Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial but exploit the block structure with the aim of being memory-efficient in the representation of the Krylov subspace basis. The problem may appear in the form of a low-degree polynomial (quartic or quintic, say) expressed in the monomial basis, or a high-degree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a nonmonomial basis. We have implemented a parallel solver in SLEPc covering both cases that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart and illustrate the robustness and performance of the solver with some numerical experiments.The first author was supported by the Spanish Ministry of Education, Culture and Sport through an FPU grant with reference AP2012-0608.Campos, C.; Román Moltó, JE. (2016). Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc. SIAM Journal on Scientific Computing. 38(5):385-411. https://doi.org/10.1137/15M1022458S38541138

Optimized analysis of isotropic high-nuclearity spin clusters with GPU acceleration

This is the author’s version of a work that was accepted for publication in Computer Physics Communications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Physics Communications, vol. 209, (2016). DOI 10.1016/j.cpc.2016.08.014.The numerical simulation of molecular clusters formed by a finite number of exchange-coupled paramagnetic centers is very relevant for many applications, modeling systems between molecules and extended solids. In the context of realistic scenarios, many centers need to be considered, and thus the required computational effort grows very fast. In a previous work (Ramos et al., 2010), a set of parallel programs were presented with standard message-passing parallelization (MPI) for both anisotropic and isotropic systems. In this work, we have further developed the code for isotropic models. On one hand, the computational cost has been significantly reduced by avoiding some of the matrix diagonalizations, corresponding to blocks with negligible contribution for the particular configuration. On the other hand, we have extended the parallelization in order to exploit available graphics processing units (GPUs). The new MPI-GPU paradigm reduces the computational time by at least one additional order of magnitude and enables the resolution of larger problems. © 2016 Elsevier B.V. All rights reserved.This work was partially supported by the Spanish Ministry of Economy and Competitiveness under grant TIN2013-41049-P. Alejandro Lamas Davina was supported by the Spanish Ministry of Education, Culture and Sports through a grant with reference FPU13-06655.Lamas Daviña, A.; Ramos Peinado, E.; Román Moltó, JE. (2016). Optimized analysis of isotropic high-nuclearity spin clusters with GPU acceleration. Computer Physics Communications. 209:70-78. https://doi.org/10.1016/j.cpc.2016.08.014S707820

Improving accuracy of parallel SLICOT model reduction routines for stable systems

© 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.This paper shows part of the work carried out to develop parallel versions of the SLICOT routines for model reduction of stable systems. In particular, the routines that have been parallelised are those based on the solution of Lyapunov equations. The goal is to be able to work with larger unreduced models and also to obtain better performance in the reduction process. New routines have been developed using standard libraries to improve portability and efficiency. A preliminary version was released previously by the authors, which achieved high performance. However, accuracy improvements have been necessary in order to make the new routines similar to the sequential ones in this aspect. Routines presented in this paper preserve good performance obtained by the previous parallel implementation while maintaining high accuracy of sequential SLICOT routines.This work was partially supported by the Spanish Ministry of Economy and Competitiveness under grant TIN2013-41049-PGuerrero López, D.; Román Moltó, JE. (2015). Improving accuracy of parallel SLICOT model reduction routines for stable systems. IEEE. https://doi.org/10.1109/MED.2015.7158781

Nevadas en la montaña alcoyana (Alicante) ¿riesgo climático?

Ponencia presentada en: I Congreso de la Asociación Española de Climatología “La climatología española en los albores del siglo XXI”, celebrado en Barcelona del 1 al 3 de diciembre de 1999.[ES]Con una periodicidad variable, que trataremos de determinar, la intensidad de las nevadas en esta área geográfica, donde se llegan a acumular más de cincuenta centímetros de espesor en pocas horas, puede traer funestas consecuencias para las infraestructuras y para las actividades económicas, esencialmente para la industria. En estos casos podemos hablar de la nieve como riesgo climático.[EN]With a various periodicity, that we shall try to determine, the intensity of the snowfalls in this geographic area, where are accumulated more of fifty centimetres during a few hours, can get terrible consequences for the infrastructures and the economic activities, essentially for the manufactures. In these cases we can talk about the snow like a climatic risk

Rainwater: from abandonment for agricultural purpose to its implementation as a non-conventional resource in urban areas

Las aguas de turbias desempeñaron un papel fundamental en los paisajes agrarios tradicionales del sureste peninsular. En ellos, la escasez de precipitaciones unido a su torrencialidad, obligó a la adopción de una sistematización de laderas y barrancos para aprovechar al máximo los caudales circulantes asociados a precipitaciones de cierta intensidad. Estos sistemas, que van a entrar en un proceso de abandono y desarticulación a partir de mediados del siglo XX, han sido objeto de un interés creciente vinculado a cuestiones ambientales y patrimoniales. A éstas se une recientemente su posible utilización como recursos no convencionales asociados a las políticas de sostenibilidad ambiental y de adecuación de la calidad de las aguas según su uso (fit for purpose). El objetivo de este artículo es examinar, teniendo en cuenta el significado patrimonial y ambiental que estos recursos de agua desempeñaron en las sociedades agrarias tradicionales, su recuperación, desde el punto de vista de iniciativas a escala local y de sostenibilidad ambiental. Para ello se analizan las actuaciones llevadas a cabo en ámbitos como los municipios litorales de la provincia de Alicante y el área metropolitana de Cataluña.Stormwater played a fundamental role in the traditional agricultural landscapes of Southeastern Spain. Relatively rare but violent outbursts of precipitation forced traditional agriculture to build a sophisticated system of slope and ravine management to take advantage of flows associated with intense rainfall. These systems, which will be entering a process of abandonment and disbandment from the middle of the 20th century onwards, have been the subject of growing interest linked to environmental and landscape conservation issues. On the other hand, rainwater flows also regain their importance in urban areas as local, alternative resources able to provide for the needs of certain uses such as garden irrigation after the concept of fit for purpose. The aim of this paper is to examine, initiatives aiming at the recovery of rainwater flows under the same philosophy of traditional agrarian societies but this time directed to the urban sphere. To this effect we will present cases in the coastal municipalities of the province of Alicante and the metropolitan area of Barcelona where rainwater and other alternative water flows are increasingly used.Este artículo se inserta en los proyectos de investigación Urbanización y metabolismo hídrico en el litoral de Alicante: análisis de tendencias para el periodo 2000-2010 (CSO2012-36997-C02-02) y Análisis de los cambios recientes en el consumo de agua en entornos de urbanización acelerada: propuesta metodológica y estudio de caso para el litoral Mediterráneo español (CSO2012-36997-C02-01) financiados por el Ministerio de Economía y Competitividad

