SMASH: Structured matrix approximation by separation and hierarchy

Abstract

Summary This paper presents an efficient method to perform structured matrix approximation by separation and hierarchy (SMASH), when the original dense matrix is associated with a kernel function. Given the points in a domain, a tree structure is first constructed based on an adaptive partition of the computational domain to facilitate subsequent approximation procedures. In contrast to existing schemes based on either analytic or purely algebraic approximations, SMASH takes advantage of both approaches and greatly improves efficiency. The algorithm follows a bottom‐up traversal of the tree and is able to perform the operations associated with each node on the same level in parallel. A strong rank‐revealing factorization is applied to the initial analytic approximation in the separation regime so that a special structure is incorporated into the final nested bases. As a consequence, the storage is significantly reduced on one hand and a hierarchy of the original grid is constructed on the other hand. Due to this hierarchy, nested bases at upper levels can be computed in a similar way as the leaf level operations but on coarser grids. The main advantages of SMASH include its simplicity of implementation, its flexibility to construct various hierarchical rank structures, and a low storage cost. The efficiency and robustness of SMASH are demonstrated through various test problems arising from integral equations, structured matrices, etc.

A Low-Rank Technique for Computing the Quasi-Stationary Distribution of Subcritical Galton--Watson Processes

“…which is only informative for h > 1. Equation (9) highlights the connection between the existence of higher order derivatives of P (z) and those of G(z). In probabilistic terms, it relates the factorial moments of the quasi-stationary distribution to those of the offspring distribution.…”

“…Adopting this strategy, still allows to store and operate with matrices with a O(n) complexity. The H 2 and HSS representations of the matrix A can be obtained by applying the algorithms described in [9,23]. The use of these more sophisticated formats is beyond the scope of this paper and might be the subject of future investigations.…”

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

“…Efforts have been devoted to preconditioning symmetric positive definite systems by exploiting the HSS matrix structure 40 and unsymmetric systems by ‐matrix computations 39,41 . To have a comprehensive summary of different hierarchical structures and techniques, as well as their applicability and limitations, we refer to 42 . For interested readers to gain more insight on the latest development in hierarchical matrices by leveraging machine learning techniques, we recommend 43 .…”

Hierarchical matrix approximations for space-fractional diffusion equations

“…Related hybrid analytical-algebraic methods have been presented in the literature, including the HCA method [49] which overcomes the unreliability of the heuristic algebraic adaptive cross approximation (ACA) [50] method, by combining it with an interpolationbased separable approximation of the kernel. SMASH [51] method for the construction of H 2 matrices using rank revealing QR factorizations. The use of linear algebra-based operations in hybrid methods generally results in better compression, smaller ranks, and more general applicability, than is possible with fast multipole methods [52].…”