A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition

In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition component by integrating the multiresolution framework into the implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm

A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition

In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition component by integrating the multiresolution framework into the implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm

Positive Definite Matrices: Compression, Decomposition, Eigensolver, and Concentration

For many decades, the study of positive-definite (PD) matrices has been one of the most popular subjects among a wide range of scientific researches. A huge mass of successful models on PD matrices has been proposed and developed in the fields of mathematics, physics, biology, etc., leading to a celebrated richness of theories and algorithms. In this thesis, we draw our attention to a general class of PD matrices that can be decomposed as the sum of a sequence of positive-semidefinite matrices. For this class of PD matrices, we will develop theories and algorithms on operator compression, multilevel decomposition, eigenpair computation, and spectrum concentration. We divide these contents into three main parts. In the first part, we propose an adaptive fast solver for the preceding class of PD matrices which includes the well-known graph Laplacians. We achieve this by establishing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which have nearly optimal performance on both complexity and well-posedness. To develop our methods, we introduce a novel notion of energy decomposition for PD matrices and two important local measurement quantities, which provide theoretical guarantee and computational guidance for the construction of an appropriate partition and a nested adaptive basis. In the second part, we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse PD matrix under the multiresolution matrix compression framework. We exploit the well-conditioned property of every decomposition components by integrating the multiresolution framework into the Implicitly Restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. In the third part, we derive concentration inequalities on partial sums of eigenvalues of random PD matrices by introducing the notion of k-trace. For this purpose, we establish a generalized Lieb's concavity theorem, which extends the original Lieb's concavity theorem from the normal trace to k-traces. Our argument employs a variety of matrix techniques and concepts, including exterior algebra, mixed discriminant, and operator interpolation.</p

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest