106 research outputs found

    Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

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    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

    NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc

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    [EN] SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.This work was partially funded by the Spanish Agencia Estatal de Investigacion AEI http://ciencia.gob.es under grants TIN2016-75985-P AEI and PID2019-107379RB-I00 AEI (including European Commission FEDER funds).Campos, C.; Roman, JE. (2021). NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc. ACM Transactions on Mathematical Software. 47(3):1-29. https://doi.org/10.1145/3447544S12947

    A polynomial Jacobi-Davidson solver with support for non-monomial bases and deflation

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    [EN] Large-scale polynomial eigenvalue problems can be solved by Krylov methods operating on an equivalent linear eigenproblem (linearization) of size d center dot n where d is the polynomial degree and n is the problem size, or by projection methods that keep the computation in the n-dimensional space. Jacobi-Davidson belongs to the latter class of methods, and, since it is a preconditioned eigensolver, it may be competitive in cases where explicitly computing a matrix factorization is exceedingly expensive. However, a fully fledged implementation of polynomial Jacobi-Davidson has to consider several issues, including deflation to compute more than one eigenpair, use of non-monomial bases for the case of large degree polynomials, and handling of complex eigenvalues when computing in real arithmetic. We discuss these aspects and present computational results of a parallel implementation in the SLEPc library.This work was supported by Agencia Estatal de Investigación (AEI) under Grant TIN2016-75985-P, which includes European Commission ERDF funds.Campos, C.; Jose E. Roman (2020). A polynomial Jacobi-Davidson solver with support for non-monomial bases and deflation. BIT Numerical Mathematics. 60(2):295-318. https://doi.org/10.1007/s10543-019-00778-zS295318602Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26(3), 640–659 (2005)Balay, S., Abhyankar, S., Adams, M., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W., Karpeyev, D., Kaushik, D., Knepley, M., May, D., McInnes, L.C., Mills, R., Munson, T., Rupp, K., Sanan, P., Smith, B., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. Technical report ANL-95/11—revision 3.10, Argonne National Laboratory (2018)Betcke, T., Kressner, D.: Perturbation, extraction and refinement of invariant pairs for matrix polynomials. Linear Algebra Appl. 435(3), 514–536 (2011)Betcke, T., Voss, H.: A Jacobi–Davidson-type projection method for nonlinear eigenvalue problems. Future Gen. Comput. Syst. 20(3), 363–372 (2004)Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: a collection of nonlinear eigenvalue problems. ACM Trans. Math. Softw. 39(2), 7:1–7:28 (2013)Campos, C., Roman, J.E.: Parallel Krylov solvers for the polynomial eigenvalue problem in SLEPc. SIAM J. Sci. Comput. 38(5), S385–S411 (2016)Effenberger, C.: Robust successive computation of eigenpairs for nonlinear eigenvalue problems. SIAM J. Matrix Anal. Appl. 34(3), 1231–1256 (2013)Effenberger, C., Kressner, D.: Chebyshev interpolation for nonlinear eigenvalue problems. BIT 52(4), 933–951 (2012)Fokkema, D.R., Sleijpen, G.L.G., 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)Guo, J.S., Lin, W.W., Wang, C.S.: Numerical solutions for large sparse quadratic eigenvalue problems. Linear Algebra Appl. 225, 57–89 (1995)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)Higham, N.J., Al-Mohy, A.H.: Computing matrix functions. Acta Numer. 19, 159–208 (2010)Higham, N.J., Mackey, D.S., Tisseur, F.: The conditioning of linearizations of matrix polynomials. SIAM J. Matrix Anal. Appl. 28(4), 1005–1028 (2006)Hochbruck, M., Lochel, D.: A multilevel Jacobi–Davidson method for polynomial PDE eigenvalue problems arising in plasma physics. SIAM J. Sci. Comput. 32(6), 3151–3169 (2010)Hochstenbach, M.E., Sleijpen, G.L.G.: Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem. Numer. Linear Algebra Appl. 15(1), 35–54 (2008)Huang, T.M., Hwang, F.N., Lai, S.H., Wang, W., Wei, Z.H.: A parallel polynomial Jacobi–Davidson approach for dissipative acoustic eigenvalue problems. Comput. Fluids 45(1), 207–214 (2011)Hwang, F.N., Wei, Z.H., Huang, T.M., Wang, W.: A parallel additive Schwarz preconditioned Jacobi–Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation. J.Comput. Phys. 229(8), 2932–2947 (2010)Kressner, D.: A block Newton method for nonlinear eigenvalue problems. Numer. Math. 114, 355–372 (2009)Kressner, D., Roman, J.E.: Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis. Numer. Linear Algebra Appl. 21(4), 569–588 (2014)Lancaster, P.: Linearization of regular matrix polynomials. Electron. J. Linear Algebra 17, 21–27 (2008)Matsuo, Y., Guo, H., Arbenz, P.: Experiments on a parallel nonlinear Jacobi–Davidson algorithm. Procedia Comput. Sci. 29, 565–575 (2014)Meerbergen, K.: Locking and restarting quadratic eigenvalue solvers. SIAM J. Sci. Comput. 22(5), 1814–1839 (2001)Roman, J.E., Campos, C., Romero, E., Tomas, A.: SLEPc users manual. Technical report DSIC-II/24/02—Revision 3.10, D. Sistemes Informàtics i Computació, Universitat Politècnica de València (2018)Romero, E., Roman, J.E.: A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Trans. Math. Softw. 40(2), 13:1–13:29 (2014)Rommes, J., Martins, N.: Computing transfer function dominant poles of large-scale second-order dynamical systems. SIAM J. Sci. Comput. 30(4), 2137–2157 (2008)Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM Publications, Philadelphia (2003)Sensiau, C., Nicoud, F., van Gijzen, M., van Leeuwen, J.W.: A comparison of solvers for quadratic eigenvalue problems from combustion. Int. J. Numer. Methods Fluids 56(8), 1481–1488 (2008)Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17(2), 401–425 (1996)Sleijpen, G.L.G., Booten, A.G.L., Fokkema, D.R., van der Vorst, H.A.: Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36(3), 595–633 (1996)Sleijpen, G.L.G., van der Vorst, H.A., Meijerink, E.: Efficient expansion of subspaces in the Jacobi–Davidson method for standard and generalized eigenproblems. Electron. Trans. Numer. Anal. 7, 75–89 (1998)Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43(2), 235–286 (2001)van Gijzen, M.B., Raeven, F.: The parallel computation of the smallest eigenpair of an acoustic problem with damping. Int. J. Numer. Methods Eng. 45(6), 765–777 (1999)van Noorden, T., Rommes, J.: Computing a partial generalized real Schur form using the Jacobi–Davidson method. Numer. Linear Algebra Appl. 14(3), 197–215 (2007)Voss, H.: A Jacobi–Davidson method for nonlinear and nonsymmetric eigenproblems. Comput. Struct. 85(17–18), 1284–1292 (2007

