A Bramble-Pasciak conjugate gradient method for discrete Stokes equations with random viscosity

We study the iterative solution of linear systems of equations arising from stochastic Galerkin finite element discretizations of saddle point problems. We focus on the Stokes model with random data parametrized by uniformly distributed random variables and discuss well-posedness of the variational formulations. We introduce a Bramble-Pasciak conjugate gradient method as a linear solver. It builds on a non-standard inner product associated with a block triangular preconditioner. The block triangular structure enables more sophisticated preconditioners than the block diagonal structure usually applied in MINRES methods. We show how the existence requirements of a conjugate gradient method can be met in our setting. We analyze the performance of the solvers depending on relevant physical and numerical parameters by means of eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of the relevant preconditioned sub-matrices. We illustrate our findings using the flow in a driven cavity as a numerical test case, where the viscosity is given by a truncated Karhunen-Lo\`eve expansion of a random field. In this example, a Bramble-Pasciak conjugate gradient method with block triangular preconditioner outperforms a MINRES method with block diagonal preconditioner in terms of iteration numbers.Comment: 19 pages, 1 figure, submitted to SIAM JU

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two-by-two structure, with one of the submatrices block diagonal. Each one of the diagonal blocks in this submatrix is closely related to the deterministic mean-value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix-vector multiplications for the off-diagonal blocks. Neither the global matrix, nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good preconditioned for the mean-value deterministic problem. We provide a condition number bound for a model elliptic problem and the performance of the method is illustrated by numerical experiments.Comment: 15 pages, 2 figures, 9 tables, (updated numerical experiments

Truncation preconditioners for stochastic Galerkin finite element discretizations

Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, these preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and non-affine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare truncation preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones.Comment: 27 pages, 6 table

Low-rank solutions to the stochastic Helmholtz equation

In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtain an advantage in terms of storage and CPU time compared to computations in full rank. We also propose a general approach to implement a preconditioner using the low-rank format efficiently