lambda_1 of a symmetric matrix A by steepest descent using a direction r = Ax-\lambda (x) x of a scaled gradient of a Rayleigh quotient \lambda(x) = (x, Ax)/(x, x) in a scalar product (x, y) = x'y, with the step size computed by minimizing the Rayleigh quotient in the linear span of the vectors x and w, i.e. in a locally optimal manner. Samokish proposed applying a preconditioner T to the residual vector r to generate the preconditioned direction w = T r and derived asymptotic, as x approaches the eigenvector, convergence rate bounds. D'yakonov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
