Spectral Methods for Parameterized Matrix Equations

We apply polynomial approximation methods—known in the numerical PDEs context as spectral methods—to approximate the vector-valued function that satisfies a linear system of equations where the matrix and the right-hand side depend on a parameter. We derive both an interpolatory pseudospectral method and a residual-minimizing Galerkin method, and we show how each can be interpreted as solving a truncated infinite system of equations; the difference between the two methods lies in where the truncation occurs. Using classical theory, we derive asymptotic error estimates related to the region of analyticity of the solution, and we present a practical residual error estimate. We verify the results with two numerical examples.

[1]  Catherine Elizabeth Powell,et al.  Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data , 2008, SIAM J. Sci. Comput..

[2]  Rajeev Motwani,et al.  The PageRank Citation Ranking : Bringing Order to the Web , 1999, WWW 1999.

[3]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[4]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[5]  G. Golub,et al.  Parameter Estimation in the Presence of Bounded Data Uncertainties , 1998, SIAM J. Matrix Anal. Appl..

[6]  W. Gautschi,et al.  THE INTERPLAY BETWEEN CLASSICAL ANALYSIS AND ( NUMERICAL ) LINEAR ALGEBRA — A TRIBUTE TO GENE , 2002 .

[7]  G. Golub,et al.  Matrices, Moments and Quadrature with Applications , 2009 .

[8]  James G. Nagy,et al.  Nonlinear least squares and super resolution , 2008 .

[9]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[10]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[11]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[12]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[13]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[14]  Zhaojun Bai,et al.  A two-directional Arnoldi process and its application to parametric model order reduction , 2009 .

[15]  Claude Brezinski,et al.  The PageRank Vector: Properties, Computation, Approximation, and Acceleration , 2006, SIAM J. Matrix Anal. Appl..

[16]  Gene H. Golub,et al.  Matrices, moments, and quadrature , 2007, Milestones in Matrix Computation.

[17]  Zhaojun Bai,et al.  The Lanczos Method for Parameterized Symmetric Linear Systems with Multiple Right-Hand Sides , 2010, SIAM J. Matrix Anal. Appl..

[18]  Howard C. Elman,et al.  Block-diagonal preconditioning for spectral stochastic finite-element systems , 2008 .

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

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

[21]  Walter Gautschi,et al.  THE INTERPLAY BETWEEN CLASSICAL ANALYSIS AND (NUMERICAL) LINEAR ALGEBRA — A TRIBUTE TO GENE H. GOLUB , 2002 .

[22]  T. J. Rivlin.  An Introduction to the Approximation of Functions , 2003 .

[23]  Luca Dieci,et al.  Lyapunov Exponents of Systems Evolving on Quadratic Groups , 2003, SIAM J. Matrix Anal. Appl..

[24]  Qiqi Wang,et al.  A Rational Interpolation Scheme with Superpolynomial Rate of Convergence , 2010, SIAM J. Numer. Anal..

[25]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[26]  David F. Gleich,et al.  Using Polynomial Chaos to Compute the Influence of Multiple Random Surfers in the PageRank Model , 2007, WAW.

[27]  Jan S. Hesthaven,et al.  Spectral Methods for Time-Dependent Problems: Contents , 2007 .

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

[29]  D. O’Leary,et al.  Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering , 2005 .

[30]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[31]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[32]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

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

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

[35]  R. Webster,et al.  Kriging: a method of interpolation for geographical information systems , 1990, Int. J. Geogr. Inf. Sci..