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A matrix decomposition MFS algorithm for certain linear elasticity problems

NUMERICAL algorithmS 2006年第2期43卷 123-149页

作者： Karageorghis, A. Smyrlis, Y. -S. Tsangaris, T. Univ Cyprus Dept Math & Stat CY-1678 Nicosia Cyprus

We propose an efficient matrix decomposition Method of Fundamental Solutions algorithm for the solution of certain two-dimensional linear elasticity problems. In particular, we consider the solution of the Cauchy-Navier equations in circular domains subject to Dirichlet boundary conditions, that is when the displacements are prescribed on the boundary. The proposed algorithm is extended to the case of annular domains. Numerical experiments for both types of problems are presented.

关键词： method of fundamental solutions Cauchy-Navier system matrix decomposition algorithm fast Fourier transform circulant matrices

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A matrix decomposition MFS algorithm for axisymmetric potential problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2004年第5期28卷 463-474页

作者： Smyrlis, YS Karageorghis, A Univ Cyprus Dept Math & Stat CY-1678 Nocosia Cyprus

The method of fundamental solutions is a boundary-type meshless method for the solution of certain elliptic boundary value problems. By exploiting the structure of the matrices appearing when this method is applied to certain three-dimensional potential problems, we develop an efficient matrix decomposition algorithm for their solution. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： method of fundamental solutions elliptic boundary value problems matrix decomposition algorithm

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matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics

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NUMERICAL algorithmS 2009年第1期52卷 1-23页

作者： Bialecki, Bernard Fairweather, Graeme Knudson, David B. Lipman, D. Abram Nguyen, Que N. Sun, Weiwei Weinberg, Gadalia M. DAPE PRP Army Mil Personnel Struct & Plans Div G1 Washington DC 20310 USA Avanade Inc Seattle WA 98121 USA City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China Colorado Sch Mines Dept Math & Comp Sci Golden CO 80401 USA

matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson's equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N (2)logN) operations on an NxN uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs.

关键词： Elliptic boundary value problems Finite element Galerkin method Piecewise Hermite cubics Generalized eigenvalue problem Eigenvalues and eigenfunctions matrix decomposition algorithm

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Multi-level method of fundamental solutions for solving polyharmonic problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 456卷

作者： Karageorghis, Andreas Chen, C. S. Univ Cyprus Dept Math & Stat POB 20537 CY-1678 Nicosia Cyprus Univ Southern Mississippi Sch Math & Nat Sci Hattiesburg MS 39406 USA

We consider a multi-level method of fundamental solutions for solving polyharmonic problems governed by 4N N u = 0 , N is an element of N\{1} in both two and three dimensions. Instead of approximating the solution with linear combinations of N fundamental solutions, we show that, with appropriate deployments of the source points, it is possible to employ an approximation involving only the fundamental solution of the operator 4N N . To determine the optimal position of the source points, we apply the recently developed effective condition number method. In addition, we show that when the proposed technique is applied to boundary value problems in circular or axisymmetric domains, with appropriate distributions of boundary and source points, it lends itself to the application of matrix decomposition algorithms. The results of several numerical tests are presented and analysed.

关键词： Polyharmonic equation Effective condition number matrix decomposition algorithm Method of fundamental solutions

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A Peculiar Approach Of Movie Rating Prediction Using Svr-algorithm Comparing To matrix decomposition With Accurate Rating 1

A Peculiar Approach Of Movie Rating Prediction Using Svr-Alg...

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1st International Conference on Technologies for Smart Green Connected Society 2021, ICTSGS 2021

作者： Ganesh, R. Velu, C.M. SaveethaUniversity Tamil Nadu Chennai602105 India

ISBN: (纸本)9781607685395

The framework has been made for a Movie Rating System to enable us to enjoy our time fruitfully. Film recommendation systems enable users to sort and select the best film fans to see without going through thousands of films which were stored in archives. A total of 392 samples were collected from movie datasets available in kaggle. For this two algorithms were used, one matrix decomposition algorithm and another Support Vector Regression(SVR) algorithm, both the algorithms were executed and compared for accuracy. SVR achieved accuracy, precision, sensitivity and specificity of 87%, 88%, 89% and 93% respectively compared to 80%, 83%, 84% and 86% by the matrix decomposition algorithm. The results were obtained with a level of significance with (p © The Electrochemical Society

关键词： Artificial Intelligence Machine Learning matrix decomposition algorithm Peculiar recommendation Popularity SVR algorithm

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A Fast Numerical algorithm for Calculating Electromagnetic Scattering from an Object above a Rough Surface

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ELECTROMAGNETICS 2013年第1期33卷 10-22页

作者： An, Yuyuan Wang, Daoxiang Chen, Ru-Shan Nanjing Univ Sci & Technol Dept Commun Engn Nanjing 210094 Jiangsu Peoples R China

