Title:
The proxy point method for rank-structured matrices

dc.contributor.advisor Chow, Edmond
dc.contributor.author Xing, Xin
dc.contributor.committeeMember Sherrill, David
dc.contributor.committeeMember Xia, Jianlin
dc.contributor.committeeMember Xi, Yuanzhe
dc.contributor.committeeMember Zhou, Haomin
dc.contributor.department Mathematics
dc.date.accessioned 2020-01-14T14:47:32Z
dc.date.available 2020-01-14T14:47:32Z
dc.date.created 2019-12
dc.date.issued 2019-11-06
dc.date.submitted December 2019
dc.date.updated 2020-01-14T14:47:32Z
dc.description.abstract Rank-structured matrix representations, e.g., $\mathcal{H}^2$ and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their application is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form. This dissertation is mainly about the study and application of a hybrid analytic-algebraic compression method, called \textit{the proxy point method}. This work uncovers the full strength of this presently underutilized method that could potentially resolve the above bottleneck for all rank-structured matrix techniques. As a result, this work could extend the applicability and improve the performance of rank-structured matrix techniques and thus facilitate dense matrix computations in a wider range of scientific computing problems, such as particle simulations, numerical solution of integral equations, and Gaussian processes. Application of the proxy point method in practice is presently very limited. Only two special instances of the method have been used heuristically to compress interaction blocks defined by specific kernel functions over points. We address several critical problems of the proxy point method which limit its applicability. A general form of the method is then proposed, paving the way for its wider application in the construction of different rank-structured matrix representations with kernel functions that are more general than those usually used. In addition to kernel-defined interactions between points, we further extend the applicability of the proxy point method to compress the interactions between charge distributions in quantum chemistry calculations. Specifically, we propose a variant of the proxy point method to efficiently construct an $\mathcal{H}^2$ matrix representation of the four-dimensional electron repulsion integral tensor. The linear-scaling matrix-vector multiplication algorithm for the constructed $\mathcal{H}^2$ matrix is then used for fast Coulomb matrix construction which is an important step in many quantum chemical methods. Two additional contributions related to $\mathcal{H}^2$ and HSS matrices are also presented. First, we explain the exact equivalence between $\mathcal{H}^2$ matrices and the fast multipole method (FMM). This equivalence has not been rigorously studied in the literature. Numerical comparisons between FMM and $\mathcal{H}^2$ matrices based on the proxy point method are also provided, showing the relative advantages and disadvantages of the two methods. Second, we consider the application of HSS approximations as preconditioners for symmetric positive definite (SPD) matrices. Preserving positive definiteness is essential for rank-structured matrix approximations to be used efficiently in various algorithms and applications, e.g., the preconditioned conjugate gradient method. We propose two methods for constructing HSS approximations to an SPD matrix that preserve positive definiteness.
dc.description.degree Ph.D.
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/62327
dc.language.iso en_US
dc.publisher Georgia Institute of Technology
dc.subject Rank-structured matrices
dc.subject Low-rank approximation
dc.subject Kernel matrices
dc.subject Numerical linear algebra
dc.title The proxy point method for rank-structured matrices
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Chow, Edmond
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isAdvisorOfPublication 655f3cad-7fa9-483b-ad38-c475e5333d8e
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
thesis.degree.level Doctoral
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