Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Novel memory-efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree $d$. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor $d$. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so called Q-Arnoldi and TOAR methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift-and-invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree $30$ arising from the interpolation of nonlinear eigenvalue problems which stem from boundary element discretizations of PDE eigenvalue problems.Daniel Kressner; Román Moltó, JE. (2014). Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis. Numerical Linear Algebra with Applications. 21(4):569-588. doi:10.1002/nla.1913S569588214Mackey, D. S., Mackey, N., Mehl, C., & Mehrmann, V. (2006). Vector Spaces of Linearizations for Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 28(4), 971-1004. doi:10.1137/050628350Mackey, D. S., Mackey, N., Mehl, C., & Mehrmann, V. (2006). Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations. SIAM Journal on Matrix Analysis and Applications, 28(4), 1029-1051. doi:10.1137/050628362Higham, N. J., Mackey, D. S., & Tisseur, F. (2006). The Conditioning of Linearizations of Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 28(4), 1005-1028. doi:10.1137/050628283Adhikari, B., Alam, R., & Kressner, D. (2011). Structured eigenvalue condition numbers and linearizations for matrix polynomials. Linear Algebra and its Applications, 435(9), 2193-2221. doi:10.1016/j.laa.2011.04.020Bai, Z., & Su, Y. (2005). SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 26(3), 640-659. doi:10.1137/s0895479803438523Meerbergen, K. (2009). The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 30(4), 1463-1482. doi:10.1137/07069273xLin, Y., Bao, L., & Wei, Y. (2010). Model-order reduction of large-scalekth-order linear dynamical systems via akth-order Arnoldi method. International Journal of Computer Mathematics, 87(2), 435-453. doi:10.1080/00207160802130164Stewart, G. W. (2001). Matrix Algorithms. doi:10.1137/1.9780898718058Kamiya, N., Andoh, E., & Nogae, K. (1993). Eigenvalue analysis by the boundary element method: new developments. Engineering Analysis with Boundary Elements, 12(3), 151-162. doi:10.1016/0955-7997(93)90011-9Bindel D Hood A Localization theorems for nonlinear eigenvalues. arXiv: 1303.4668 2013Botchev, M. A., Sleijpen, G. L. G., & Sopaheluwakan, A. (2009). An SVD-approach to Jacobi–Davidson solution of nonlinear Helmholtz eigenvalue problems. Linear Algebra and its Applications, 431(3-4), 427-440. doi:10.1016/j.laa.2009.03.024Effenberger, C., & Kressner, D. (2012). Chebyshev interpolation for nonlinear eigenvalue problems. BIT Numerical Mathematics, 52(4), 933-951. doi:10.1007/s10543-012-0381-5Van Beeumen, R., Meerbergen, K., & Michiels, W. (2013). A Rational Krylov Method Based on Hermite Interpolation for Nonlinear Eigenvalue Problems. SIAM Journal on Scientific Computing, 35(1), A327-A350. doi:10.1137/120877556Sitton, G. A., Burrus, C. S., Fox, J. W., & Treitel, S. (2003). Factoring very-high-degree polynomials. IEEE Signal Processing Magazine, 20(6), 27-42. doi:10.1109/msp.2003.1253552Amiraslani, A., Corless, R. M., & Lancaster, P. (2008). Linearization of matrix polynomials expressed in polynomial bases. IMA Journal of Numerical Analysis, 29(1), 141-157. doi:10.1093/imanum/drm051Betcke, T., & Kressner, D. (2011). Perturbation, extraction and refinement of invariant pairs for matrix polynomials. Linear Algebra and its Applications, 435(3), 514-536. doi:10.1016/j.laa.2010.06.029Beyn, W. J., & Thümmler, V. (2010). Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 31(3), 1361-1381. doi:10.1137/080723107Kressner, D. (2009). A block Newton method for nonlinear eigenvalue problems. Numerische Mathematik, 114(2), 355-372. doi:10.1007/s00211-009-0259-xLehoucq, R. B., Sorensen, D. C., & Yang, C. (1998). ARPACK Users’ Guide. doi:10.1137/1.9780898719628Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Clenshaw, C. W. (1955). A note on the summation of Chebyshev series. Mathematics of Computation, 9(51), 118-118. doi:10.1090/s0025-5718-1955-0071856-0Stewart, G. W. (2002). A Krylov--Schur Algorithm for Large Eigenproblems. SIAM Journal on Matrix Analysis and Applications, 23(3), 601-614. doi:10.1137/s0895479800371529Su Y A compact Arnoldi algorithm for polynomial eigenvalue problems 2008 http://math.cts.nthu.edu.tw/Mathematics/RANMEP%20Slides/Yangfeng%20Su.pdfSteinbach, O., & Unger, G. (2009). A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator. Numerische Mathematik, 113(2), 281-298. doi:10.1007/s00211-009-0239-1Effenberger, C., Kressner, D., Steinbach, O., & Unger, G. (2012). Interpolation-based solution of a nonlinear eigenvalue problem in fluid-structure interaction. PAMM, 12(1), 633-634. doi:10.1002/pamm.201210305Betcke, T., Higham, N. J., Mehrmann, V., Schröder, C., & Tisseur, F. (2013). NLEVP. ACM Transactions on Mathematical Software, 39(2), 1-28. doi:10.1145/2427023.2427024Grammont, L., Higham, N. J., & Tisseur, F. (2011). A framework for analyzing nonlinear eigenproblems and parametrized linear systems. Linear Algebra and its Applications, 435(3), 623-640. doi:10.1016/j.laa.2009.12.03

