An efficient and memory free algorithm for subdiffusion equation using incremental singular value decomposition

TL;DR

This paper introduces an efficient, memory-free algorithm for solving time-fractional PDEs using incremental SVD, significantly reducing memory needs while maintaining accuracy and competitiveness.

Contribution

The paper develops a novel incremental SVD-based algorithm that minimizes memory usage in solving TFPDEs, with rigorous error analysis and validated numerical experiments.

Findings

Reduces memory requirements compared to direct methods

Maintains competitive performance with fast evaluation methods

Provides rigorous error analysis and validation through numerical experiments

Abstract

In this paper, we address the well-known challenge in the numerical solution of time-fractional partial differential equations (TFPDEs), namely, that the dependence on all previous time levels leads to storage requirements that grow linearly with the number of time steps. To overcome this difficulty, we develop an efficient algorithm based on incremental singular value decomposition (ISVD), which avoids the excessive memory demands associated with storing the full solution history. A rigorous error analysis is established, and numerical experiments are presented to validate the theoretical results. Comparisons with the direct method and a representative fast evaluation method show that the proposed ISVD approach dramatically reduces memory usage relative to the direct method and remains competitive with the fast method over the tested parameter regimes.