Roger G Ghanem
Parallelization and preconditioning of iterative solvers for linear systems arising in the stochastic finite element method

Civil and Environmental Engineering
University of Southern California
3620 S Vermont Street
KAP 254C
Los Angeles
CA 90089-2531
Tel: (213) 740-9528
Fax: (213) 740-2037
ghanem@usc.edu
Ramakrishna Tipireddy
Maarten Arnst

This communication investigates the use of preconditioning and parallelization methods to improve the computational efficiency of iterative solvers for linear systems arising in the Stochastic Finite Element Method (SFEM).

Linear matrix systems obtained in the SFEM are typically of much larger dimension than those obtained in the classical deterministic FEM. Indeed, SFEM systems are typically [1] of form:

$$\sum_{i=0}^{N}\sum_{j=0}^{L}c_{ijk}\boldsymbol{K}_{j}\boldsymbol{u}_{i}=\boldsymbol{f}_{k}\;, k = 0,\ldots,N \quad,$$ (1)

where $N$ is the number of terms retained in the polynomial chaos expansion of the random response, $L$ is the number of terms in the Karhunen-Loeve expansion of the random material properties, and the dimension of all submatrices  $\boldsymbol{K}_{i}$ is equal to the number of spatial degrees of freedom. Since the coefficients $c_{ijk}$ vanish for certain combinations of the indices $i$, $j$ and $k$, the system matrix derived from the above formulation has a particular block-sparsity structure, see e.g. [3]. Other important properties include the dominance of  $\boldsymbol{K}_0$ over the other  $\boldsymbol{K}_i$'s, and the fact that all matrices  $\boldsymbol{K}_i$ have the same sparsity pattern and are usually symmetric.

The objective of this communication is to compare several preconditioning and parallelization methods which capitalize on the aforementioned properties to improve the computational efficiency of iterative solvers for SFEM systems of form (). Based on the dominance of  $\boldsymbol{K}_0$ over the other  $\boldsymbol{K}_i$'s, a first preconditioning method consists in reformulating the problem as a system of linear equations with multiple right-hand sides:

$$c_{k0k}\boldsymbol{K}_{0}\boldsymbol{u}_{k}=\boldsymbol{f}_{k}-\s... ...}-\left(\sum_{j\not=0}^{L}c_{kjk}\boldsymbol{K}_{j}\right)\boldsymbol{u}_{k}\;, k = 0,\ldots,N .$$ (2)

A second preconditioning method consists in keeping only the diagonal block matrices on the left-hand side and moving the remaining blocks to the right-hand side:

$$\sum_{j=0}^{L}c_{kjk}\boldsymbol{K}_{j}\boldsymbol{u}_{k}=\boldsy... ...-\sum_{i\not=k}^{N}\sum_{j=0}^{L}c_{ijk}\boldsymbol{K}_{j}\boldsymbol{u}_{i}\;, k = 0,\ldots,N .$$ (3)

This method is expected to need fewer iterations to converge, but entails a higher computational effort at each iteration since the diagonal blocks are not identical.

The SANDIA developed Trilinos library [2] is used to implement the framework. The package Epetra for mat-vec operations, and the AztecOO, IFPACK and Belos preconditioner packages are used in particular. The Belos package provides a solver manager for solving linear systems simultaneously on multiple right-hand sides. All packages use the Message Passing Interface (MPI) to allow for execution on parallel platforms. At the conference, the methods discussed above will be presented and then compared based on their application to a case history in stochastic structural mechanics.