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

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

[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

Acoustic modal analysis with heat release fluctuations using nonlinear eigensolvers

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

Thick-restarted joint Lanczos bidiagonalization for the GSVD

The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov subspaces. We consider the joint Lanczos bidiagonalization method, and analyze the feasibility of adapting the thick restart technique that is being used successfully in the context of other linear algebra problems. Numerical experiments illustrate the effectiveness of the proposed method. We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance. The analysis is done using a parallel implementation in the SLEPc library

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

[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

Multigroup neutron diffusion equation with the finite volume method in reactors using MOX fuels

[EN] The use of mixed oxide (MOX) fuel to partially fill the cores of commercial light water reactors (LWRs) gives rise to a reduction of the radioactive waste and production of more energy. However, the use of MOX fuels in LWRs changes the physics characteristics of the reactor core, since the variation with energy of the cross sections for the plutonium isotopes is more complex than for the uranium isotopes. Although the neutron diffusion theory could be applied to reactors using MOX fuels, more emphasis on treatment of the energy discretization should be placed. This energy discretization could be typically 4¿8 energy groups, instead of the standard 2-energy group approach. In this work, the authors developed a finite volume method for discretizing the general multigroup neutron diffusion equation. This method solves the eigenvalue problem by using Krylov projection methods, in which the size of the vectors used for building the Krylov subspace does not depend on the number of energy groups, but it can solve the multigroup formulation with upscattering and fission production terms in several energy groups. The method was applied to MOX reactors for its validation. © 2017 Atomic Energy Society of Japan. All rights reserved.This work has been partially supported by the Spanish Ministerio de Eduacion Cultura y Deporte [grant number FPU13/01009]; the Spanish Ministerio de Ciencia e Innovacion [project ENE2014-59442-P]; the Spanish Ministerio de Economia y Competitividad and the European Fondo Europeo de Desarrollo Regional (MINECO/FEDER) [project ENE2015-68353-P]; the Generalitat Valenciana [project PROMETEOII/2014/008]; and the Spanish Ministerio de Economia y Competitividad [project TIN2016-75985-P].Bernal-Garcia, A.; Roman, JE.; Miró Herrero, R.; Verdú Martín, GJ. (2017). Multigroup neutron diffusion equation with the finite volume method in reactors using MOX fuels. Journal of Nuclear Science and Technology. 54(11):1251-1260. https://doi.org/10.1080/00223131.2017.1359120S12511260541