Xiaoli Li

Xiaoli Li, Jie Shen, Hongxing Rui

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

Zhengguang Liu, Xiaoli Li

In this paper, we propose a novel Lagrange Multiplier approach, named zero-factor (ZF) approach to solve a series of gradient flow problems. The numerical schemes based on the new algorithm are unconditionally energy stable with the original energy and do not require any extra assumption conditions. We also prove that the ZF schemes with specific zero factors lead to the popular SAV-type method. To reduce the computation cost and improve the accuracy and consistency, we propose a zero-factor approach with relaxation, which we named the relaxed zero-factor (RZF) method, to design unconditional energy stable schemes for gradient flows. The RZF schemes can be proved to be unconditionally energy stable with respect to a modified energy that is closer to the original energy, and provide a very simple calculation process. The variation of the introduced zero factor is highly consistent with the nonlinear free energy which implies that the introduced ZF method is a very efficient way to capture the sharp dissipation of nonlinear free energy. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

Zhengguang Liu, Aijie Cheng, Xiaoli Li et al.

In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference scheme to solve a two-sided fractional ordinary equation. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from $O(N^{3})$ required by traditional methods to $O(Nlog^{2}N)$ and the memory requirement from $O(N^{2})$ to $O(N)$ without using any lossy compression, where $N$ is the number of unknowns. Two finite difference schemes to solve time fractional hyperbolic equations with different fractional order $γ$ are considered. We present a fast solution technique depending on the special structure of coefficient matrices by rearranging the order of unknowns. It helps to reduce the computational work from $O(N^2M)$ required by traditional methods to $O(N$log$^{2}N)$ and the memory requirement from $O(NM)$ to $O(N)$ without using any lossy compression, where $N=τ^{-1}$ and $τ$ is the size of time step, $M=h^{-1}$ and $h$ is the size of space step. Importantly, a fast method is employed to solve the classical time fractional diffusion equation with a lower coast at $O(MN$log$^2N)$, where the direct method requires an overall computational complexity of $O(N^2M)$. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Peiliang Gong, Mohamed Ragab, Min Wu et al.

Test-time adaptation aims to adapt pre-trained deep neural networks using solely online unlabelled test data during inference. Although TTA has shown promise in visual applications, its potential in time series contexts remains largely unexplored. Existing TTA methods, originally designed for visual tasks, may not effectively handle the complex temporal dynamics of real-world time series data, resulting in suboptimal adaptation performance. To address this gap, we propose Augmented Contrastive Clustering with Uncertainty-aware Prototyping (ACCUP), a straightforward yet effective TTA method for time series data. Initially, our approach employs augmentation ensemble on the time series data to capture diverse temporal information and variations, incorporating uncertainty-aware prototypes to distill essential characteristics. Additionally, we introduce an entropy comparison scheme to selectively acquire more confident predictions, enhancing the reliability of pseudo labels. Furthermore, we utilize augmented contrastive clustering to enhance feature discriminability and mitigate error accumulation from noisy pseudo labels, promoting cohesive clustering within the same class while facilitating clear separation between different classes. Extensive experiments conducted on three real-world time series datasets and an additional visual dataset demonstrate the effectiveness and generalization potential of the proposed method, advancing the underexplored realm of TTA for time series data.

Mohamed Ragab, Peiliang Gong, Emadeldeen Eldele et al.

Source-free domain adaptation (SFDA) aims to adapt a model pre-trained on a labeled source domain to an unlabeled target domain without access to source data, preserving the source domain's privacy. While SFDA is prevalent in computer vision, it remains largely unexplored in time series analysis. Existing SFDA methods, designed for visual data, struggle to capture the inherent temporal dynamics of time series, hindering adaptation performance. This paper proposes MAsk And imPUte (MAPU), a novel and effective approach for time series SFDA. MAPU addresses the critical challenge of temporal consistency by introducing a novel temporal imputation task. This task involves randomly masking time series signals and leveraging a dedicated temporal imputer to recover the original signal within the learned embedding space, bypassing the complexities of noisy raw data. Notably, MAPU is the first method to explicitly address temporal consistency in the context of time series SFDA. Additionally, it offers seamless integration with existing SFDA methods, providing greater flexibility. We further introduce E-MAPU, which incorporates evidential uncertainty estimation to address the overconfidence issue inherent in softmax predictions. To achieve that, we leverage evidential deep learning to obtain a better-calibrated pre-trained model and adapt the target encoder to map out-of-support target samples to a new feature representation closer to the source domain's support. This fosters better alignment, ultimately enhancing adaptation performance. Extensive experiments on five real-world time series datasets demonstrate that both MAPU and E-MAPU achieve significant performance gains compared to existing methods. These results highlight the effectiveness of our proposed approaches for tackling various time series domain adaptation problems.