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Analysis of FEAST Spectral Approximations Using the DPG Discretization

Jay Gopalakrishnan ORCID logo , Luka Grubišić ORCID logo , Jeffrey Ovall ORCID logo and Benjamin Parker ORCID logo EMAIL logo


A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as “FEAST”, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov–Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.

Award Identifier / Grant number: FA9451-18-2-0031

Award Identifier / Grant number: HRZZ-9345

Award Identifier / Grant number: DMS-1522471

Award Identifier / Grant number: DMS-1624776

Funding statement: This work was supported by the AFOSR (through AFRL Cooperative Agreement #18RDCOR018, under grant FA9451-18-2-0031), the Croatian Science Foundation grant HRZZ-9345, bilateral Croatian-USA grant (administered jointly by Croatian-MZO and NSF), and NSF grant DMS-1522471. The numerical studies were facilitated by the equipment acquired using NSF’s Major Research Instrumentation grant DMS-1624776.


[1] W.-J. Beyn, An integral method for solving nonlinear eigenvalue problems, Linear Algebra Appl. 436 (2012), no. 10, 3839–3863. 10.1016/j.laa.2011.03.030Search in Google Scholar

[2] T. Bouma, J. Gopalakrishnan and A. Harb, Convergence rates of the DPG method with reduced test space degree, Comput. Math. Appl. 68 (2014), no. 11, 1550–1561. 10.1016/j.camwa.2014.08.004Search in Google Scholar

[3] T. Bühler and D. A. Salamon, Functional Analysis, Grad. Stud. Math. 191, American Mathematical Society, Providence, 2018. 10.1090/gsm/191Search in Google Scholar

[4] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, A posteriori error control for DPG methods, SIAM J. Numer. Anal. 52 (2014), no. 3, 1335–1353. 10.1137/130924913Search in Google Scholar

[5] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl. 72 (2016), no. 3, 494–522. 10.1016/j.camwa.2016.05.004Search in Google Scholar

[6] C. Carstensen and F. Hellwig, Optimal convergence rates for adaptive lowest-order discontinuous Petrov–Galerkin schemes, SIAM J. Numer. Anal. 56 (2018), no. 2, 1091–1111. 10.1137/17M1146671Search in Google Scholar

[7] L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov–Galerkin methods. II. Optimal test functions, Numer. Methods Partial Differential Equations 27 (2011), no. 1, 70–105. 10.1002/num.20640Search in Google Scholar

[8] L. Demkowicz and J. Gopalakrishnan, A primal DPG method without a first-order reformulation, Comput. Math. Appl. 66 (2013), no. 6, 1058–1064. 10.21236/ADA587915Search in Google Scholar

[9] J. Gopalakrishnan, L. Grubišić and J. Ovall, Spectral discretization errors in filtered subspace iteration, preprint (2018), 10.1090/mcom/3483Search in Google Scholar

[10] J. Gopalakrishnan and B. Q. Parker, Pythonic FEAST, Software hosted at Bitbucket, Search in Google Scholar

[11] J. Gopalakrishnan and W. Qiu, An analysis of the practical DPG method, Math. Comp. 83 (2014), no. 286, 537–552. 10.1090/S0025-5718-2013-02721-4Search in Google Scholar

[12] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. Search in Google Scholar

[13] S. Güttel, E. Polizzi, P. T. P. Tang and G. Viaud, Zolotarev quadrature rules and load balancing for the FEAST eigensolver, SIAM J. Sci. Comput. 37 (2015), no. 4, A2100–A2122. 10.1137/140980090Search in Google Scholar

[14] A. Horning and A. Townsend, Feast for differential eigenvalue problems, preprint (2019), 10.1137/19M1238708Search in Google Scholar

[15] T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995. 10.1007/978-3-642-66282-9Search in Google Scholar

[16] E. Polizzi, A density matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B 79 (2009), Article ID 115112. 10.1103/PhysRevB.79.115112Search in Google Scholar

[17] G. A. Reider, Photonics: An Introduction, Springer, Cham, 2016. 10.1007/978-3-319-26076-1Search in Google Scholar

[18] Y. Saad, Analysis of subspace iteration for eigenvalue problems with evolving matrices, SIAM J. Matrix Anal. Appl. 37 (2016), no. 1, 103–122. 10.1137/141002037Search in Google Scholar

[19] T. Sakurai and H. Sugiura, A projection method for generalized eigenvalue problems using numerical integration, Comput. Appl. Math. 159 (2003), no. 1, 119–128. 10.1016/S0377-0427(03)00565-XSearch in Google Scholar

[20] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Grad. Texts in Math. 265, Springer, Dordrecht, 2012. 10.1007/978-94-007-4753-1Search in Google Scholar

[21] J. Schöberl, NGSolve, Search in Google Scholar

[22] L. N. Trefethen and T. Betcke, Computed eigenmodes of planar regions, Recent Advances in Differential Equations and Mathematical Physics, Contemp. Math. 412, American Mathematical Society, Providence (2006), 297–314. 10.1090/conm/412/07783Search in Google Scholar

Received: 2019-01-24
Revised: 2019-02-08
Accepted: 2019-02-08
Published Online: 2019-03-27
Published in Print: 2019-04-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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