A Fast Collocation Method for Eigen-Problems of Weakly Singular Integral Operators

AbstractA multiscale collocation method is developed for solving the eigen-problem of weakly singular integral operators. We employ a matrix truncation strategy of Chen, Micchelli and Xu to compress the collocation matrix, which the compressed matrix has only $\mathcal{O}(N\log N)$ nonzero entries, where N denotes the order of the matrix. This truncation leads to a fast collocation method for solving the eigen-problem. We prove that the fast collocation method has the optimal convergence order for approximation of the eigenvalues and eigenvectors. The power iteration method is used for solving the corresponding discrete eigen-problem. We present a numerical example to demonstrate how the methods can be used to compute a nonzero eigenvalue rapidly and efficiently.

[1]  Wolfgang Dahmen,et al.  Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution , 1993, Adv. Comput. Math..

[2]  J. Bramble,et al.  Rate of convergence estimates for nonselfadjoint eigenvalue approximations , 1973 .

[3]  Charles A. Micchelli,et al.  A Construction of Refinable Sets for Interpolating Wavelets , 1998 .

[4]  Maher Mnif,et al.  Perturbation theory of lower semi-Browder multivalued linear operators , 2011 .

[5]  Ian H. Sloan,et al.  Piecewise Continuous Collocation for Integral Equations , 1983 .

[6]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[7]  Arvet Pedas,et al.  Quadratic spline collocation method for weakly singular integral equations and corresponding eigenvalue problem 1 , 2002 .

[8]  Yuesheng Xu,et al.  Fast Collocation Methods for Second Kind Integral Equations , 2002, SIAM J. Numer. Anal..

[9]  J. Osborn.  Spectral approximation for compact operators , 1975 .

[10]  I. Babuska,et al.  Estimates for the errors in Eigenvalue and Eigenvector approximation by Galerkinmethods, with particular attention to the case of multiple Eigenvalues , 1987 .

[11]  Yuesheng Xu,et al.  Reconstruction and Decomposition Algorithms for Biorthogonal Multiwavelets , 1997, Multidimens. Syst. Signal Process..

[12]  K. Atkinson.  THE NUMERICAL SOLUTION OF THE EIGENVALUE PROBLEM FOR COMPACT INTEGRAL OPERATORS , 2008 .

[13]  Weifu Fang,et al.  A fast collocation method for an inverse boundary value problem , 2004 .

[14]  Charles A. Micchelli,et al.  A Multilevel Method for Solving Operator Equations , 2001 .

[15]  Yuesheng Xu,et al.  The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes , 1997, Adv. Comput. Math..

[16]  Bin Wu,et al.  Fast Collocation Methods for High-Dimensional Weakly Singular Integral Equations , 2008 .

[17]  F. Chatelin.  Spectral approximation of linear operators , 2011 .

[18]  Alain Largillier,et al.  Spectral Computations for Bounded Operators , 2001 .

[19]  Balmohan V. Limaye.  Spectral Approximation for Compact Integral Operators , 2004 .

[20]  Tosio Kato,et al.  Perturbation theory for nullity, deficiency and other quantities of linear operators , 1958 .

[21]  Charles A. Micchelli,et al.  Wavelet Galerkin methods for second-kind integral equations , 1997 .

[22]  I. Babuska,et al.  Finite element-galerkin approximation of the eigenvalues and Eigenvectors of selfadjoint problems , 1989 .

[23]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[24]  L. Hacia.  Approximate Solutions of a Certain Class of Operator Equations , 1989 .

[25]  C. Micchelli,et al.  Using the Matrix Refinement Equation for the Construction of Wavelets on Invariant Sets , 1994 .

[26]  K. Atkinson.  Convergence Rates for Approximate Eigenvalues of Compact Integral Operators , 1975 .

[27]  Yuesheng Xu,et al.  A construction of interpolating wavelets on invariant sets , 1999, Math. Comput..

[28]  N. Gnaneshwar,et al.  A degenerate kernel method for eigenvalue problems of compact integral operators , 2007, Adv. Comput. Math..

[29]  Yuesheng Xu,et al.  Discrete Wavelet Petrov–Galerkin Methods , 2002, Adv. Comput. Math..

[30]  M. Thamban Nair.  On strongly stable approximations , 1992 .

[31]  Guanrong Chen,et al.  Approximate Solutions of Operator Equations , 1997 .

[32]  Ian H. Sloan.  Iterated Galerkin Method for Eigenvalue Problems , 1976 .

[33]  Yi Wang,et al.  A Fast Wavelet Collocation Method for Integral Equations on Polygons , 2005 .