Sparse Matrix Factorization Using Overlapped Localizing LOGOS Modes on a Shifted Grid

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

“…Different factorization algorithms can be developed using localizing transformations [8][9][10][11][12][13]. This section briefly reviews a triangular factorization [10][11][12].…”

“…The set of approaches considered here are based on the use of localizing functions [8][9][10][11][12][13]. Localization-based direct solvers incorporate source functions having support in individual groups (of an underlying multilevel tree) that localize their scattered fields to the group that contains the source.…”

“…Localization-based direct solvers incorporate source functions having support in individual groups (of an underlying multilevel tree) that localize their scattered fields to the group that contains the source. In one approach, localizing sources have been combined with orthogonally matched receivers to obtain an upper-triangular factorization [10][11][12][13]. In the following, this approach is extended to obtain a data-sparse, block-diagonal representation of the system matrix.…”

“…The system matrix, Z, is determined using a moment method to discretize the underlying integral operator [15]. As described in [12,13], a multilevel tree is used to organize interactions on the simulation domain. The number of levels in the tree is L, and tree levels are indexed from 1 to L. Level-1 is the coarsest (root) level of the tree, and level-L is the finest level.…”

“…In the following, a multilevel, data-sparse, H2 representation [16] is used to represent Z. The procedure used to construct the H2 representation is discussed in [13].…”