An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems

Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. \ud This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method

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

Parallel iterative refinement in polynomial eigenvalue problems

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

Computing the common zeros of two bivariate functions via Bezout resultants

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (�$\ge$ 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions

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. 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

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

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