Pietsch Factorization and Grothendieck Factorization are the two landmark theorems in modern functional analysis. They were first introduced to the numerical linear algebra community by the work of Joel A. Tropp in the column subset selection problem, which seeks to extract from a matrix a column submatrix that has lower spectral norm. Despite their broad application in functional analysis, the...

We develop a preconditioner for systems arising from space-time finite element discretizations of parabolic equations. The preconditioner is based on a transformation of the coupled system into block diagonal form and an efficient solution strategy for the arising 2 × 2 blocks. The suggested strategy makes use of an inexact factorization of the Schur complement of these blocks, for which unifor...

In this paper, we address the problem of preconditioning sequences of large sparse nonsymmetric systems of linear equations and present two new strategies to construct approximate updates of factorized preconditioners. Both updates are based on the availability of an incomplete LU (ILU) factorization for one matrix of the sequence and differ in the approximation of the so-called ideal updates. ...

In this paper, we design interconnected hierarchical rank-structured methods for the fast direct solution and effective preconditioning of the Helmholtz equation. First, for essentially elliptic PDEs, a direct method is proposed to exploit interconnected structures within two hierarchical layers: the hierarchical partitioning of a large problem into sub-problems for smaller subdomains, and the ...

This paper describes the design and implementation of three core factorization routines—LU, QR, and Cholesky—included in the out-of-core extension of ScaLAPACK. These routines allow the factorization and solution of a dense system that is too large to fit entirely in physical memory. The full matrix is stored on disk and the factorization routines transfer sub-matrice panels into memory. The ‘l...

Abstract. Numerical homogenization is used for upscaling of the linear elasticity tensor of strongly heterogeneous microstructures. The implemented 3D algorithm is described in terms of six auxiliary elastic problems for the reference volume element (RVE). Rotated trilinear RannacherTurek finite elements are used for discretization of the involved subproblems. A parallel PCG method is implement...

We investigate and compare a few parallel preconditioning techniques in the iterative solution of large sparse linear systems arising from solid Earth simulation with and without using contact information in the domain partitioning process. Previous studies are focused on using static or matrix pattern based incomplete LU (ILU) preconditioners in a localized preconditioner implementation. Our c...

We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible Navier-Stokes equations. The spatial discretization is carried out using a Discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear sy...