Construction and Arithmetics of H-Matrices

By Lars Grasedyck, Wolfgang Hackbusch. Computing, 2003

477 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/45219705c08c14dbfe565289596d8f58713cc8a3

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In Atomic and Molecular Physics, and Optics

In Computational Mechanics

In Computational Theory and Mathematics

Citations in the corpus

Efficient methods for large‐scale linear inversion using a geostatistical approach (2012, background): “[18] Hierarchical matrices [Börm et al., 2003; Grasedyck and Hackbusch, 2003; Rjasanow and Steinbach, 2007; Bebendorf, 2008] (or H-matrices, for short) are efficient data-sparse representations of certain densely populated matrices.”
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions (2009, methodology): “• Many fast numerical algorithms for solving PDEs and for rapidly evaluating potential fields such as the fast multipole method [66] and H-matrices [65], rely on low-rank approximations of continuum operators.”
Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators (2005, unknown): “[10], [13]-[15]).”
Randomized numerical linear algebra: Foundations and algorithms (2020, methodology): “These methods were generalized by Hackbusch and co-workers, who developed the H- and H2-matrix frameworks (Hackbusch 1999, Grasedyck and Hackbusch 2003, Hackbusch, Khoromskij and Sauter 2002).”
Introduction to Hierarchical Matrices with Applications (2003, methodology): “The starting index sets are τ = σ = I. The complexity for the matrix-vector multiplication (sparse block tree) is O(nk log(n)) ([10], [ 11 ]).”
ℋ︁‐LU factorization in preconditioners for augmented Lagrangian and grad‐div stabilized saddle point systems (2012, methodology): “In this paper, we propose to use an approximate LU factorization of A computed in the hierarchical (H-)matrix format [19, 20] to precondition this augmented block.”
Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling (2008, methodology): “1 Hierarchical matrix techniques The hierarchical matrix (H-matrix) technique [46, 50, 51, 55] (see also the mosaic-skeleton method [82]) allows an efficient treatment of dense matrices arising, e.”
Tensor-Product Approximation to Multidimensional Integral Operators and Green's Functions (2008, background): “The class of hierarchical (H) matrices allows an approximate matrix arithmetic with almost linear complexity [14, 15, 16, 12].”
Hierarchical Tensor-Product Approximation to the Inverse and Related Operators for High-Dimensional Elliptic Problems (2005, background): “A class of hierarchical ðHÞ matrices has been introduced and developed in [15][19], [11].”
Likelihood approximation with hierarchical matrices for large spatial datasets (2017, background): “2 Hierarchical Approximation of Covariance Matrices Hierarchical matrices have been described in detail [28, 27, 29, 24, 30, 43].”
A fast dual boundary element method for 3D anisotropic crack problems (2009, background): “The approximate representation allows storage savings with respect to the full rank representation and speeds up the matrix-vector product [43].”
Efficient Computation of Lead Field Bases and Influence Matrix for the FEM-based EEG and MEG Inverse Problem. Part I: Complexity Considerations (2003, background): “Standard geometrically balanced clustering [49] yields a partition of the matrix U∞ as shown in figure 4.”
A New Scheme for the Tensor Representation (2009, unknown): “[5]).”
Domain-decomposition Based ℌ-LU Preconditioners (2007, methodology): “constant (which depends on the block cluster tree T ) and p denotes the depth of the tree, cf. Definition 3. Estimates that show that the H-matrix addition and multiplication have almost optimal complexity are provided in [ 9 ] along with details on the efficient implementation of these operations.”
Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part I. Separable Approximation of Multi-variate Functions (2005, background): “The class of hierarchical (H) matrices allows an approximate matrix arithmetic with almost linear complexity [11]–[14], [9].”
Fast Direct Methods for Gaussian Processes (2014, background): “We refer the readers to [2], [10], [14], [15], [27], [33], [34],”