Nonlinear approximation theory
Nonlinear approximation theory
A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
Algorithms for Numerical Analysis in High Dimensions
SIAM Journal on Scientific Computing
Karhunen-Loève approximation of random fields by generalized fast multipole methods
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
On tensor approximation of Green iterations for Kohn-Sham equations
Computing and Visualization in Science
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time
SIAM Journal on Matrix Analysis and Applications
Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays
SIAM Journal on Scientific Computing
Tensor Decompositions and Applications
SIAM Review
Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
SIAM Journal on Scientific Computing
Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs
Foundations of Computational Mathematics
Hierarchical Singular Value Decomposition of Tensors
SIAM Journal on Matrix Analysis and Applications
Sparse Tensor Discretization of Elliptic sPDEs
SIAM Journal on Scientific Computing
Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-Structured Format
SIAM Journal on Scientific Computing
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems
SIAM Journal on Matrix Analysis and Applications
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats
Computers & Mathematics with Applications
Approximation rates for the hierarchical tensor format in periodic Sobolev spaces
Journal of Complexity
Segmentation of Stochastic Images using Level Set Propagation with Uncertain Speed
Journal of Mathematical Imaging and Vision
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We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multiparametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the $M$-term truncated Karhunen-Loève expansion. Our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high-dimensional, parametric, elliptic PDEs providing linear scaling complexity in the dimension $M$ of the parameter space. It is based on rank-reduced, tensor-formatted separable approximations of the high-dimensional tensors and matrices involved in the iterative process, combined with the use of spectrally equivalent low-rank tensor-structured preconditioners to the parametric matrices resulting from a finite element discretization of the high-dimensional parametric, deterministic problems. Numerical illustrations for the $M$-dimensional parametric elliptic PDEs resulting from sPDEs on parameter spaces of dimensions $M\leq100$ indicate the advantages of employing low-rank tensor-structured matrix formats in the numerical solution of such problems.