Stefan Reitzinger

Parallel Algebraic Multigrid Methods on Distributed Memory Computers

Algebraic multigrid methods are well suited as preconditioners for iterative solvers. We consider linear systems of equations which are sparse and symmetric positive definite and which stem from a finite element discretization of a second order self-adjoint elliptic partial differential equation or a system of them. Since preconditioners based on algebraic multigrid are very efficient, additional speedup can only be achieved by parallelization. In this paper, we propose a general parallel algebraic multigrid algorithm for finite element discretizations based on domain decomposition ideas which is well suited for distributed memory computers. This paper pays special attention to the coarsening strategy which has to be adapted in the parallel case. Moreover, a general framework of data distribution gives rise to a construction scheme for the prolongation operators. nResults of numerical studies on parallel computers with distributed memory are presented which show the high efficiency of the approach.

Algebraic multigrid methods based on element preconditioning

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the “region of robustness“ between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy. For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem. The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.

Cache Issues of Algebraic Multigrid Methods for Linear Systems with Multiple Right-Hand Sides

This paper is concerned with the solution of a linear system with multiple right-hand sides. Such problems arise from nonlinear, time-dependent, inverse, or optimization problems. In order to solve these problems efficiently we use block variants of the conjugate gradient method and combine them with cache aware algorithms. Further, we describe the acceleration of algebraic multigrid (AMG) methods by using such cache aware algorithms. Finally, numerical studies are presented that show the high efficiency of the proposed methods.

Symbolic methods for the element preconditioning technique

The method of element preconditioning requires the construction of an M-matrix which is as close as possible to a given symmetric positive definite matrix in the spectral sense. In this paper we give a symbolic solution of the arising optimization problem for various subclasses. This improves the performance of the resulting algorithm considerably.

A Symmetric Low-Frequency Stable Broadband Maxwell Formulation for Industrial Applications

A new monolithic, symmetric, low-frequency stable, broadband formulation of the time-harmonic Maxwells equations is presented. The approach is based on a variational formulation of Maxwells equations which accounts for the frequency dependencies of the underlying physical phenomena and does not require auxiliary variables. The formulation allows for stable evaluation at low frequencies, even if the frequency equals zero. It has a broadband validity and correctly comprises both high-frequency effects and (quasi-)static behaviors. This is demonstrated by using different test examples from scientific to large-scale real-world applications.

A reduced basis approach for broadband Maxwell simulations with validity in any frequency

Based on a low-frequency stable formulation of Maxwells equations, we propose a finite element and reduced basis scheme which allows for a robust and accurate evaluation of the system in any frequency. The approach considers in particular the limit cases, as the frequency tends to zero or to a maximal frequency for which a discretization is reasonable.

Sensitivity Analysis for Maxwell Eigenvalue Problems in Industrial Applications

In this paper we focus on the sensitivity analysis of Maxwell’s eigenvalue problem, where the derivatives of the eigenvalues are calculated with respect to design parameters (i.e., material or geometrical parameters). Utilizing the adjoint approach the derivatives can be calculated at almost no additional cost. The challenge consists in the computation of the required derivatives (i.e., derivatives of bilinear forms with respect to the design parameters) from a higher order, curved finite element discretization. Numerical studies show the application for a real life electromagnetic filter application where the sensitivities of the eigenvalues give a better insight into the characteristics of the underlying filter. The benefit is apparent if the adjoint method is compared to a standard finite difference approach.