units of storage by storing only its non-zero entries, a non-sparse matrix would require
Hierarchical matrices provide an approximation requiring only
Compared to many other data-sparse representations of non-sparse matrices, hierarchical matrices offer a major advantage: the results of matrix arithmetic operations like matrix multiplication, factorization or inversion can be approximated in
[2] Hierarchical matrices rely on local low-rank approximations: let
units of storage, the factorized representation requires only
is not too large, the storage requirements are reduced significantly.
Large submatrices are stored in factorized representation, while small submatrices are stored in standard representation in order to improve efficiency.
Low-rank matrices are closely related to degenerate expansions used in panel clustering and the fast multipole method to approximate integral operators.
In this sense, hierarchical matrices can be considered the algebraic counterparts of these techniques.
Hierarchical matrices are successfully used to treat integral equations, e.g., the single and double layer potential operators appearing in the boundary element method.
are families of finite element basis functions.
is sufficiently smooth, we can approximate it by polynomial interpolation to obtain where
Of particular interest are cross approximation techniques[6][7][15] that use only the entries of the original matrix
Since the solution operator of an elliptic partial differential equation can be expressed as an integral operator involving Green's function, it is not surprising that the inverse of the stiffness matrix arising from the finite element method and spectral method can be approximated by a hierarchical matrix.
Green's function depends on the shape of the computational domain, therefore it is usually not known.
Surprisingly, it is possible to prove[11][12][13][14] that the inverse can be approximated even if the differential operator involves non-smooth coefficients and Green's function is therefore not smooth.
The most important innovation of the hierarchical matrix method is the development of efficient algorithms for performing (approximate) matrix arithmetic operations on non-sparse matrices, e.g., to compute approximate inverses, LU decompositions and solutions to matrix equations.
The central algorithm is the efficient matrix-matrix multiplication, i.e., the computation of
The algorithm requires the submatrices of the hierarchical matrices to be organized in a block tree structure and takes advantage of the properties of factorized low-rank matrices to compute the updated
Taking advantage of the block structure, the inverse can be computed by using recursion to compute inverses and Schur complements of diagonal blocks and combining both using the matrix-matrix multiplication.
In a similar way, the LU decomposition[16][17] can be constructed using only recursion and multiplication.
In order to treat very large problems, the structure of hierarchical matrices can be improved: H2-matrices[18][19] replace the general low-rank structure of the blocks by a hierarchical representation closely related to the fast multipole method in order to reduce the storage complexity to
In the context of boundary integral operators, replacing the fixed rank
by block-dependent ranks leads to approximations that preserve the rate of convergence of the underlying boundary element method at a complexity of
[20][21] Arithmetic operations like multiplication, inversion, and Cholesky or LR factorization of H2-matrices can be implemented based on two fundamental operations: the matrix-vector multiplication with submatrices and the low-rank update of submatrices.
While the matrix-vector multiplication is straightforward, implementing efficient low-rank updates with adaptively optimized cluster bases poses a significant challenge.
[22] HLib is a C software library implementing the most important algorithms for hierarchical and
AHMED is a C++ software library that can be downloaded for educational purposes.
HLIBpro is an implementation of the core hierarchical matrix algorithms for commercial applications.
H2Lib is an open source implementation of hierarchical matrix algorithms intended for research and teaching.