Tensor Rank bounds for Point Singularities in $\mathbb{R}^3$

Abstract

A. We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in R 3 . We consider functions in countably normed Sobolev spaces with radial weights and analytic-or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces.It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ε ∈ (0, 1) in the Sobolev space H 1 . We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker-QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hpfinite element approximations for Gevrey-regular functions with point singularities in the unit cube Q = (0, 1) 3 . Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results. C 1. Introduction 1.1. Tensor Structured Function Approximation 1.2. Problem Formulation 1.3. Contributions 1.4. Structure of this paper 2. Tensor structured representations 2.1. Notation 2.2. Tensor Train (TT) Format 2.3. Rank bound analysis of TT representations 2.4. Quantized Tensor Train (QTT) format in one physical dimension 2.5. Classic QTT format in three physical space dimensions 2.6. Transposed order QTT format in three physical space dimensions 2.7. Tucker-QTT 2.8. Degrees of freedom 3. Functional setting 3.1. Low order virtual FE space X 3.2. Auxiliary hp space 3.3. Quasi interpolation operator P 4. Quasi interpolation error 5. QTT formatted approximation of u ∈ J γ (Q) 5.1. Tensor Rank Bounds for QTT Approximation 5.2. Rank bounds for transposed order QTT representations 5.

Prospects of tensor-based numerical modeling of the collective electrostatic potential in many-particle systems

“…The quantized tensor train (QTT) decomposition introduced in [43,63] allows to further reduce the representation complexity for multidimensional tensors to logarithmic scale in the volume size, O(d log n). QTT tensor decompositions are nowadays widely used in solution of the multidimensional problems in numerical analysis, see for example, [65,17,37,38,53,11], and [49,19,66,58,1,54,58], as well as the recent books on tensor numerical methods in computational quantum chemistry and in scientific computing [41,44] and references therein.…”

Ubiquitous Nature of the Reduced Higher Order SVD in Tensor-Based Scientific Computing

“…Tensor numerical methods [1,2] provide means to overcome the problem of the exponential increase of numerical complexity in the dimension of the problem d, due to their intrinsic feature of reducing the computational costs of multi-linear algebra on rank-structured data to merely linear scaling in both the gridsize n and dimension d. They appeared as bridging of the algebraic tensor decompositions initiated in chemometrics [3][4][5][6][7][8][9][10] and of the nonlinear approximation theory on separable low-rank representation of multi-variate functions and operators [11][12][13]. The canonical [14,15], Tucker [16], tensor train (TT) [17,18], and hierarchical Tucker (HT) [19] formats are the most commonly used rank-structured parametrizations in applications of modern tensor numerical methods.…”