Tensor methods for the numerical solution of high-dimensional parametric partial differential equations
Year: 2018
Type: dissertation
Abstract: This thesis deals with tensor methods for the numerical solution of parametric partial differential equations (PDEs). Because of their parametric dependence, these differential equations exhibit a high dimensionality. Numerical methods like the Galerkin method quickly reach their limits, since the resulting linear system grows exponentially with the number of parameters. This is known as the curse of dimensionality. The work can be roughly divided into three parts. The first part covers the theory of parametric PDEs using the leading example of a diffusion equation with parametric coefficients. We consider both a so called affine coefficient as well as a log-normal one. The necessary theoretical foundations for the numerical treatment of these problems are laid out. The second part is a review of the state of the art of tensor methods. These have gained a lot of recognition in recent years. We introduce a number of tensor decomposition formats and discuss their advantages and disadvantages. In particular, we devise a general tensor representation and unite some concepts from mathematics and quantum physics where these formats have been known for a while. We highlight especially the methods for low rank approximation with tensors, because these will be used for the numerical solution of the parametric diffusion equation. In the last part, we present our own results. We developed a number of numerical methods in order to facilitate the use of tensor methods especially in the log-normal case. Additionally, we present error estimators, both for the affine and the log-normal case, that allow for the comparison of the errors resulting from the different discretisations. These enable us to formulate an adaptive algorithm for the numerical solution of these equations. Finally, we illustrate the applicability of our methods with some numerical experiments.
Source: DepositOnce
Author: Pfeffer, Max
Language: English
Subfield: Computational Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2948452995