An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem

[EN] Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient.The authors would like to express their heartfelt thanks to the editor and anonymous referees for their useful comments and constructive suggestions that substantially improved the quality and presentation of this article. This research was developed during a visit of Z.D. to Universitat Politecnica de Valencia. Z.D. would like to thank the hospitality shown by D. Sistemes Informatics i Computacio, Universitat Politecnica de Valencia. J.E.R. was partially supported by the Spanish Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. The authors thank Carmen Campos for useful comments on an initial draft of the article.Dalvand, Z.; Hajarian, M.; Román Moltó, JE. (2020). An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem. Numerical Linear Algebra with Applications. 27(6):1-24. https://doi.org/10.1002/nla.2327S124276Chu, M., & Golub, G. (2005). Inverse Eigenvalue Problems. doi:10.1093/acprof:oso/9780198566649.001.0001Hajarian, M., & Abbas, H. (2018). Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints. Computers & Mathematics with Applications, 76(6), 1458-1475. doi:10.1016/j.camwa.2018.06.038Hajarian, M. (2019). An efficient algorithm based on Lanczos type of BCR to solve constrained quadratic inverse eigenvalue problems. Journal of Computational and Applied Mathematics, 346, 418-431. doi:10.1016/j.cam.2018.07.025Hajarian, M. (2018). Solving constrained quadratic inverse eigenvalue problem via conjugate direction method. 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Strategies for spectrum slicing based on restarted Lanczos methods

In the context of symmetric-definite generalized eigenvalue problems, it is often required to compute all eigenvalues contained in a prescribed interval. For large-scale problems, the method of choice is the so-called spectrum slicing technique: a shift-and-invert Lanczos method combined with a dynamic shift selection that sweeps the interval in a smart way. This kind of strategies were proposed initially in the context of unrestarted Lanczos methods, back in the 1990's. We propose variations that try to incorporate recent developments in the field of Krylov methods, including thick restarting in the Lanczos solver and a rational Krylov update when moving from one shift to the next. We discuss a parallel implementation in the SLEPc library and provide performance results. © 2012 Springer Science+Business Media, LLC.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under grant TIN2009-07519.Campos González, MC.; Román Moltó, JE. (2012). Strategies for spectrum slicing based on restarted Lanczos methods. Numerical Algorithms. 60(2):279-295. https://doi.org/10.1007/s11075-012-9564-z279295602Amestoy, P.R, Duff, I.S., L’Excellent, J.Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184(2–4), 501–520 (2000)Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.2, Argonne National Laboratory (2011)Ericsson, T., Ruhe, A.: The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comput. 35(152), 1251–1268 (1980)Grimes, R.G., Lewis, J.G., Simon, H.D.: A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems. SIAM J. Matrix Anal. Appl. 15(1), 228–272 (1994)Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31(3), 351–362 (2005)Hernandez, V., Roman, J.E., Tomas, A.: Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33(7–8), 521–540 (2007)Marques, O.A.: BLZPACK: description and user’s guide. Tech. Rep. TR/PA/95/30, CERFACS, Toulouse, France (1995)Meerbergen, K.: Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem. Numer. Linear Algebra Appl. 8(1), 33–52 (2001)Meerbergen, K., Scott, J.: The design of a block rational Lanczos code with partial reorthogonalization and implicit restarting. Tech. Rep. RAL-TR-2000-011, Rutherford Appleton Laboratory (2000)Nour-Omid, B., Parlett, B.N., Ericsson, T., Jensen, P.S.: How to implement the spectral transformation. Math. Comput. 48(178), 663–673 (1987)Olsson, K.H.A., Ruhe, A.: Rational Krylov for eigenvalue computation and model order reduction. BIT Numer. Math. 46, 99–111 (2006)Ruhe, A.: Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl. 58, 391–405 (1984)Ruhe, A.: Rational Krylov subspace method. In: Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.) Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Society for Industrial and Applied Mathematics, pp. 246–249. Philadelphia (2000)Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)Stewart, G.W.: A Krylov–Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601–614 (2001)Vidal, AM., Garcia, V.M., Alonso, P., Bernabeu, M.O.: Parallel computation of the eigenvalues of symmetric Toeplitz matrices through iterative methods. J. Parallel Distrib. Comput. 68(8), 1113–1121 (2008)Wu, K., Simon, H.: Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22(2), 602–616 (2000)Zhang, H., Smith, B., Sternberg, M., Zapol, P.: SIPs: Shift-and-invert parallel spectral transformations. ACM Trans. Math. Softw. 33(2), 1–19 (2007