## Spectral discretization errors in filtered subspace iteration

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Jay Gopalakrishnan, Luka Grubišić and Jeffrey Ovall
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## Abstract:

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite-dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates.## References

- P. M. Anselone and T. W. Palmer,
*Spectral analysis of collectively compact, strongly convergent operator sequences*, Pacific J. Math.**25**(1968), 423–431. MR**227807** - Anthony P. Austin and Lloyd N. Trefethen,
*Computing eigenvalues of real symmetric matrices with rational filters in real arithmetic*, SIAM J. Sci. Comput.**37**(2015), no. 3, A1365–A1387. MR**3352612**, DOI 10.1137/140984129 - I. Babuška and J. Osborn,
*Eigenvalue problems*, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR**1115240** - Wolf-Jürgen Beyn,
*An integral method for solving nonlinear eigenvalue problems*, Linear Algebra Appl.**436**(2012), no. 10, 3839–3863. MR**2914550**, DOI 10.1016/j.laa.2011.03.030 - Theo Bühler and Dietmar A. Salamon,
*Functional analysis*, Graduate Studies in Mathematics, vol. 191, American Mathematical Society, Providence, RI, 2018. MR**3823238**, DOI 10.1090/gsm/191 - Alexandre Ern and Jean-Luc Guermond,
*Theory and practice of finite elements*, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR**2050138**, DOI 10.1007/978-1-4757-4355-5 - Jay Gopalakrishnan, Luka Grubišić, Jeffrey Ovall, and Benjamin Parker,
*Analysis of FEAST spectral approximations using the DPG discretization*, Comput. Methods Appl. Math.**19**(2019), no. 2, 251–266. MR**3935888**, DOI 10.1515/cmam-2019-0030 - J. Gopalakrishnan and B. Q. Parker,
*Pythonic FEAST*. Software hosted at Bitbucket: https://bitbucket.org/jayggg/pyeigfeast, 2017. - P. Grisvard,
*Singularities in boundary value problems*, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR**1173209** - Pierre Grisvard,
*Elliptic problems in nonsmooth domains*, Classics in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original [ MR0775683]; With a foreword by Susanne C. Brenner. MR**3396210**, DOI 10.1137/1.9781611972030.ch1 - Luka Grubišić, Vadim Kostrykin, Konstantin A. Makarov, and Krešimir Veselić,
*Representation theorems for indefinite quadratic forms revisited*, Mathematika**59**(2013), no. 1, 169–189. MR**3028178**, DOI 10.1112/S0025579312000125 - S. Güttel,
*Rational Krylov methods for operator functions*, Dissertation Thesis, Bergakademie Freiberg, (2010). - Stefan Güttel, Eric Polizzi, Ping Tak Peter Tang, and Gautier Viaud,
*Zolotarev quadrature rules and load balancing for the FEAST eigensolver*, SIAM J. Sci. Comput.**37**(2015), no. 4, A2100–A2122. MR**3384838**, DOI 10.1137/140980090 - Ruihao Huang, Allan A. Struthers, Jiguang Sun, and Ruming Zhang,
*Recursive integral method for transmission eigenvalues*, J. Comput. Phys.**327**(2016), 830–840. MR**3564365**, DOI 10.1016/j.jcp.2016.10.001 - Akira Imakura, Lei Du, and Tetsuya Sakurai,
*Relationships among contour integral-based methods for solving generalized eigenvalue problems*, Jpn. J. Ind. Appl. Math.**33**(2016), no. 3, 721–750. MR**3579284**, DOI 10.1007/s13160-016-0224-x - David S. Jerison and Carlos E. Kenig,
*The Neumann problem on Lipschitz domains*, Bull. Amer. Math. Soc. (N.S.)**4**(1981), no. 2, 203–207. MR**598688**, DOI 10.1090/S0273-0979-1981-14884-9 - Tosio Kato,
*Perturbation theory for linear operators*, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR**1335452** - Andrew Knyazev, Abram Jujunashvili, and Merico Argentati,
*Angles between infinite dimensional subspaces with applications to the Rayleigh-Ritz and alternating projectors methods*, J. Funct. Anal.**259**(2010), no. 6, 1323–1345. MR**2659762**, DOI 10.1016/j.jfa.2010.05.018 - Andrew V. Knyazev and Merico E. Argentati,
*Rayleigh-Ritz majorization error bounds with applications to FEM*, SIAM J. Matrix Anal. Appl.**31**(2009), no. 3, 1521–1537. MR**2587790**, DOI 10.1137/08072574X - E. Polizzi,
*Density-matrix-based algorithms for solving eigenvalue problems*, Phys. Rev. B., 79 (2009), p. 115112. - Michael Reed and Barry Simon,
*Methods of modern mathematical physics. III*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR**529429** - Yousef Saad,
*Analysis of subspace iteration for eigenvalue problems with evolving matrices*, SIAM J. Matrix Anal. Appl.**37**(2016), no. 1, 103–122. MR**3530393**, DOI 10.1137/141002037 - Tetsuya Sakurai and Hiroshi Sugiura,
*A projection method for generalized eigenvalue problems using numerical integration*, Proceedings of the 6th Japan-China Joint Seminar on Numerical Mathematics (Tsukuba, 2002), 2003, pp. 119–128. MR**2022322**, DOI 10.1016/S0377-0427(03)00565-X - Alfred H. Schatz,
*An observation concerning Ritz-Galerkin methods with indefinite bilinear forms*, Math. Comp.**28**(1974), 959–962. MR**373326**, DOI 10.1090/S0025-5718-1974-0373326-0 - Konrad Schmüdgen,
*Unbounded self-adjoint operators on Hilbert space*, Graduate Texts in Mathematics, vol. 265, Springer, Dordrecht, 2012. MR**2953553**, DOI 10.1007/978-94-007-4753-1 - J. Schöberl,
*NGSolve*. http://ngsolve.org, 2017. - Ping Tak Peter Tang and Eric Polizzi,
*FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection*, SIAM J. Matrix Anal. Appl.**35**(2014), no. 2, 354–390. MR**3188390**, DOI 10.1137/13090866X - Lloyd N. Trefethen and Timo Betcke,
*Computed eigenmodes of planar regions*, Recent advances in differential equations and mathematical physics, Contemp. Math., vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 297–314. MR**2259116**, DOI 10.1090/conm/412/07783

## Additional Information

**Jay Gopalakrishnan**- Affiliation: Fariborz Maseeh Department of Mathematics, Portland State University, PO Box 751, Portland, Oregon 97207-0751
- MR Author ID: 661361
- Email: gjay@pdx.edu
**Luka Grubišić**- Affiliation: Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
- Email: luka.grubisic@math.hr
**Jeffrey Ovall**- Affiliation: Fariborz Maseeh Department of Mathematics, Portland State University, PO Box 751, Portland, Oregon 97207-0751
- MR Author ID: 728623
- Email: jovall@pdx.edu
- Received by editor(s): July 7, 2018
- Received by editor(s) in revised form: March 11, 2019
- Published electronically: September 4, 2019
- Additional Notes: This work was partially supported by the AFOSR (through AFRL Cooperative Agreement #18RDCOR018 and 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.
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp.
**89**(2020), 203-228 - MSC (2010): Primary 35P15, 65F15; Secondary 65M12, 47A75
- DOI: https://doi.org/10.1090/mcom/3483
- MathSciNet review: 4011540