TY - JOUR

T1 - On solving the Ill-conditioned system Ax = b

T2 - General-purpose conditioners obtained from the boundary-collocation solution of the laplace equation, using trefftz expansionswith multiple length scales

AU - Liu, Chein Shan

AU - Yeih, Weichung

AU - Atluri, Satya N.

PY - 2009

Y1 - 2009

N2 - Here we develop a general purpose pre/post conditioner T, to solve an ill-posed system of linear equations, Ax = b. The conditioner T is obtained in the course of the solution of the Laplace equation, through a boundary-collocation Trefftz method, leading to: Ty = x, where y is the vector of coefficients in the Trefftz expansion, and x is the boundary data at the discrete points on a unit circle. We show that the quality of the conditioner T is greatly enhanced by using multiple characteristic lengths (Multiple Length Scales) in the Trefftz expansion. We further show that T can be multiplicatively decomposed into a dilation TD and a rotation TR. For an odd-ordered A, we develop four conditioners based on the solution of the Laplace equation for Dirichlet boundary conditions, while for an even-ordered A we develop four conditioners employing the Neumann boundary conditions. All these conditioners are well-behaved and easily invertible. Several examples involving ill-conditioned A, 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, when Ax = b is highly ill-conditioned, and b is noisy.

AB - Here we develop a general purpose pre/post conditioner T, to solve an ill-posed system of linear equations, Ax = b. The conditioner T is obtained in the course of the solution of the Laplace equation, through a boundary-collocation Trefftz method, leading to: Ty = x, where y is the vector of coefficients in the Trefftz expansion, and x is the boundary data at the discrete points on a unit circle. We show that the quality of the conditioner T is greatly enhanced by using multiple characteristic lengths (Multiple Length Scales) in the Trefftz expansion. We further show that T can be multiplicatively decomposed into a dilation TD and a rotation TR. For an odd-ordered A, we develop four conditioners based on the solution of the Laplace equation for Dirichlet boundary conditions, while for an even-ordered A we develop four conditioners employing the Neumann boundary conditions. All these conditioners are well-behaved and easily invertible. Several examples involving ill-conditioned A, 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, when Ax = b is highly ill-conditioned, and b is noisy.

KW - Dilation matrix

KW - Ill-posed linear equations

KW - Multi-scale trefftz method (MSTM)

KW - Multi-scale trefftz-collocation laplacian conditioner (MSTCLC)

KW - Rotation matrix

KW - Transformation matrix

UR - http://www.scopus.com/inward/record.url?scp=76249124079&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:76249124079

VL - 44

SP - 281

EP - 311

JO - CMES - Computer Modeling in Engineering and Sciences

JF - CMES - Computer Modeling in Engineering and Sciences

SN - 1526-1492

IS - 3

ER -