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On Solving the Ill-Conditioned System Ax=b: General-Purpose Conditioners Obtained From the Boundary-Collocation Solution of the Laplace Equation, Using Trefftz Expansions With Multiple Length Scales

Chein-Shan Liu1, Weichung Yeih2, Satya N. Atluri3
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail:liuchei.nshan@msa.hinet.net
Department of Harbor & River Engineering and Computation and Simulation Center, National Taiwan Ocean University, Keelung, Taiwan
Center for Aerospace Research & Education, University of California, Irvine

Computer Modeling in Engineering & Sciences 2009, 44(3), 281-312. https://doi.org/10.3970/cmes.2009.044.281

Abstract

Here we develop a general purpose pre/post conditionerT, to solve an ill-posed system of linear equations,Ax=b. The conditionerTis obtained in the course of the solution of the Laplace equation, through a boundary-collocation Trefftz method, leading to:Ty=x, whereyis the vector of coefficients in the Trefftz expansion, andxis the boundary data at the discrete points on a unit circle. We show that the quality of the conditionerTis greatly enhanced by using multiple characteristic lengths (Multiple Length Scales) in the Trefftz expansion. We further show thatTcan be multiplicatively decomposed into a dilationTDand a rotationTR. For an odd-orderedA, we develop four conditioners based on the solution of the Laplace equation for Dirichlet boundary conditions, while for an even-orderedAwe develop four conditioners employing the Neumann boundary conditions. All these conditioners are well-behaved and easily invertible. Several examples involving ill-conditionedA, such as the Hilbert matrices, those arising from the Method of Fundamental Solutions, those arising from very-high order polynomial interpolations, and those resulting from the solution of the first-kind Fredholm integral equations, are presented. The results demonstrate that the presently proposed conditioners result in very high computational efficiency and accuracy, whenAx=bis highly ill-conditioned, andbis noisy.

Keywords

Ill-posed linear equations, Multi-Scale Trefftz Method (MSTM), Multi-Scale Trefftz-Collocation Laplacian Conditioner (MSTCLC), Transformation matrix, Dilation matrix, Rotation matrix

Cite This Article

Liu, C., Yeih, W., Atluri, S. N. (2009). On Solving the Ill-Conditioned System Ax=b: General-Purpose Conditioners Obtained From the Boundary-Collocation Solution of the Laplace Equation, Using Trefftz Expansions With Multiple Length Scales. CMES-Computer Modeling in Engineering & Sciences, 44(3), 281–312.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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