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An Incremental SVD Method for Non-Fickian Flows in Porous Media: Addressing Storage and Computational Challenges

Gang Chen, Yangwen Zhang, Dujin Zuo

TL;DR

This paper tackles the memory and computational bottlenecks in solving Non-Fickian flow models governed by integro-differential equations with memory kernels. It introduces a memory-free, data-driven approach based on incremental SVD to compress and update solution data online, achieving linear growth in computation with respect to time steps under a low-rank data assumption. A rigorous error analysis demonstrates that the method preserves convergence rates comparable to standard finite element schemes, with additional error terms governed by the truncation tolerance and the data rank. Numerical experiments across non-singular and singular kernels, including variable-order cases, show substantial memory savings and competitive accuracy, highlighting the method's potential for large-scale memory-dependent PDEs and broader memory-embedded systems.

Abstract

It is well known that the numerical solution of the Non-Fickian flows at the current stage depends on all previous time instances. Consequently, the storage requirement increases linearly, while the computational complexity grows quadratically with the number of time steps. This presents a significant challenge for numerical simulations. While numerous existing methods address this issue, our proposed approach stems from a data science perspective and maintains uniformity. Our method relies solely on the rank of the solution data, dissociating itself from dependency on any specific partial differential equation (PDE). In this paper, we make the assumption that the solution data exhibits approximate low rank. Here, we present a memory-free algorithm, based on the incremental SVD technique, that exhibits only linear growth in computational complexity as the number of time steps increases. We prove that the error between the solutions generated by the conventional algorithm and our innovative approach lies within the scope of machine error. Numerical experiments are showcased to affirm the accuracy and efficiency gains in terms of both memory usage and computational expenses.

An Incremental SVD Method for Non-Fickian Flows in Porous Media: Addressing Storage and Computational Challenges

TL;DR

This paper tackles the memory and computational bottlenecks in solving Non-Fickian flow models governed by integro-differential equations with memory kernels. It introduces a memory-free, data-driven approach based on incremental SVD to compress and update solution data online, achieving linear growth in computation with respect to time steps under a low-rank data assumption. A rigorous error analysis demonstrates that the method preserves convergence rates comparable to standard finite element schemes, with additional error terms governed by the truncation tolerance and the data rank. Numerical experiments across non-singular and singular kernels, including variable-order cases, show substantial memory savings and competitive accuracy, highlighting the method's potential for large-scale memory-dependent PDEs and broader memory-embedded systems.

Abstract

It is well known that the numerical solution of the Non-Fickian flows at the current stage depends on all previous time instances. Consequently, the storage requirement increases linearly, while the computational complexity grows quadratically with the number of time steps. This presents a significant challenge for numerical simulations. While numerous existing methods address this issue, our proposed approach stems from a data science perspective and maintains uniformity. Our method relies solely on the rank of the solution data, dissociating itself from dependency on any specific partial differential equation (PDE). In this paper, we make the assumption that the solution data exhibits approximate low rank. Here, we present a memory-free algorithm, based on the incremental SVD technique, that exhibits only linear growth in computational complexity as the number of time steps increases. We prove that the error between the solutions generated by the conventional algorithm and our innovative approach lies within the scope of machine error. Numerical experiments are showcased to affirm the accuracy and efficiency gains in terms of both memory usage and computational expenses.
Paper Structure (10 sections, 14 theorems, 107 equations, 4 figures, 3 tables, 4 algorithms)

This paper contains 10 sections, 14 theorems, 107 equations, 4 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. The finite element method for integro-differential euqations
  3. The incremental SVD method
  4. Incremental SVD method for the integro-differential equation
  5. Error estimate
  6. Assumptions and Main Result
  7. Proof of \ref{['main_res']}
  8. Singular Kernel Case
  9. Numerical experiments
  10. Conclusion

Key Result

Lemma 1

zhang2022answer Assume that $\Sigma=\operatorname{diag}\left(\sigma_1, \sigma_2, \ldots, \sigma_k\right)$ with $\sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_k$, and $\bar{\Sigma}=\operatorname{diag}\left(\mu_1, \mu_2, \ldots, \mu_{k+1}\right)$ with $\mu_1 \geq \mu_2 \geq \ldots \geq \mu_{k+1}$. Th

Figures (4)

  • Figure 1: The process of using the incremental SVD to solve the integro-differential equation.
  • Figure 2: A comparison of wall time and memory costs between the two algorithms is conducted for various time steps and $K(t)=\log {(1+t)}$, specifically when $h=1/250$ and $\Delta t=10^{-4}$.
  • Figure 3: A comparison of wall time and memory costs between the two algorithms is conducted for various time steps when $K(t)=e^{-\lambda t}\frac{1}{\Gamma(\alpha)}t^{\alpha-1}$ with $\alpha=0.8,\lambda=0.2$, specifically when $h=1/128$ and $\Delta t=10^{-4}$.
  • Figure 4: rank changes with respect to number of time steps $N$

Theorems & Definitions (26)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Theorem 1
  • Remark 3
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 16 more