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

Using spectral low rank preconditioners for large electromagnetic calculations

SUMMARY For solving large dense complex linear systems that arise in electromagnetic calculations, we perform experiments using a general purpose spectral low rank update preconditioner in the context of the GMRES method preconditioned by an approximate inverse preconditioner. The goal of the spectral preconditioner is to improve the convergence properties by shifting by one the smallest eigenvalues of the original preconditioned system. Numerical experiments on parallel distributed memory computers are presented to illustrate the efficiency of this technique on large and challenging real-life industrial problems

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

Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics

We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising in industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pattern. Some strategies for determining the nonzero pattern of an approximate inverse are described. The results of numerical experiments suggest that sparse approximate inverse preconditioning is a viable approach for the solution of large-scale dense linear systems on parallel computers

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

Spectral two-level preconditioners for sequences of linear systems

De nombreuses simulations numériques nécessitent la résolution d'une série de systèmes linéaires impliquant une même matrice mais des second-membres différents. Des méthodes efficaces pour ce type de problèmes cherchent à tirer bénéfice des résolutions précédentes pour accélérer les résolutions restantes. Deux grandes classes se distinguent dans la façon de procéder: la première vise à réutiliser une partie du sous-espace de Krylov, la deuxième à construire une mise à jour du préconditionneur à partir de vecteurs approximant un espace invariant. Dans cette thèse, nous nous sommes intéressés à cette dernière approche en cherchant à améliorer le préconditionneur d'origine. Dans une première partie, une seule mise à jour du préconditionneur est considérée pour tous les systèmes. Cette mise à jour consiste en une correction spectrale de rang faible qui permet de translater de un la position des plus petites valeurs propres en module de la matrice du système préconditionné de départ. Des expérimentations numériques sont réalisées en utilisant la méthode GMRES couplée à un préconditionneur de type inverse approchée. L'information spectrale est obtenue par un solveur de valeurs propres lors d'une phase préliminaire au calcul. Dans une deuxième partie, on autorise une possible mise à jour entre chaque système. Une correction spectrale incrémentale est proposée. Des expérimentations numériques sont réalisées en utilisant la méthode GMRES-DR, d'une part parce qu'elle est efficace en tant que solveur linéaire, et d'autre part parce qu'elle permet une bonne approximation des petites valeurs propres au cours de la résolution linéaire. Des stratégies sont développées afin de sélectionner l'information spectrale la plus pertinente. Ces approches ont été validées sur des problèmes de grande taille issus de simulations industrielles en électromagnétisme. Dans ce but, elles ont été implantées dans un code parallèle développé par EADS-CCR. ABSTRACT : Many numerical simulations in scientific and engineering applications require the solution of a set of large linear systems involving the same coefficient matrix but different right-hand sides. Efficient methods for tackling this problem attempt to benefit from the previously solved right-hand sides for the solution of the next ones. This goal can be achieved either by recycling Krylov subspaces or by building preconditioner updates based on near invariant subspace information. In this thesis, we focus our attention on this last approach that attempts to improve a selected preconditioner. In the first part, we consider only one update of the preconditioner for all the systems. This update consists of a spectral low-rank correction that shifts by one the smallest eigenvalues in magnitude of the matrix of the original preconditioned system. We perform experiments in the context of the GMRES method preconditioned by an approximate inverse preconditioner. The spectral information is computed by an eigensolver in a preprocessing phase. In the second part, we consider an update of the preconditioner between each system. An incremental spectral correction of the preconditioner is proposed. We perform experiments using the GMRES-DR method, thanks to its efficiency as a linear solver and its ability to recover reliable approximations of the desired eigenpairs at run time. Suitable strategies are investigated for selecting reliable eigenpairs. The efficiency of the proposed approaches is in particular assessed for the solution of large and challenging problems in electromagnetic applications. For this purpose, they have been implemented in a parallel industrial code developed by EADS-CCR

Sparse approximate inverse preconditioners for electromagnetic surface scattering simulations

Simulation of electromagnetic waves scattered by a connected three dimensional non-convex obstacle at medium frequencies (where the size of the obstacle is 10 to 100 times the incident wavelength) requires a non-asymptotic approach. Standard boundary element schemes at such frequencies require millions of unknowns. However, recently developed high-order algorithms require only tens of thousands of unknowns at medium frequencies for a class of three dimensional obstacles. At such frequencies we use a sparse approximation to the scattering matrix and so iterative solvers are required. We describe an efficient scheme to solve the associated linear systems using sparse approximate inverse preconditioners. The sparse preconditioners developed in this work facilitate efficient solutions of complex dense linear systems arising in electromagnetic scattering simulations. References B. Carpentieri, I. S. Duff, and L. Giraud. Sparse pattern selection strategies for robust frobenius-norm minimization preconditioners in electromagnetism. Numer. Linear Algebra Appl., 7:667--685, 2000. doi:10.1002/1099-1506(200010/12)7:7/8<667::AID-NLA218>3.0.CO;2-X. B. Carpentieri, I. S. Duff, L. Giraud, and G. Sylvand. Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations. SIAM J. Sci. Comput., 27:774--792, 2005. doi:10.1137/040603917. E. Chow. A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput., 21:1804--1822, 2000. doi:10.1137/S106482759833913X. D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. Wiley, 1983. M. Ganesh and S. C. Hawkins. A spectrally accurate algorithm for electromagnetic scattering in three dimensions. Numerical Algorithms, 43:25--60, 2006. doi:10.1007/s11075-006-9033-7. M. Ganesh and S. C. Hawkins. A hybrid high-order algorithm for radar cross section computations. SIAM J. Sci. Comput., 29:1217--1243, 2007. doi:10.1137/060664859. M. Ganesh and S. C. Hawkins. Sparse preconditioners for dense complex linear systems arising in some radar cross section computations. ANZIAM J., 48:C233--C248, 2007. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/61. L. Y. Kolotolina. Explicit preconditioning of systems of linear algebraic equations. J. Sov. Math., 43:2566--2573, 1988. R. B. Melrose and M. E. Taylor. Near peak scattering and the corrected kirchhoff approximation for a convex obstacle. Adv. in Math., 55:242--315, 1985. doi:10.1016/0001-8708(85)90093-3. Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3):856--869, July 1986. doi:10.1137/0907058. G. Alleon, M. Benzi, and L. Giraud. Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics. Numerical Algorithms, 16:1--15, 1997. doi:10.1023/A:1019170609950. J. M. Song, C. C. Lu, W. C. Chew, and S. W. Lee. Fast Illinois solver code (FISC). IEEE Antennas Propag. Mag., 40:27--34, 1998. doi:10.1109/74.706067. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users' Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999. B. Carpentieri. Fast iterative solution methods in electromagnetic scattering. Technical Report 17/2006, University of Graz, 2006. http://www.uni-graz.at/~carpenti/papers/IMA17-06.pdf. B. Carpentieri, I. S. Duff, and L. Giraud. Robust preconditioning of dense problems from electromagnetics. In Numer. Anal. and App. Lecture Notes in Computer Science 1988, pages 170--178. Springer, 2000. http://www.uni-graz.at/~carpenti/papers/rousse.pdf