Low-rank approximate inverse for preconditioning tensor-structured linear systems

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator. It provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators. These subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank-one corrections. In order to handle high-order tensors, approximate projections are computed in low-rank Hierarchical Tucker subsets of the successive subspaces of operators. Some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step, where an optimal rank-one correction is searched as the tensor product of operators with the desired properties. Numerical examples illustrate the ability of this algorithm to provide efficient preconditioners for linear systems in tensor format that improve the convergence of iterative solvers and also the quality of the resulting low-rank approximations of the solution

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors

A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with residual norm corresponding to the error in a specified solution norm. We introduce and analyze an iterative algorithm that is able to provide a controlled approximation of the optimal approximation of the solution in a given low-rank subset, without any a priori information on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy corrections in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The residual norm can be designed such that the resulting low-rank approximations are quasi-optimal with respect to particular norms of interest, thus yielding to goal-oriented order reduction strategies for the approximation of high-dimensional problems. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces

Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in L2L_2

Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such approximations, not with respect to a fixed finite dimensional discrete background problem, but with respect to the exact solution of the continuous problem. While the energy norm offers a natural error measure corresponding to the underlying operator considered as an isomorphism from the energy space onto its dual, this norm requires a careful treatment in its interplay with the tensor structure of the problem. In this paper we build on our previous work on energy norm-convergent subspace-based tensor schemes contriving, however, a modified formulation which now enforces convergence only in $L_2$. In order to still be able to exploit the mapping properties of elliptic operators, a crucial ingredient of our approach is the development and analysis of a suitable asymmetric preconditioning scheme. We provide estimates for the computational complexity of the resulting method in terms of the solution error and study the practical performance of the scheme in numerical experiments. In both regards, we find that controlling solution errors in this weaker norm leads to substantial simplifications and to a reduction of the actual numerical work required for a certain error tolerance.Comment: 26 pages, 7 figure

A continuous analogue of the tensor-train decomposition

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities