A Sparse Factorization for Fast Computation of Localizing Modes

Abstract

Localizing modes provide an effective basis for developing efficient, error-controlled factorizations of the dense matrices encountered in integral equation formulations of wave phenomena at low to moderate frequencies. An essential component of these factorization algorithms is the numerical determination of the underlying localizing modes. This communication describes the details of an efficient procedure for computing the so-called non-overlapping, localizing modes. The principle component of the procedure is a QR-like factorization of the sparse multilevel data structure used to compress discrete integral operators. Numerical examples demonstrate the performance of the algorithm.

Diagonal Factorization of Integral Equation Matrices via Localizing Sources and Orthogonally Matched Receivers

“…As discussed elsewhere [12,13], this is accomplished by first building a data-sparse H2 representation of Z [16]. The methods used to efficiently manipulate the H2 representation to determine the submatrices Λ l,L , Λ l,R , P l,L , and P l,R ofX l andȲ l are detailed in [12,17]. When the error control procedure is used, the modification indicated by Eq.…”

“…The matrices Λ l and P l in (2) are square, permuted block-diagonal matrices (e.g., see Figure 1 are represented using an l-level H2 data structure [16]. The procedure used to efficiently manipulate the H2 data structures to compute the localizing functions in Λ (L) l is discussed in detail in [12,17]. All other matrices, including Λ l and P l above, and X l and Y l below, are either block diagonal or permuted block diagonal matrices.…”

“…(2), and P l,L is the sub-matrix P (N ) l . Both Λ l,R and P l,L are obtained analyzing column blocks of Z (NN) l using the algebraic methods detailed in [12,17]. The sub-matrices Λ l,L and P l,R are obtained by performing the same algebraic analysis on row-blocks of Z l .…”

“…In computing the diagonal blocks of Λ l,L and Λ l,R from the H2 representation of Z l+1 using the algebraic procedure of [12,17], the number of localizing vectors in λ m l,L and λ m l,R may be different due to the asymmetry of Z l+1 . To maintain square b m l and Z l in Eq.…”

Error Analysis for Matrix Factorizations Using Non-Overlapped Localizing Basis Functions

“…The principle component of that procedure is a QR-like factorization strategy for a particular sparse representation of a system matrix. This procedure is referred to as a 2R factorization [32]. Additional improvements that reduce the CPU costs associated with the NL-LOGOS factorization have also been described elsewhere [33], [34].…”