Core Concepts
An efficient memory-free algorithm based on the incremental SVD technique is presented to solve non-Fickian flows in porous media, exhibiting only linear growth in computational complexity as the number of time steps increases.
Abstract
The content discusses an efficient numerical approach for solving non-Fickian flows in porous media, which are characterized by an integro-differential equation with a memory term. The key highlights are:
The standard finite element method coupled with a time discretization scheme for solving the integro-differential equation leads to a linear increase in storage requirements and quadratic growth in computational complexity as the number of time steps increases.
To address this challenge, the authors propose a novel algorithm that leverages the incremental Singular Value Decomposition (SVD) technique. This approach enables storing the solution data in a compressed form, using several smaller matrices instead of a large dense matrix.
The authors make the assumption that the solution data exhibits approximate low rank, which allows them to apply the incremental SVD method. This results in a memory-efficient algorithm with only linear growth in computational complexity as the number of time steps increases.
The authors provide a rigorous error analysis, proving that the error between the solutions generated by the conventional algorithm and their innovative approach lies within the scope of machine error.
Numerical experiments are presented to validate the accuracy and efficiency gains of the proposed method in terms of both memory usage and computational expenses, even for cases with weakly singular kernels or variable-order time-fractional equations.
Stats
The storage cost of the history term in the standard finite element method is O(mn).
The computational cost of the standard finite element method is O(mn^2).
Quotes
"The incremental SVD algorithm can be easily used in conjunction with a time stepping code for simulating equation (1.1). This approach enables storing solution data in several smaller matrices, alleviating the need for a huge dense matrix as often seen in traditional methods."
"By presuming that the solution data demonstrates an approximate low-rank characteristic, we are able to address the issue of data storage in solving the integro-differential equation (1.1)."