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Mathematics of Computation

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Spectral discretization errors in filtered subspace iteration


Authors: Jay Gopalakrishnan, Luka Grubišić and Jeffrey Ovall
Journal: Math. Comp. 89 (2020), 203-228
MSC (2010): Primary 35P15, 65F15; Secondary 65M12, 47A75
DOI: https://doi.org/10.1090/mcom/3483
Published electronically: September 4, 2019
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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.


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Additional Information

Jay Gopalakrishnan
Affiliation: Fariborz Maseeh Department of Mathematics, Portland State University, PO Box 751, Portland, Oregon 97207-0751
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
Email: jovall@pdx.edu

DOI: https://doi.org/10.1090/mcom/3483
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.
Article copyright: © Copyright 2019 American Mathematical Society