On sparse tensor structures in lattice theory and applications of the polynomial collocation method based on sparse grids

Vesa Kaarnioja

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

Due to the exponential increase in computational power ever since the invention of the computer, the use of tensors has become a more viable way to approach problems involving many variables. However, the efficient treatment of high-dimensional problems still requires special techniques such as tensor decompositions and utilizing sparsity. The first part of this dissertation considers the properties of symmetric meet and join tensors arising in lattice theory, which can be understood as generalizations of meet and join matrices such as classically studied GCD and LCM matrices, respectively. New low-parametric tensor decompositions are developed for general classes of lattice-theoretic tensors in both polyadic and tensor-train formats. The compressed representations endowed by these decompositions enable numerical computations involving high dimensionality and order, and the efficient application of tensor eigenvalue solution algorithms is studied for tensors belonging to these classes. The second part of this dissertation considers the application of sparse grid collocation algorithms for the solution of parameter-dependent partial differential equations involving high dimensionality. We consider as applications a class of stochastic eigenvalue problems and a parameter-dependent complete electrode model of electrical impedance tomography. A novel basis selection technique based on the maximum volume principle is introduced for multivariate polynomial interpolation over arbitrary node configurations.
Translated title of the contributionHarvat hilateoreettiset tensorirakenteet ja harvoja hiloja hyödyntävän polynomisen kollokaatiomenetelmän sovelluksia
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Ilmonen, Pauliina, Supervising Professor
  • Hakula, Harri, Thesis Advisor
Publisher
Print ISBNs978-952-60-7520-4
Electronic ISBNs978-952-60-7519-8
Publication statusPublished - 2017
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • tensor-train
  • tensor eigenvalue
  • semilattice
  • sparse grid
  • polynomial collocation
  • multivariate interpolation

Fingerprint Dive into the research topics of 'On sparse tensor structures in lattice theory and applications of the polynomial collocation method based on sparse grids'. Together they form a unique fingerprint.

Cite this