In this paper, we consider a fully discrete finite element method (FEM) to solve the two-dimensional nonlinear Fisher' equation with Riesz fractional derivatives in space. This method is chiefly performed by using Crank–Nicolson discretization in conjunction with a linearized approach in time and FEM in space. The existence, uniqueness of the weak solution, and the numerical stability of the scheme are proved in great detail. The optimal error estimate computed by $L^2$-norm showed both in time and space is derived by introducing a fractional orthogonal projection. Moreover, several numerical examples are conducted on unstructured triangular meshes by a properly designed algorithm.

Compared with side-scan sonar (SSS) and forward-looking sonar (FLS), synthetic aperture sonar (SAS) device can generate high-resolution underwater images, which is important for marine topographic mapping. Nevertheless, existing deep learning methods face challenges in extracting detailed feature information from underwater SAS images for semantic segmentation tasks, primarily due to the significant interference from the complex underwater environment and seabed reverberation noise. To address these challenges, we propose a novel network named U-SAS for SAS image semantic segmentation, which uses the hybrid convolution and attention mechanism architecture to extract rich multi-level features from SAS images. Specifically, U-SAS separates the extracted features into complementary representation, namely salient features and abstract location features. To enhance the distinctive representation of multi-level features, U-SAS incorporates the max-min module (MMM) and convolutional block attention mechanism (CBAM). The MMM effectively emphasizes the robust and significant underwater acoustic features, while the CBAM suppresses seabed reverberation noise and enhances the representation of positional information. Besides, we construct the global decoding module (GDM) to fuse salient features and abstract location features to enhance the correlation between local features and achieve global semantic understanding. To verify the effectiveness and feasibility of the proposed U-SAS network, we conducted extensive experiments on the SAS image dataset of complex underwater scenes. The experimental results show that U-SAS achieves 78.58% and 58.59% of mAcc and mIoU, respectively, which outperforms other state-of-the-art methods. Implementation codes will be available on https://github.com/darkseid-arch/U-SAS.

Based on existing research, we hypothesize that the positioning performance of Time Difference of Arrival (TDOA) systems may exhibit specific spatial distribution characteristics. Through two approaches—performance metric function construction and derivation and geometric intersection analysis—this study reveals the spatial distribution characteristics of positioning performance in minimal 3D-TDOA systems. We systematically establish and refine the theoretical framework for this research. The study confirms the hypothesis, demonstrating that positioning accuracy (PA) is a function of input parameter precision, target location, and sensor placement. Furthermore, we propose and develop core concepts, fundamental theorems, and key properties, including PA boundary (PAB), algorithm shadow zones (regions with poor localization performance), and tangential conditions. This theoretical framework exhibits strong generalizability and applicability. Simulations and experiments validate the theoretical findings. Additionally, we explore the practical applications of this research in target location optimization and sensor deployment optimization. This study provides new theoretical insights for system optimization and performance evaluation, advancing the field of positioning theory.