    Parallel iterative refinement in polynomial eigenvalue problems

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    This is the peer reviewed version of the following article: Campos, C., and Roman, J. E. (2016) Parallel iterative refinement in polynomial eigenvalue problems. Numer. Linear Algebra Appl., 23: 730–745, which has been published in final form at http://dx.doi.org/10.1002/nla.2052. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-ArchivingMethods for the polynomial eigenvalue problem sometimes need to be followed by an iterative refinement process to improve the accuracy of the computed solutions. This can be accomplished by means of a Newton iteration tailored to matrix polynomials. The computational cost of this step is usually higher than the cost of computing the initial approximations, due to the need of solving multiple linear systems of equations with a bordered coefficient matrix. An effective parallelization is thus important, and we propose different approaches for the message-passing scenario. Some schemes use a subcommunicator strategy in order to improve the scalability whenever direct linear solvers are used. We show performance results for the various alternatives implemented in the context of SLEPc, the Scalable Library for Eigenvalue Problem Computations.This work was partially supported by the Spanish Ministry of Economy and Competitiveness under grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU grant with reference AP2012-0608. The computational experiments of Section 5 were carried out on the supercomputer Tirant at Universitat de Valencia.Campos, C.; Román Moltó, JE. (2016). Parallel iterative refinement in polynomial eigenvalue problems. Numerical Linear Algebra with Applications. 23(4):730-745. https://doi.org/10.1002/nla.2052S73074523

    Acoustic modal analysis with heat release fluctuations using nonlinear eigensolvers

