8 research outputs found
Approximate Real Symmetric Tensor Rank
We investigate the effect of an -room of perturbation tolerance
on symmetric tensor decomposition. To be more precise, suppose a real symmetric
-tensor , a norm on the space of symmetric -tensors, and
are given. What is the smallest symmetric tensor rank in the
-neighborhood of ? In other words, what is the symmetric tensor
rank of after a clever -perturbation? We prove two theorems
and develop three corresponding algorithms that give constructive upper bounds
for this question. With expository goals in mind; we present probabilistic and
convex geometric ideas behind our results, reproduce some known results, and
point out open problems.Comment: Fixed few typos and error in writing of Algorithm 1. To appear in
Arnold Mathematical Journa
Quantifying low rank approximations of third order symmetric tensors
In this paper, we present a method to certify the approximation quality of a
low rank tensor to a given third order symmetric tensor. Under mild
assumptions, best low rank approximation is attained if a control parameter is
zero or quantified quasi-optimal low rank approximation is obtained if the
control parameter is positive.This is based on a primal-dual method for
computing a low rank approximation for a given tensor. The certification is
derived from the global optimality of the primal and dual problems, and is
characterized by easily checkable relations between the primal and the dual
solutions together with another rank condition. The theory is verified
theoretically for orthogonally decomposable tensors as well as numerically
through examples in the general case.Comment: 46page
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described