A fast numerical algorithm is proposed for calculating the difference field radar cross-section of an object above a rough surface. The electric-field integral equation of the difference of the induced current on the rough surface and the induced current on the object is derived, which is solved by an iterative solver. The characteristic basis functions are used to calculate the induced current on the rough surface, which is part of the right-hand side of the system. It is observed that the coupling matrices (nonself-block interactions of the rough surface and the interactions between the object and the rough surface) are rank deficient. The matrix decomposition algorithm is used and extended to compress the coupling matrices. Numerical examples show that the characteristic basis functions method combined with the matrix decomposition algorithm can greatly reduce the CPU time and memory cost. The accuracy and efficiency are discussed by numerical examples.

关键词： characteristic basis functions difference scattering matrix decomposition algorithm rough surface

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The plane waves method for axisymmetric Helmholtz problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2016年 69卷 46-56页

作者： Karageorghis, Andreas Univ Cyprus Dept Math & Stat POB 20537 CY-1678 Nicosia Cyprus

The plane waves method is employed for the solution of Dirichlet and Neumann boundary value problems for the homogeneous Helmholtz equation in two- and three-dimensional domains possessing radial symmetry. The appropriate selection of collocation points and unitary direction vectors in the method leads to circulant and block circulant coefficient matrices in two and three dimensions, respectively. We propose efficient matrix decomposition algorithms which make use of fast Fourier transforms for the solution of the systems resulting from such a discretization. In conjunction with the method of particular solutions, the method is extended to the solution of inhomogeneous axisymmetric Helmholtz problems. Several numerical examples are presented. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Plane waves method Collocation matrix decomposition algorithm Fast Fourier transforms Helmholtz equation

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A preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2003年第2期41卷 589-604页

作者： Aitbayev, R Bialecki, B New Mexico Inst Min & Technol Dept Math Socorro NM 87801 USA Colorado Sch Mines Dept Math & Comp Sci Golden CO 80401 USA

We study the computation of the orthogonal spline collocation solution of a linear Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form Lu = Sigmaa(ij) (x) u(xixj) + Sigma b(i)(x) u(xi) + c(x) u. We apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator, and prove that the resulting algorithm has a convergence rate independent of the partition step size. We solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm. On a uniform N x N partition, the cost of the algorithm for computing the collocation solution within a tolerance epsilon is O(N-2 lnN| ln epsilon|).

关键词： nonselfadjoint or indefinite elliptic boundary value problem orthogonal spline collocation conjugate gradient method preconditioner matrix decomposition algorithm

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Finite Difference Schemes for the Cauchy-Navier Equations of Elasticity with Variable Coefficients

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JOURNAL OF SCIENTIFIC COMPUTING 2015年第1期62卷 78-121页

作者： Bialecki, Bernard Karageorghis, Andreas Colorado Sch Mines Dept Appl Math & Stat Golden CO 80401 USA Univ Cyprus Dept Math & Stat CY-1678 Nicosia Cyprus

We solve the variable coefficient Cauchy-Navier equations of elasticity in the unit square, for Dirichlet and Dirichlet-Neumann boundary conditions, using second order finite difference schemes. The resulting linear systems are solved by the preconditioned conjugate gradient (PCG) method with preconditioners corresponding to to the Laplace operator. The multiplication of a vector by the matrices of the resulting systems and the solution of systems with the preconditioners are performed at optimal and nearly optimal costs, respectively. For the case of Dirichlet boundary conditions, we prove the second order accuracy of the scheme in the discrete norm, symmetry of the resulting matrix and its spectral equivalence to the preconditioner. For the case of Dirichlet-Neumann boundary conditions, we prove symmetry of the resulting matrix. Numerical tests demonstrating the convergence properties of the schemes and PCG are presented.

关键词： Cauchy-Navier equations Finite difference scheme matrix decomposition algorithm Preconditioned conjugate gradient method Fast Fourier transforms

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A fourth-order orthogonal spline collocation method for two-dimensional Helmholtz problems with interfaces

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2020年第6期36卷 1811-1829页

作者： Bhal, Santosh Kumar Danumjaya, Palla Fairweather, Graeme BITS Goa Campus Dept Math Pilani 403726 Goa India Amer Math Soc Math Reviews Ann Arbor MI USA

Orthogonal spline collocation is implemented for the numerical solution of two-dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost ofO(N-2 log N)on anN x Npartition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting optimal global estimates in various norms and superconvergence phenomena for a broad spectrum of wave numbers.

关键词： almost block diagonal linear systems discontinuous data fast Fourier transforms Helmholtz problems matrix decomposition algorithm optimal global convergence rates orthogonal spline collocation superconvergence

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