Optimal design of optical analog solvers of linear systems

AbstractIn this paper, given a linear system of equations $$\mathbf {A}\, \mathbf {x}= \mathbf {b}$$ A x = b , we are finding locations in the plane to place objects such that sending waves from the source points and gathering them at the receiving points solves that linear system of equations. The ultimate goal is to have a fast physical method for solving linear systems. The issue discussed in this paper is to apply a fast and accurate algorithm to find the optimal locations of the scattering objects. We tackle this issue by using asymptotic expansions for the solution of the underlying partial differential equation. This also yields a potentially faster algorithm than the classical BEM for finding solutions to the Helmholtz equation.

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H-Matrices in Fuzzy Linear Systems

International Journal of Computational Mathematics ◽

10.1155/2014/394135 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 2

Author(s):

H. Saberi Najafi ◽

S. A. Edalatpanah

Keyword(s):

Linear System ◽

Theoretical Analysis ◽

Linear Systems ◽

Numerical Experiments ◽

Coefficient Matrix ◽

System Of Equations ◽

Linear System Of Equations ◽

Fuzzy Linear System ◽

Fuzzy Linear Systems

We consider a class of fuzzy linear system of equations and demonstrate some of the existing challenges. Furthermore, we explain the efficiency of this model when the coefficient matrix is an H-matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis.

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New iterative methods for dense linear systems

E3S Web of Conferences ◽

10.1051/e3sconf/202129302013 ◽

2021 ◽

Vol 293 ◽

pp. 02013

Author(s):

Jinmei Wang ◽

Lizi Yin ◽

Ke Wang

Keyword(s):

Linear System ◽

Linear Systems ◽

Iterative Methods ◽

Parallel Implementation ◽

System Of Equations ◽

Systems Of Equations ◽

Linear System Of Equations ◽

Dense Linear System ◽

Linear Systems Of Equations

Solving dense linear systems of equations is quite time consuming and requires an efficient parallel implementation on powerful supercomputers. Du, Zheng and Wang presented some new iterative methods for linear systems [Journal of Applied Analysis and Computation, 2011, 1(3): 351-360]. This paper shows that their methods are suitable for solving dense linear system of equations, compared with the classical Jacobi and Gauss-Seidel iterative methods.

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An Optimal Algorithm for the Solution of the Helmholtz Equation

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2020/v35i530286 ◽

2020 ◽

pp. 121-133

Author(s):

Richard O. Akinola ◽

Blessing Okwudo Ogbeh ◽

Isaac Chukle

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Helmholtz Equation ◽

Numerical Solution ◽

Optimal Algorithm ◽

Weather Prediction ◽

System Of Equations ◽

Partial Differential ◽

Linear System Of Equations ◽

Seidel Method

Aims/Objectives: The Helmholtz equation is a partial differential equation which is used in numerical weather prediction. Angwenyi et. al., used a five point finite difference stencil in discretizing the partial differential equation and solved the resulting square system of equations using eight iterative methods and concluded that the BICGSTAB was the most computationally efficient using just one example. However, based on a comparison of the norm of the residual and CPU time of four methods presented in this work on the same example in their paper and others; we not only discovered that the Gauss Seidel method out performed the BICGSTAB contradicting the claim of the authors but also the Thomas Block Tridiagonal Algorithm (TBTA)in the absence of round off errors.Methodology: We compared the performance of the Gauss Seidel Method, BICGSTAB, Matlab backslash, and the Thomas Block Tridiagonal Algorithm (TBTA) for the numerical solution of the Helmholtz equation with different step sizes. Results: We discovered that in the absence of round off errors, not only did the Gauss Seidel method but also the Thomas Block Tridiagonal Algorithm (TBTA) out performed the BICGSTAB contradicting the claim of Angwenyi et. al.Conclusion: We do not recommend the BICGSTAB for the solution of the linear system of equations arising from the discretization of the Helmholtz equation as claimed by Angwenyi et al. Rather, the Thomas Block Tridiagonal Algorithm should be used and if one is thinking of an iterative method for the numerical solution of the Helmholtz equation, the Gauss-Seidel method should be the method of choice rather than the BICGSTAB.

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Properweak regular splitting and its application to convergence of alternating iterations

Filomat ◽

10.2298/fil1819563m ◽

2018 ◽

Vol 32 (19) ◽

pp. 6563-6573 ◽

Cited By ~ 1

Author(s):

Debasisha Mishra

Keyword(s):

Linear System ◽

Convergence Rate ◽

Iterative Method ◽

Linear Systems ◽

Iterative Scheme ◽

Convergence Theory ◽

System Of Equations ◽

Comparison Result ◽

Linear System Of Equations ◽

Regular Splitting

Theory of matrix splittings is a useful tool for finding the solution of a rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit the theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld [Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which ensures the faster convergence rate of the proposed alternating iterative scheme.

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