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- W1890964192 abstract "N. Saito recently proposed a new method to analyze and represent data recorded on a domain Ω of general shape in Rd by expanding the data into a set of the eigenfunctions of Laplacian defined over Ω. Instead of directly solving the Laplacian eigenvalue problem on Ω via the Helmholtz equation (which can be quite complicated and costly), he found an integral operator commuting with the Laplacian, which shares the eigenfunctions with the Laplacian. After discretization, the eigenvalue problem for this integral operator is converted to that of a symmetric kernel matrix. A conventional approach of computing selected eigenvalues of such a matrix is, for instance, an implicitly re-started Lanezos method (IRLM), which costs O( kN2) floating point operations where k is related to the number of the selected eigenvalues and the distribution of the eigenvalues. This approach is prohibitively expensive for a large kernel matrix that is often generated by a dense sampling of the domain. The kernel function of this integral operator for a domain in R2 , however, is of the special form K( x, y) = – 12p log ||x – y||, i.e., the fundamental solution of the Laplacian. This kernel function represents the potential at point x due to a charge located at point y. This fact allows us to apply the ideas of Fast Multipole Method (FMM) and Hierarchical Semi-Separable (HSS) representation to exploit the structure of the kernel matrix. In this thesis, we investigate a hierarchical matrix-splitting scheme and the low rank property of the submatrices generated by the splitting scheme. By doing so, we design a hierarchical matrix decomposition method that leads to a fast algorithm of matrix-vector multiplication which costs O(N). By supplying our fast matrix-vector multiplication routine to IRLM, we obtain an eigenvalue solver that costs O(kN)." @default.
- W1890964192 created "2016-06-24" @default.
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- W1890964192 date "2007-01-01" @default.
- W1890964192 modified "2023-09-27" @default.
- W1890964192 title "On a fast algorithm for computing the laplacian eigenpairs via commuting integral operators" @default.
- W1890964192 hasPublicationYear "2007" @default.
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