This work presents a parallel implementation of the implicitly restarted Arnoldi/Lanczos method for the solution of eigenproblems approximated by the finite element method. The implicitly restarted Arnoldi/Lanczos uses a restart scheme in order to improve the convergence of the desired portion of the spectrum, addressing issues such as memory requirements and computational costs related to the generation and storage of the Krylov basis. The presented implementation is suitable for distributed memory architectures, especially PC clusters. In the parallel solution, a subdomain by subdomain approach was implemented and overlapping and non-overlapping mesh partitions were tested. Compressed data structures in the formats CSRC and CSRC/CSR were used to store the coefficient matrices. The parallelization of numerical linear algebra operations present in both Krylov and implicitly restarted methods are discussed. Numerical examples are shown, in order to point out the efficiency and applicability of the proposed method.
We consider the problem of developing an efficient multi-threaded implementation of the matrix-vector multiplication algorithm for sparse matrices with structural symmetry. Matrices are stored using the compressed sparse row-column format (CSRC), designed for profiting from the symmetric non-zero pattern observed in global finite element matrices. Unlike classical compressed storage formats, performing the sparse matrix-vector product using the CSRC requires thread-safe access to the destination vector. To avoid race conditions, we have implemented two partitioning strategies. In the first one, each thread allocates an array for storing its contributions, which are later combined in an accumulation step. We analyze how to perform this accumulation in four different ways. The second strategy employs a coloring algorithm for grouping rows that can be concurrently processed by threads. Our results indicate that, although incurring an increase in the working set size, the former approach leads to the best performance improvements for most matrices.
This paper presents a parallel implementation of the implicitly restarted Lanczos method for the solution of large and sparse eigenproblems that occur in modal analysis of complex structures using the finite element method. The implicitly restarted technique improves convergence of the desired eigenvalues without the penalty of lost of orthogonality keeping the number of factorization steps in a modest size. In the parallel solution, a subdomain by subdomain approach was implemented and overlapping and non-overlapping mesh partitions were used. Compressed data structures in the formats CSRC and CSRC/CSR were employed to store the global matrices coefficients. The parallelization of numerical linear algebra operations presented in both Krylov and implicitly restarted methods are discussed.
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