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Parallel Symmetric Eigenvalue Problem Solvers

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Technical Report

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Purdue University West Lafayette

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Sparse symmetric eigenvalue problems arise in many computational science and engineering applications such as structural mechanics, nanoelectronics, and spectral reordering. Often, the large size of these problems requires the development of Eigensolvers that scale well on parallel computing platforms. In this thesis, we describe two such eigen solvers TraceMin and TraceMin-Davidson. These methods are different from many existing eigensolvers in that they do not require accurate linear solvers to be performed in each iteration in order to obtain accurate estimates of the smallest eigenvalues and their corresponding eigenvectors. We also develop effective solvers for the saddle-point problems that arise in each outer TraceMin iteration. In addition, we present parallel implementations for both solvers for seeking either few of the smallest eigenpairs or seeking a large number of eigenpairs in any interval of the spectrum. Numerical experiments demonstrate clearly that Trace Minimization is a very effective parallel eigensolver compared to i Krylov-Schur, ii LOBPCG, iii Jacobi-Davidson a TraceMin-like scheme developed 15 years after TraceMin, and iv FEAST.

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