Application of the Jacobi Davidson method for spectral low-rank preconditioning in computational electromagnetics problems

[EN] We consider the numerical solution of linear systems arising from computational electromagnetics applications. For large scale problems the solution is usually obtained iteratively with a Krylov subspace method. It is well known that for ill conditioned problems the convergence of these methods can be very slow or even it may be impossible to obtain a satisfactory solution. To improve the convergence a preconditioner can be used, but in some cases additional strategies are needed. In this work we study the application of spectral lowrank updates (SLRU) to a previously computed sparse approximate inverse preconditioner.The updates are based on the computation of a small subset of the eigenpairs closest to the origin. Thus, the performance of the SLRU technique depends on the method available to compute the eigenpairs of interest. The SLRU method was first used using the IRA s method implemented in ARPACK. In this work we investigate the use of a Jacobi Davidson method, in particular its JDQR variant. The results of the numerical experiments show that the application of the JDQR method to obtain the spectral low-rank updates can be quite competitive compared with the IRA s method.Mas Marí, J.; Cerdán Soriano, JM.; Malla Martínez, N.; Marín Mateos-Aparicio, J. (2015). Application of the Jacobi Davidson method for spectral low-rank preconditioning in computational electromagnetics problems. Journal of the Spanish Society of Applied Mathematics. 67:39-50. doi:10.1007/s40324-014-0025-6S395067Bergamaschi, L., Pini, G., Sartoretto, F.: Computational experience with sequential, and parallel, preconditioned Jacobi–Davidson for large sparse symmetric matrices. J. Comput. Phys. 188(1), 318–331 (2003)Carpentieri, B.: Sparse preconditioners for dense linear systems from electromagnetics applications. PhD thesis, Institut National Polytechnique de Toulouse, CERFACS (2002)Carpentieri, B., Duff, I.S., Giraud, L.: Sparse pattern selection strategies for robust Frobenius-norm minimization preconditioners in electromagnetism. Numer. Linear Algebr. Appl. 7(7–8), 667–685 (2000)Carpentieri, B., Duff, I.S., Giraud, L.: A class of spectral two-level preconditioners. SIAM J. Sci. Comput. 25(2), 749–765 (2003)Carpentieri, B., Duff, I.S., Giraud, L., Magolu monga Made, M.: Sparse symmetric preconditioners for dense linear systems in electromagnetism. Numer. Linear Algebr. Appl. 11(8–9), 753–771 (2004)Carpentieri, B., Duff, I.S., Giraud, L., Sylvand, G.: Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations. SIAM J. Sci. Comput. 27(3), 774–792 (2005)Darve, E.: The fast multipole method I: error analysis and asymptotic complexity. SIAM J. Numer. Anal. 38(1), 98–128 (2000)Fokkema, D.R., Sleijpen, G.L., Van der Vorst, H.A.: Jacobi–Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20(1), 94–125 (1998)Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(3), 325–348 (1987)Grote, M., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18(3), 838–853 (1997)Harrington, R.: Origin and development of the method of moments for field computation. IEEE Antenna Propag. Mag. (1990)Kunz, K.S., Luebbers, R.J.: The finite difference time domain method for electromagnetics. SIAM J. Sci. Comput. 18(3), 838–853 (1997)Maxwell, J.C.: A dynamical theory of the electromagnetic field. Roy. S. Trans. CLV, (1864). Reprinted in Tricker, R. A. R. The Contributions of Faraday and Maxwell to Electrial Science, Pergamon Press (1966)Marín, J., Malla M.: Some experiments preconditioning via spectral low rank updates for electromagnetism applications. In: Proceedings of the international conference on preconditioning techniques for large sparse matrix problems in scientific and industrial applications (Preconditioning 2007), Toulouse (2007)Meijerink, J.A., van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31, 148–162 (1977)Sorensen, D.C., Lehoucq, R.B., Yang, C.: ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia (1998)Rao, S.M., Wilton, D.R., Glisson, A.W.: Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antenna Propag. 30, 409–418 (1982)Saad, Y.: Iterative methods for sparse linear systems. PWS Publishing Company, Boston (1996)Silvester, P.P., Ferrari, R.L.: Finite elements for electrical engineers. Cambridge University Press, Cambridge (1990)Sleijpen, S.L., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401–425 (1996)van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 12(6), 631–644 (1992

