Incremental SVD Compression for Nonlinear Oldroyd Equations with General Memory Kernels

TL;DR

This paper introduces an incremental SVD compression method for nonlinear Oldroyd equations with general memory kernels, significantly reducing memory and computational costs while maintaining accuracy.

Contribution

It develops an online low-rank compression technique for history terms in Oldroyd equations, enabling efficient simulations with controlled accuracy.

Findings

Memory usage reduced to O((m+N)r) from O(mN)

Computational work decreased to O(mNr + rN^2)

Numerical tests confirm accuracy and efficiency gains

Abstract

We study mixed finite element/Crank--Nicolson discretizations of a nonlinear Oldroyd problem with general nonsingular and weakly singular memory kernels. Direct evaluation of the history term requires storing all previous velocity snapshots, which leads to $\mathcal{O}(mN)$ memory and $\mathcal{O}(mN^2)$ work over $N$ time steps, where $m$ denotes the number of spatial degrees of freedom. To reduce this burden, we compress the velocity history online by an incremental singular value decomposition and use the compressed representation in the discrete memory term. Under an approximate low-rank assumption of numerical rank $r$, the storage decreases to $\mathcal{O}((m+N)r)$, while the total history-evaluation work becomes $\mathcal{O}(mNr+rN^2)$. For nonsingular kernels, we derive a tolerance-dependent perturbation estimate showing that the baseline finite element accuracy is retained when the compression tolerance is sufficiently small. We also extend the approach to tempered weakly singular kernels via convolution quadrature. Numerical tests show near-indistinguishable solutions from the uncompressed scheme for the reported tolerances, together with substantial memory savings and reduced wall-clock time.