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 contribution||Harvat hilateoreettiset tensorirakenteet ja harvoja hiloja hyödyntävän polynomisen kollokaatiomenetelmän sovelluksia|
|Publication status||Published - 2017|
|MoE publication type||G5 Doctoral dissertation (article)|
- tensor eigenvalue
- sparse grid
- polynomial collocation
- multivariate interpolation