Zhidong Zhang

Yikan Liu, Zhidong Zhang

In this article, we consider the reconstruction of $ρ(t)$ in the (time-fractional) diffusion equation $(\partial_t^α-\triangle)u(x,t)=ρ(t)g(x)$ ($0<α\le 1$) by the observation at a single point $x_0$. We are mainly concerned with the situation of $x_0 \notin$ supp g, which is practically important but has not been well investigated in literature. Assuming the finite sign changes of $ρ$ and an extra observation interval, we establish the multiple logarithmic stability for the problem based on a reverse convolution inequality and a lower estimate for positive solutions. Meanwhile, we develop a fixed point iteration for the numerical reconstruction and prove its convergence. The performance of the proposed method is illustrated by several numerical examples.

Liuying Zhang, Wenlong Zhang, Zhidong Zhang

We study an inverse drift problem for a two-dimensional parabolic equation on the unit square with mixed boundary conditions, where the drift coefficient is recovered from terminal observation data $g=u(\cdot,T)$. A monotone operator is constructed whose fixed point coincides with the unknown drift, yielding uniqueness in an admissible class and a constructive iterative reconstruction scheme. Numerical experiments illustrate the monotone convergence and the effectiveness of the proposed method, and show that it remains effective for noisy terminal data under the denoising strategy.

Qiling Gu, Wenlong Zhang, Zhidong Zhang

This paper considers the inverse problem of identifying the source term of parabolic equations from sparse boundary measurements. We used data from moving sensors to locate the unknown source term. This work first proves the uniqueness of the inverse problem under such measurements. Then the movement strategy of the sensor is given, from which the authors build the reconstruction algorithm. Finally, some numerical experiments are performed and the corresponding results are generated, which indicate the effectiveness of the algorithms.

Tianming Cai, Guoying Zhao, Junbin Zang et al.

In recent years, the preliminary diagnosis of ADHD using EEG has attracted the attention from researchers. EEG, known for its expediency and efficiency, plays a pivotal role in the diagnosis and treatment of ADHD. However, the non-stationarity of EEG signals and inter-subject variability pose challenges to the diagnostic and classification processes. Topological Data Analysis offers a novel perspective for ADHD classification, diverging from traditional time-frequency domain features. However, conventional TDA models are restricted to single-channel time series and are susceptible to noise, leading to the loss of topological features in persistence diagrams.This paper presents an enhanced TDA approach applicable to multi-channel EEG in ADHD. Initially, optimal input parameters for multi-channel EEG are determined. Subsequently, each channel's EEG undergoes phase space reconstruction (PSR) followed by the utilization of k-Power Distance to Measure for approximating ideal point clouds. Then, multi-dimensional time series are re-embedded, and TDA is applied to obtain topological feature information. Gaussian function-based Multivariate Kernel Density Estimation is employed in the merger persistence diagram to filter out desired topological feature mappings. Finally, the persistence image method is employed to extract topological features, and the influence of various weighting functions on the results is discussed.The effectiveness of our method is evaluated using the IEEE ADHD dataset. Results demonstrate that the accuracy, sensitivity, and specificity reach 78.27%, 80.62%, and 75.63%, respectively. Compared to traditional TDA methods, our method was effectively improved and outperforms typical nonlinear descriptors. These findings indicate that our method exhibits higher precision and robustness.

William Rundell, Zhidong Zhang

A standard inverse problem is to determine a source which is supported in an unknown domain $D$ from external boundary measurements. Here we consider the case of a time-dependent situation where the source is equal to unity in an unknown subdomain $D$ of a larger given domain $Ω$. Overposed measurements consist of time traces of the solution or its flux values on a set of discrete points on the boundary $\partialΩ$. The case of a parabolic equation was considered in [HettlichRundell:2001]. In our situation we extend this to cover the subdiffusion case based on an anomalous diffusion model and leading to a fractional order differential operator. We will show a uniqueness result and examine a reconstruction algorithm. One of the main motives for this work is to examine the dependence of the reconstructions on the parameter $α$, the exponent of the fractional operator which controls the degree of anomalous behaviour of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.