Adaptive GMRES(m) for the Electromagnetic Scattering Problem

In this article, an adaptive version of the restarted GMRES (GMRES(m)) is introduced for the resolution of the finite difference approximation of the Helmholtz equation. It has been observed that the choice of the restart parameter m strongly affects the convergence of standard GMRES(m). To overcome this problem, the GMRES(m) is formulated as a control problem in order to adaptively combine two strategies: a) the appropriate variation of the restarted parameter m, if a stagnation in the convergence is detected; and b) the augmentation of the search subspace using vectors obtained at previous cycles. The proposal is compared with similar iterative methods of the literature based on standard GMRES(m) with fixed parameters. Numerical results for selected matrices suggest that the switching adaptive proposal method could overcome the stagnation observed in standard methods, and even improve the performance in terms of computational time and memory requirements.

Anderson acceleration for electromagnetic nonlinear problems

Purpose The purpose of this paper is to implement the Anderson acceleration for different formulations of eletromagnetic nonlinear problems and analyze the method efficiency and strategies to obtain a fast convergence. Design/methodology/approach The paper is structured as follows: the general class of fixed point nonlinear problems is shown at first, highlighting the requirements for convergence. The acceleration method is then shown with the associated pseudo-code. Finally, the algorithm is tested on different formulations (finite element, finite element/boundary element) and material properties (nonlinear iron, hysteresis models for laminates). The results in terms of convergence and iterations required are compared to the non-accelerated case. Findings The Anderson acceleration provides accelerations up to 75 per cent in the test cases that have been analyzed. For the hysteresis test case, a restart technique is proven to be helpful in analogy to the restarted GMRES technique. Originality/value The acceleration that has been suggested in this paper is rarely adopted for the electromagnetic case (it is normally adopted in the electronic simulation case). The procedure is general and works with different magneto-quasi static formulations as shown in the paper. The obtained accelerations allow to reduce the number of iterations required up to 75 per cent in the benchmark cases. The method is also a good candidate in the hysteresis case, where normally the fixed point schemes are preferred to the Newton ones.

GCRO with dynamic deflated restarting for solving adjoint systems of equations for aerodynamic shape optimization

Purpose This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method. Design/methodology/approach The proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗. Findings The authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver. Originality/value This paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix.

Using an ILU/Deflation Preconditioner for Simulation of a PEM Fuel Cell Cathode Catalyst Layer

Numerical aspects of a pore scale model are investigated for the simulation of catalyst layers of polymer electrolyte membrane fuel cells. Coupled heat, mass and charged species transport together with reaction kinetics are taken into account using parallelized finite volume simulations for a range of nanostructured, computationally reconstructed catalyst layer samples. The effectiveness of implementing deflation as a second stage preconditioner generally improves convergence and results in better convergence behavior than more sophisticated first stage pre-conditioners. This behavior is attributed to the fact that the two stage preconditioner updates the preconditioning matrix at every GMRES restart, reducing the stalling effects that are commonly observed in restarted GMRES when a single stage preconditioner is used. In addition, the effectiveness of the deflation preconditioner is independent of the number of processors, whereas the localized block ILU preconditioner deteriorates in quality as the number of processors is increased. The total number of GMRES search directions required for convergence varies considerably depending on the preconditioner, but also depends on the catalyst layer microstructure, with low porosity microstructures requiring a smaller number of iterations. The improved model and numerical solution strategy should allow simulations for larger computational domains and improve the reliability of the predicted transport parameters. The preconditioning strategies presented in the paper are scalable and should prove effective for massively parallel simulations of other problems involving nonlinear equations.

A New GMRES(m) Method for Markov Chains

This paper presents a class of new accelerated restarted GMRES method for calculating the stationary probability vector of an irreducible Markov chain. We focus on the mechanism of this new hybrid method by showing how to periodically combine the GMRES and vector extrapolation method into a much efficient one for improving the convergence rate in Markov chain problems. Numerical experiments are carried out to demonstrate the efficiency of our new algorithm on several typical Markov chain problems.