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    Closed combustion devices like gas turbines and rockets are prone to thermoacoustic instabilities. Design engineers in the industry need tools to accurately identify and remove instabilities early in the design cycle. Many different approaches have been developed by the researchers over the years. In this work we focus on the Helmholtz wave equation based solver which is found to be relatively fast and accurate for most applications. This solver has been a subject of study in many previous works. The Helmholtz wave equation in frequency space reduces to a nonlinear eigenvalue problem which needs to be solved to compute the acoustic modes. Most previous implementations of this solver have relied on linearized solvers and iterative methods which as shown in this work are not very efficient and sometimes inaccurate. In this work we make use of specialized algorithms implemented in SLEPc that are accurate and efficient for computing eigenvalues of nonlinear eigenvalue problems. We make use of the n-tau model to compute the reacting source terms in the Helmholtz equation and describe the steps involved in deriving the Helmholtz eigenvalue equation and obtaining its solution using the SLEPc library

    Calculation of multiple eigenvalues of the neutron diffusion equation discretized with a parallelized finite volume method

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    [EN] The spatial distribution of the neutron flux within the core of nuclear reactors is a key factor in nuclear safety. The easiest and fastest way to determine it is by solving the eigenvalue problem of the neutron diffusion equation, which only contains spatial derivatives. The approximation of these derivatives is performed by discretizing the geometry and using numerical methods. In this work, the authors used a finite volume method based on a polynomial expansion of the neutron flux. Once these terms are discretized, a set of matrix equations is obtained, which constitutes the eigenvalue problem. A very effective class of methods for the solution of eigenvalue problems are those based on projection onto a low-dimensional subspace, such as Krylov subspaces. Thus, the SLEPc library was used for solving the eigenvalue problem by means of the Krylov-Schur method, which also uses projection methods of PETSc for solving linear systems. This work includes a complete sensitivity analysis of different issues: mesh, polynomial terms, linear systems solvers and parallelization.This work has been partially supported by the Spanish Ministerio de Eduacion Cultura y Deporte under the grant FPU13/01009, the Spanish Ministerio de Ciencia e Innovacion under the project ENE2014-59442-P, the Spanish Ministerio de Economia y Competitividad and the European Fondo Europeo de Desarrollo Regional (FEDER) under the project ENE2015-68353-P (MINECO/FEDER), the Generalitat Valenciana under the project PROMETEOII/2014/008, and the Spanish Ministerio de Economia y Competitividad and the European Fondo Europeo de Desarrollo Regional (FEDER) under the project TIN2016-075985-P.Bernal-Garcia, A.; Roman, JE.; Miró Herrero, R.; Verdú Martín, GJ. (2018). Calculation of multiple eigenvalues of the neutron diffusion equation discretized with a parallelized finite volume method. Progress in Nuclear Energy. 105:271-278. https://doi.org/10.1016/j.pnucene.2018.02.006S27127810

    A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

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    In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. 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    Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems

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    [EN] In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos-based method for computing all real eigenvalues contained in a given interval of large-scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory-efficient by representing the computed pseudo-Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.Agencia Estatal de Investigacion, Grant/Award Number: TIN2016-75985-PCampos, C.; Román Moltó, JE. (2020). 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    Eigenvalue computations in the context of data-sparse approximations of integral operators

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    In this work, we consider the numerical solution of a large eigenvalue problem resulting from a finite rank discretization of an integral operator. We are interested in computing a few eigenpairs, with an iterative method, so a matrix representation that allows for fast matrix-vector products is required. Hierarchical matrices are appropriate for this setting, and also provide cheap LU decompositions required in the spectral transformation technique. We illustrate the use of freely available software tools to address the problem, in particular SLEPc for the eigensolvers and HLib for the construction of H-matrices. The numerical tests are performed using an astrophysics application. Results show the benefits of the data-sparse representation compared to standard storage schemes, in terms of computational cost as well as memory requirements.This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion under projects TIN2009-07519, TIN2012-32846 and AIC10-D-000600 and by Fundacao para a Ciencia e a Tecnologia - FCT under project FCT/MICINN proc 441.00.Román Moltó, JE.; Vasconcelos, PB.; Nunes, AL. (2013). Eigenvalue computations in the context of data-sparse approximations of integral operators. Journal of Computational and Applied Mathematics. 237(1):171-181. doi:10.1016/j.cam.2012.07.021S171181237
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