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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Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems

We investigate multi-grid methods for solving linear systems arising from arc-length continuation techniques applied to nonlinear elliptic eigenvalue problems. We find that the usual multi-grid methods diverge in the neighborhood of singular points of the solution branches. As a result, the continuation method is unable to continue past a limit point in the Bratu problem. This divergence is analyzed and a modified multi-grid algorithm has been devised based on this analysis. In principle, this new multi-grid algorithm converges for elliptic systems, arbitrarily close to singularity and has been used successfully in conjunction with arc-length continuation procedures on the model problem. In the worst situation, both the storage and the computational work are only about a factor of two more than the unmodified multi-grid methods

Spectra, thresholds, and modal fields of a circular microcavity laser transforming into a square

© 2015 IEEE. We investigate the lasing spectra, threshold gain values, and modal fields for a two-dimensional microcavity laser with a square contour defined by the super-circle equation. The cavity modes are considered accurately using the linear electromagnetic formalism of the Lasing Eigenvalue Problem (LEP) with exact boundary and radiation conditions. We reduce the original problem to a nonlinear eigenvalue problem for the Muller boundary integral equation and build a numerical algorithm of its solution. Computations demonstrate that under the deformation of microcavity from the circle to a square there exist modes that preserve low thresholds

Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method

The partial Schur factorization can be used to represent several eigenpairs of a matrix in a numerically robust way. Different adaptions of the Arnoldi method are often used to compute partial Schur factorizations. We propose here a technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP). The technique is inspired by the algorithm in [8], now called the infinite Arnoldi method. The infinite Arnoldi method is a method designed for NEPs, and can be interpreted as Arnoldi's method applied to a linear infinite-dimensional operator, whose reciprocal eigenvalues are the solutions to the NEP. As a first result we show that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. We characterize the structure of the invariant pairs of the operator and show how one can carry out a modification of the infinite Arnoldi method by respecting the structure. This also allows us to naturally add the feature known as locking. We nest this algorithm with an outer iteration, where the infinite Arnoldi method for a particular type of structured functions is appropriately restarted. The restarting exploits the structure and is inspired by the well-known implicitly restarted Arnoldi method for standard eigenvalue problems. The final algorithm is applied to examples from a benchmark collection, showing that both processing time and memory consumption can be considerably reduced with the restarting technique