Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients

Abstract We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen–Loeve expansion of a stochastic PDE posed in a one-dimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tensor-structured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the tensor train and quantized tensor train formats. We suggest an efficient tensor-structured preconditioning of the entire multiparametric family of one-dimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensor-structured preconditioned GMRES solver in a series of numerical experiments.

[1]  Hermann G. Matthies,et al.  Solving stochastic systems with low-rank tensor compression , 2012 .

[2]  VLADIMIR A. KAZEEV,et al.  Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse , 2012, SIAM J. Matrix Anal. Appl..

[3]  R. DeVore,et al.  Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs , 2010 .

[4]  Boris N. Khoromskij,et al.  Superfast Fourier Transform Using QTT Approximation , 2012 .

[5]  Dmitry V. Savostyanov.  QTT-rank-one vectors with QTT-rank-one and full-rank Fourier images , 2012 .

[6]  S. V. Dolgov,et al.  ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ , 2014 .

[7]  B. Khoromskij.  Tensor numerical methods for multidimensional PDES: theoretical analysis and initial applications , 2015 .

[8]  Hermann G. Matthies,et al.  Efficient Analysis of High Dimensional Data in Tensor Formats , 2012 .

[9]  I. Oseledets.  Constructive Representation of Functions in Low-Rank Tensor Formats , 2012, Constructive Approximation.

[10]  Boris N. Khoromskij,et al.  Fast and accurate 3D tensor calculation of the Fock operator in a general basis , 2012, Comput. Phys. Commun..

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

[12]  Albert Cohen,et al.  Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , 2010, Found. Comput. Math..

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

[14]  B. Khoromskij.  Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances , 2012 .

[15]  B. Khoromskij,et al.  DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator , 2010 .

[16]  B. Khoromskij.  Tensor Numerical Methods for High-dimensional PDEs: Basic Theory and Initial Applications , 2014, 1408.4053.

[17]  Ivan V. Oseledets,et al.  Solution of Linear Systems and Matrix Inversion in the TT-Format , 2012, SIAM J. Sci. Comput..

[18]  Valeria Simoncini,et al.  Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing , 2003, SIAM J. Sci. Comput..

[19]  I. V. Oseledets,et al.  Tensors inside of matrices give logarithmic complexity , 2009 .

[20]  Ivan V. Oseledets,et al.  Approximation of 2d˟2d Matrices Using Tensor Decomposition , 2010, SIAM J. Matrix Anal. Appl..

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

[22]  Elisabeth Ullmann,et al.  Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients , 2012, Numerische Mathematik.

[23]  Claude Jeffrey Gittelson,et al.  Adaptive stochastic Galerkin FEM , 2013 .

[24]  S. Dolgov.  TT-GMRES: solution to a linear system in the structured tensor format , 2012, 1206.5512.

[25]  C. Schwab,et al.  Sparse high order FEM for elliptic sPDEs , 2009 .

[26]  Peter Benner,et al.  Low-Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients , 2015, SIAM/ASA J. Uncertain. Quantification.

[27]  Christoph Schwab,et al.  Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients , 2007 .

[28]  M. Bartlett.  An Inverse Matrix Adjustment Arising in Discriminant Analysis , 1951 .

[29]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2010, SIAM Rev..

[30]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[31]  Boris N. Khoromskij,et al.  Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝd , 2009 .

[32]  B. Khoromskij,et al.  Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs , 2010 .

[33]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[34]  Frances Y. Kuo,et al.  Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications , 2011, J. Comput. Phys..

[35]  Boris N. Khoromskij,et al.  Numerical Solution of the Hartree - Fock Equation in Multilevel Tensor-Structured Format , 2011, SIAM J. Sci. Comput..

[36]  Daniel Kressner,et al.  A literature survey of low‐rank tensor approximation techniques , 2013, 1302.7121.

[37]  Lars Grasedyck,et al.  Hierarchical Tensor Approximation of Output Quantities of Parameter-Dependent PDEs , 2015, SIAM/ASA J. Uncertain. Quantification.

[38]  Vladimir A. Kazeev,et al.  Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity , 2013, SIAM J. Sci. Comput..

[39]  Boris N. Khoromskij,et al.  Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation , 2014, Comput. Phys. Commun..

[40]  R. DeVore,et al.  ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S , 2011 .

[41]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[42]  Ivan V. Oseledets,et al.  DMRG Approach to Fast Linear Algebra in the TT-Format , 2011, Comput. Methods Appl. Math..

[43]  Reinhold Schneider,et al.  Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis , 2013, SIAM J. Sci. Comput..

[44]  Lei Tang,et al.  Efficiency Based Adaptive Local Refinement for First-Order System Least-Squares Formulations , 2011, SIAM J. Sci. Comput..

[45]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[46]  Daniel Kressner,et al.  Reduced Basis Methods: From Low-Rank Matrices to Low-Rank Tensors , 2016, SIAM J. Sci. Comput..

[47]  Е Е Тыртышников,et al.  A reciprocal preconditioner for structured matrices arising from elliptic problems with jumping coefficients , 2012 .

[48]  Daniel Kressner,et al.  Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems , 2011, SIAM J. Matrix Anal. Appl..

[49]  Ivan Oseledets,et al.  Tensor-Train Decomposition , 2011, SIAM J. Sci. Comput..

[50]  Boris N. Khoromskij,et al.  Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs , 2010, Comput. Methods Appl. Math..

[51]  B. Khoromskij.  O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling , 2011 .

[52]  J. Sherman,et al.  Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix , 1950 .