Matthias Bollhöfer

On Large-Scale Diagonalization Techniques for the Anderson Model of Localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete

Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media

LDL^T

A robust ILU with pivoting based on monitoring the growth of the inverse factors

factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices

An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete

On the Relations between ILUs and Factored Approximate Inverses

LDL^T

A Robust and Efficient ILU that Incorporates the Growth of the Inverse Triangular Factors

factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls

Exploiting thread-level parallelism in the iterative solution of sparse linear systems

\|L^{-1}\|

Parallelization of multilevel ILU preconditioners on distributed-memory multiprocessors

for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a “black-box” solver to a series of challenging two- and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.

A Factored Approximate Inverse Preconditioner with Pivoting

Abstract An incomplete LU decomposition with pivoting is presented that progressively monitors the growth of the inverse factors of L , U . The information on the growth of the inverse factors is used as feedback for dropping entries in L and U . This method often yields a robust and effective preconditioner especially when the system is highly indefinite. Numerical examples demonstrate the effectiveness of this approach.

Algebraic Multilevel Methods and Sparse Approximate Inverses

A new software code for computing selected eigenvalues and associated eigenvectors of a real symmetric matrix is described. The eigenvalues are either the smallest or those closest to some specified target, which may be in the interior of the spectrum. The underlying algorithm combines the Jacobi–Davidson method with efficient multilevel incomplete LU (ILU) preconditioning. Key features are modest memory requirements and robust convergence to accurate solutions. Parameters needed for incomplete LU preconditioning are automatically computed and may be updated at run time depending on the convergence pattern. The software is easy to use by non-experts and its top level routines are written in FORTRAN 77. Its potentialities are demonstrated on a few applications taken from computational physics.