H-matrix (iterative method)

An H-matrix, in the context of iterative methods for solving large linear systems, refers to a matrix with a hierarchical low-rank structure. These matrices arise frequently in the discretization of boundary integral equations and other problems involving non-local operators. The "H" designation stands for "hierarchical," emphasizing the tree-like structure used to represent the matrix.

The key idea behind H-matrix methods is to approximate certain off-diagonal blocks of the matrix by low-rank representations, such as singular value decompositions (SVD) or related approximations. This leverages the fact that, in many physical problems, interactions between well-separated regions are often smooth and can be accurately represented with relatively few degrees of freedom.

An H-matrix is constructed by recursively partitioning the index set of the matrix into clusters. Each block of the matrix corresponding to interactions between distant clusters is approximated by a low-rank matrix. The remaining blocks, typically corresponding to nearby interactions, are treated as dense matrices. This hierarchical decomposition significantly reduces the storage requirements and computational complexity of matrix-vector multiplications and other linear algebra operations compared to storing and manipulating the full matrix.

When used within an iterative solver, such as the Conjugate Gradient method or GMRES, the H-matrix structure allows for efficient approximate matrix-vector multiplications. Instead of performing a dense matrix-vector product (which would have a complexity of O(N^2) for an N x N matrix), the hierarchical structure is exploited to achieve a computational complexity closer to O(N log^p N), where p is a small integer. This reduction in complexity makes it feasible to solve very large-scale problems that would be intractable with dense matrix representations.

The construction of the H-matrix requires careful consideration of the partitioning strategy, the choice of low-rank approximation method, and the selection of appropriate ranks for each block. Adaptive methods can be used to determine the rank based on a specified error tolerance. Different variants of H-matrices exist, such as H^2-matrices and HSS-matrices, which offer different trade-offs between storage, complexity, and accuracy.

H-matrix methods are particularly well-suited for problems arising in computational electromagnetics, acoustics, fluid dynamics, and other areas where boundary integral equation formulations are used. They provide a powerful tool for solving large-scale scientific and engineering problems efficiently and accurately.