A two-level ILU preconditioner for electromagnetic applications

[EN] Computational electromagnetics based on the solution of the integral form of Maxwell s equations with boundary element methods require the solution of large and dense linear systems. For large-scale problems the solution is obtained by using iterative Krylov-type methods provided that a fast method for performing matrix vector products is available. In addition, for ill-conditioned problems some kind of preconditioning technique must be applied to the linear system in order to accelerate the convergence of the iterative method and improve its performance. For many applications it has been reported that incomplete factorizations often suffer from numerical instability due to the indefiniteness of the coefficient matrix. In this context, approximate inverse preconditioners based on Frobenius-norm minimization have emerged as a robust and highly parallel alternative. In this work we propose a two-level ILU preconditioner for the preconditioned GMRES method. The computation and application of the preconditioner is based on graph partitioning techniques. Numerical experiments are presented for different problems and show that with this technique it is possible to obtain robust ILU preconditioners that perform competitively compared with Frobenius-norm minimization preconditioners.This work was supported by the Spanish Ministerio de Economía y Competitividad under grant MTM2014-58159-P and MTM2015-68805-REDT.Cerdán Soriano, JM.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2017). A two-level ILU preconditioner for electromagnetic applications. Journal of Computational and Applied Mathematics. 309:371-382. https://doi.org/10.1016/j.cam.2016.03.012S37138230

Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm

Cataloged from PDF version of article.Surface integral-equation methods accelerated with the multilevel fast multipole algorithm (MLFMA) provide a suitable mechanism for electromagnetic analysis of real-life dielectric problems. Unlike the perfect-electric-conductor case, discretizations of surface formulations of dielectric problems yield 2 x 2 partitioned linear systems. Among various surface formulations, the combined tangential formulation (CTF) is the closest to the category of first-kind integral equations, and hence it yields the most accurate results, particularly when the dielectric constant is high and/or the dielectric problem involves sharp edges and corners. However, matrix equations of CTF are highly ill-conditioned, and their iterative solutions require powerful preconditioners for convergence. Second-kind surface integral-equation formulations yield better conditioned systems, but their conditionings significantly degrade when real-life problems include high dielectric constants. In this paper, for the first time in the context of surface integral-equation methods of dielectric objects, we propose Schur complement preconditioners to increase their robustness and efficiency. First, we approximate the dense system matrix by a sparse near-field matrix, which is formed naturally by MLFMA. The Schur complement preconditioning requires approximate solutions of systems involving the (1,1) partition and the Schur complement. We approximate the inverse of the (1,1) partition with a sparse approximate inverse (SAI) based on the Frobenius norm minimization. For the Schur complement, we first approximate it via incomplete sparse matrix-matrix multiplications, and then we generate its approximate inverse with the same SAI technique. Numerical experiments on sphere, lens, and photonic crystal problems demonstrate the effectiveness of the proposed preconditioners. In particular, the results for the photonic crystal problem, which has both surface singularity and a high dielectric constant, shows that accurate CTF solutions for such problems can be obtained even faster than with second-kind integral equation formulations, with the acceleration provided by the proposed Schur complement preconditioners

Study of preconditioners based on Markov Chain Monte Carlo methods

Nowadays, analysis and design of novel scalable methods and algorithms for fundamental linear algebra problems such as solving Systems of Linear Algebraic Equations with focus on large scale systems is a subject of study. This research focuses on the study of novel mathematical methods and scalable algorithms for computationally intensive problems such as Monte Carlo and Hybrid Methods and Algorithms

Communication-aware sparse patterns for the factorized approximate inverse preconditioner

