A multilevel finite element method for Fredholm integral eigenvalue problems

In this work, we proposed a multigrid finite element (MFE) method for solving the Fredholm integral eigenvalue problems. The main motivation for such studies is to compute the Karhunen-Loeve expansions of random fields, which play an important role in the applications of uncertainty quantification. In our MFE framework, solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh. Then, any existing efficient integration scheme can be used for the associated integration process. The error estimates are provided, and the computational complexity is analyzed. It is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh. Several numerical experiments are presented to validate the efficiency of the proposed numerical method.

[1]  Mark Girolami,et al.  Orthogonal Series Density Estimation and the Kernel Eigenvalue Problem , 2002, Neural Computation.

[2]  Tao Tang,et al.  On Discrete Least-Squares Projection in Unbounded Domain with Random Evaluations and its Application to Parametric Uncertainty Quantification , 2014, SIAM J. Sci. Comput..

[3]  Kedar Khare,et al.  Sampling theorem, bandlimited integral kernels and inverse problems , 2007 .

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

[5]  R. Tempone,et al.  ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS , 2012 .

[6]  Tao Zhou,et al.  Multivariate Discrete Least-Squares Approximations with a New Type of Collocation Grid , 2014, SIAM J. Sci. Comput..

[7]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[8]  Arieh Iserles,et al.  The spectral problem for a class of highly oscillatory Fredholm integral operators , 2010 .

[9]  K. Phoon,et al.  Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme , 2002 .

[10]  Radu Alexandru Todor,et al.  Robust Eigenvalue Computation for Smoothing Operators , 2006, SIAM J. Numer. Anal..

[11]  Jinchao Xu,et al.  A two-grid discretization scheme for eigenvalue problems , 2001, Math. Comput..

[12]  F. H. Middleton,et al.  Surface‐Wave Propagation in a Continuously Stratified Medium , 1966 .

[13]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[14]  N. Wiener.  The Homogeneous Chaos , 1938 .

[15]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

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

[17]  Ernst Rank,et al.  Finite cell method , 2007 .

[18]  Tao Tang,et al.  Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials , 2012, J. Sci. Comput..

[19]  S. P. Oliveira,et al.  Spectral element approximation of Fredholm integral eigenvalue problems , 2014, J. Comput. Appl. Math..

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

[21]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[22]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[23]  Christoph Schwab,et al.  Karhunen-Loève approximation of random fields by generalized fast multipole methods , 2006, J. Comput. Phys..

[24]  Philippe G. Ciarlet,et al.  A NEW DUALITY APPROACH TO ELASTICITY , 2012 .

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

[26]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[27]  Herbert B. Keller.  On the accuracy of finite difference approximations to the eigenvalues of differential and integral operators , 1965 .

[28]  Hehu Xie,et al.  A type of multilevel method for the Steklov eigenvalue problem , 2014 .

[29]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[30]  Hehu Xie,et al.  A multi-level correction scheme for eigenvalue problems , 2011, Math. Comput..

[31]  Tao Tang,et al.  Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed , 2010 .

[32]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[33]  Hehu Xie,et al.  A full multigrid method for eigenvalue problems , 2014, J. Comput. Phys..