An efficient and memory free algorithm for subdiffusion equation using incremental singular value decomposition
This addresses a storage bottleneck for researchers and practitioners in computational mathematics dealing with subdiffusion equations, though it is incremental as it builds on existing fast evaluation methods.
The paper tackles the memory-intensive challenge in solving time-fractional partial differential equations by developing an incremental singular value decomposition algorithm, which dramatically reduces memory usage compared to direct methods while remaining competitive with fast evaluation methods.
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.