The Conjugate Gradient (CG) method is an iterative solver targeting linear systems of equations Ax=b where A is a symmetric and positive definite matrix. CG convergence properties improve when preconditioning is applied to reduce the condition number of matrix A. While many different options can be found in the literature, the Factorized Sparse Approximate Inverse (FSAI) preconditioner constitutes a highly parallel option based on approximating A-1. This paper proposes the Communication-aware Factorized Sparse Approximate Inverse preconditioner (FSAIE-Comm), a method to generate extensions of the FSAI sparse pattern that are not only cache friendly, but also avoid increasing communication costs in distributed memory systems. We also propose a filtering strategy to reduce inter-process imbalance. We evaluate FSAIE-Comm on a heterogeneous set of 39 matrices achieving an average solution time decrease of 17.98%, 26.44% and 16.74% on three different architectures, respectively, Intel Skylake, Fujitsu A64FX and AMD Zen 2 with respect to FSAI. In addition, we consider a set of 8 large matrices running on up to 32,768 CPU cores, and we achieve an average solution time decrease of 12.59%.Marc Casas is supported by Grant RYC-2017-23269 funded by MCIN/AEI/ 10.13039/501100011033 and by “ESF Investing in your future”. This work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 955606. This work has been supported by the Computación de Altas Prestaciones VIII (BSC-HPC8) project. It has also been partially supported by the EXCELLERAT project funded by the European Commission’s ICT activity of the H2020 Programme under grant agreement number: 823691 and by the Spanish Ministry of Science and Innovation (Nucleate, Project PID2020-117001GB-I00).Peer ReviewedPostprint (author's final draft

A class of linear solvers based on multilevel and supernodal factorization

De oplossing van grote en schaarse lineaire systemen is een kritieke component van moderne wetenschap en technische simulaties. Iteratieve methoden, namelijk de klasse van moderne Krylov-subruimtemethoden, worden vaak gebruikt om grootschalige lineaire systemen op te lossen. Om de robuustheid en de convergentiesnelheid van de iteratieve methoden te verbeteren, worden preconditioneringstechnieken vaak beschouwd als cruciale componenten van de lineaire systeemoplossing. In dit proefschrift wordt een klasse van algebraïsche multilevel oplossers gepresenteerd voor het conditioneren van algemene lineaire systeemvergelijkingen die voortkomen uit computationele wetenschap en technische toepassingen. Ze kunnen spaarzame patronen produceren en geheugenkosten besparen door recursieve combinatorische algoritmen toe te passen. Robuustheid wordt verbeterd door de factorisatie te combineren met recent ontwikkelde overlappende en compressiestrategieën en door efficiënte lokale oplossers te gebruiken. We hebben de goede prestaties van de voorgestelde strategieën aangetoond met numerieke experimenten op realistische matrixproblemen, ook in vergelijking met enkele van de meest populaire algebraïsche preconditioners die tegenwoordig worden gebruikt

Effective preconditioners for iterative solutions of large-scale surface-integral-equation problems

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 2010.Thesis (Ph.D.) -- Bilkent University, 2010.Includes bibliographical references leaves 171-187.A popular method to study electromagnetic scattering and radiation of threedimensional electromagnetics problems is to solve discretized surface integral equations, which give rise to dense linear systems. Iterative solution of such linear systems using Krylov subspace iterative methods and the multilevel fast multipole algorithm (MLFMA) has been a very attractive approach for large problems because of the reduced complexity of the solution. This scheme works well, however, only if the number of iterations required for convergence of the iterative solver is not too high. Unfortunately, this is not the case for many practical problems. In particular, discretizations of open-surface problems and complex real-life targets yield ill-conditioned linear systems. The iterative solutions of such problems are not tractable without preconditioners, which can be roughly defined as easily invertible approximations of the system matrices. In this dissertation, we present our efforts to design effective preconditioners for large-scale surface-integral-equation problems. We first address incomplete LU (ILU) preconditioning, which is the most commonly used and well-established preconditioning method. We show how to use these preconditioners in a blackbox form and safe manner. Despite their important advantages, ILU preconditioners are inherently sequential. Hence, for parallel solutions, a sparseapproximate-inverse (SAI) preconditioner has been developed. We propose a novel load-balancing scheme for SAI, which is crucial for parallel scalability. Then, we improve the performance of the SAI preconditioner by using it for the iterative solution of the near-field matrix system, which is used to precondition the dense linear system in an inner-outer solution scheme. The last preconditioner we develop for perfectly-electric-conductor (PEC) problems uses the same inner-outer solution scheme, but employs an approximate version of MLFMA for inner solutions. In this way, we succeed to solve many complex real-life problems including helicopters and metamaterial structures with moderate iteration counts and short solution times. Finally, we consider preconditioning of linear systems obtained from the discretization of dielectric problems. Unlike the PEC case, those linear systems are in a partitioned structure. We exploit the partitioned structure for preconditioning by employing Schur complement reduction. In this way, we develop effective preconditioners, which render the solution of difficult real-life problems solvable, such as dielectric photonic crystals.Malas, TahirPh.D