A Domain Decomposition Rayleigh-Ritz Algorithm For Symmetric Generalized Eigenvalue Problems

SIAM JOURNAL ON SCIENTIFIC COMPUTING(2020)

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Abstract
This paper proposes a parallel domain decomposition Rayleigh-Ritz projection scheme to compute a selected number of eigenvalues (and, optionally, associated eigenvectors) of large and sparse symmetric pencils. The projection subspace associated with interface variables is built by computing a few of the eigenvectors and associated leading derivatives of a zeroth-order approximation of the nonlinear matrix-valued interface operator. On the other hand, the projection subspace associated with interior variables is built independently in each subdomain by exploiting local eigenmodes and matrix resolvent approximations. The sought eigenpairs are then approximated by a Rayleigh-Ritz projection onto the subspace formed by the union of these two subspaces. Several theoretical and practical details are discussed, and upper bounds of the approximation errors are provided. Our numerical experiments demonstrate the efficiency of the proposed technique on sequential/distributed memory architectures as well as its competitiveness against schemes such as shift-and-invert Lanczos and automated multilevel substructuring combined with p-way vertex-based partitionings.
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Key words
symmetric generalized eigenvalue problem, domain decomposition, high-performance computing, spectral Schur complement, Rayleigh-Ritz
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