Dölz, Jürgen. Hierarchical matrix techniques for partial differential equations with random input data. 2017, Doctoral Thesis, University of Basel, Faculty of Science.

PDF
8Mb 
Official URL: http://edoc.unibas.ch/diss/DissB_12427
Downloads: Statistics Overview
Abstract
We consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal.
If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated righthand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve.
Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified.
The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.
If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated righthand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve.
Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified.
The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.
Advisors:  Harbrecht, Helmut and Börm, Steffen 

Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht) 
UniBasel Contributors:  Dölz, Jürgen and Harbrecht, Helmut 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  12427 
Thesis status:  Complete 
Number of Pages:  1 OnlineRessource (125 Seiten) 
Language:  English 
Identification Number: 

edoc DOI:  
Last Modified:  22 Apr 2018 04:32 
Deposited On:  01 Feb 2018 13:18 
Repository Staff Only: item control page