An adaptive fast solver for a general class of positive definite matrices via energy decomposition

TL;DR

This paper introduces an adaptive, energy-based multiresolution solver for symmetric positive definite matrices, including graph Laplacians, achieving near-optimal complexity and stability through localized basis construction.

Contribution

It develops a novel energy decomposition framework and an adaptive operator compression scheme that reflect the operator's geometric structure, enabling efficient matrix factorization.

Findings

Achieves nearly optimal performance in complexity and well-posedness.

Introduces a new energy decomposition concept for SPD matrices.

Provides a systematic approach for localized basis construction with controlled error.

Abstract

In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (SPD) matrices which include the well-known graph Laplacian. We achieve this by developing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which achieve nearly optimal performance on both complexity and well-posedness. To develop our adaptive operator compression and multiresolution matrix factorization methods, we first introduce a novel notion of energy decomposition for SPD matrix $A$ using the representation of energy elements. The interaction between these energy elements depicts the underlying topological structure of the operator. This concept of decomposition naturally reflects the hidden geometric structure of the operator which inherits the localities of the structure. By utilizing the intrinsic geometric information under this Energy framework, we propose a systematic operator compression scheme for the inverse operator $A^{-1}$. In particular, with an appropriate partition of the underlying geometric structure, we can construct localized basis by using the concept of interior and closed energy. Meanwhile, two important localized quantities are introduced, namely the error factor and the condition factor. Our error analysis results show that these two factors will be the guidelines for finding the appropriate partition of the basis functions such that prescribed compression error and acceptable condition number can be achieved. By virtue of this insight, we propose the Patch Pairing algorithm to realize our energy partition framework for operator compression with controllable compression error and condition number.