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arXiv 2022年

作者： Zhu, Hui Shu, Shi Zhang, Jianping The School of Mathematics and Computational Science Xiangtan University Key Laboratory of Intelligent Computing & Information Processing Ministry of Education China The School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center China

Solving variational image segmentation problems with hidden physics is often expensive and requires different algorithms and manually tunes model parameter. The deep learning methods based on the U-Net structure have obtained outstanding performances in many different medical image segmentation tasks, but designing such networks requires a lot of parameters and training data, not always available for practical problems. In this paper, inspired by traditional multi-phase convexity Mumford-Shah variational model and full approximation scheme (FAS) solving the nonlinear systems, we propose a novel variational-model-informed network (denoted as FAS-Unet) that exploits the model and algorithm priors to extract the multi-scale features. The proposed model-informed network integrates image data and mathematical models, and implements them through learning a few convolution kernels. Based on the variational theory and FAS algorithm, we first design a feature extraction sub-network (FAS-Solution module) to solve the model-driven nonlinear systems, where a skip-connection is employed to fuse the multi-scale features. Secondly, we further design a convolution block to fuse the extracted features from the previous stage, resulting in the final segmentation possibility. Experimental results on three different medical image segmentation tasks show that the proposed FAS-Unet is very competitive with other state-of-the-art methods in qualitative, quantitative and model complexity evaluations. Moreover, it may also be possible to train specialized network architectures that automatically satisfy some of the mathematical and physical laws in other image problems for better accuracy, faster training and improved generalization. The code is available at https://***/zhuhui100/FASUNet. Copyright © 2.2., The Authors. All rights reserved.

关键词： Image segmentation

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A hybrid iterative neural solver based on spectral analysis for parametric PDEs

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Journal of computational Physics 2025年 538卷

作者： Chen Cui Kai Jiang Yun Liu Shi Shu Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 Hunan China

Deep learning-based hybrid iterative methods (DL-HIM) have emerged as a promising approach for designing fast neural solvers to tackle large-scale sparse linear systems. DL-HIM combine the smoothing effect of simple iterative methods with the spectral bias of neural networks, which allows them to effectively eliminate both high-frequency and low-frequency error components. However, their efficiency may decrease if simple iterative methods can not provide effective smoothing, making it difficult for the neural network to learn mid-frequency and high-frequency components. This paper first conducts a convergence analysis for general DL-HIM from a spectral viewpoint, concluding that under reasonable assumptions, DL-HIM exhibit a convergence rate independent of grid size h and physical parameters μ . To meet these assumptions, we design a neural network from an eigen perspective, focusing on learning the eigenvalues and eigenvectors corresponding to error components that simple iterative methods struggle to eliminate. Specifically, the eigenvalues are learned by a meta subnet, while the eigenvectors are approximated using Fourier modes with a transition matrix provided by another meta subnet. The resulting DL-HIM, termed the Fourier Neural Solver (FNS), can be trained to achieve a convergence rate independent of PDE parameters and grid size within a local neighborhood of the training scale by designing a loss function that ensures the neural network complements the smoothing effect of the damped Jacobi iterative methods. We verify the performance of FNS on five types of linear parametric PDEs.

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Kernel-learning parameter prediction and evaluation in algebraic multigrid method for several PDEs

arXiv

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arXiv 2025年

作者： Zhang, Juan Luo, Junyue Zhang, Fangfang Yue, Xiaoqiang Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper explores the application of kernel learning methods for parameter prediction and evaluation in the Algebraic Multigrid Method (AMG), focusing on several Partial Differential Equation (PDE) problems. AMG is an efficient iterative solver for large-scale sparse linear systems, particularly those derived from elliptic and parabolic PDE discretizations. However, its performance heavily relies on numerous parameters, which are often set empirically and are highly sensitive to AMG’s effectiveness. Traditional parameter optimization methods are either computationally expensive or lack theoretical support. To address this, we propose a Gaussian Process Regression (GPR)-based strategy to optimize AMG parameters and introduce evaluation metrics to assess their effectiveness. Trained on small-scale datasets, GPR predicts nearly optimal parameters, bypassing the time-consuming parameter sweeping process. We also use kernel learning techniques to build a kernel function library and determine the optimal kernel function through linear combination, enhancing prediction accuracy. In numerical experiments, we tested typical PDEs such as the constant-coefficient Poisson equation, variable-coefficient Poisson equation, diffusion equation, and Helmholtz equation. Results show that GPR-predicted parameters match grid search results in iteration counts while significantly reducing computational time. A comprehensive analysis using metrics like mean squared error, prediction interval coverage, and Bayesian information criterion confirms GPR’s efficiency and reliability. These findings validate GPR’s effectiveness in AMG parameter optimization and provide theoretical support for AMG’s practical application. © 2.2., CC BY.

关键词： Poisson equation

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Finite element method with Grünwald-Letnikov type approximation in time for a constant time delay subdiffusion equation

arXiv

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arXiv 2025年

作者： Bu, Weiping Zhang, Xueqin Liao, Weizhi Zhao, Yue School of Mathematics and Computational Science Xiangtan University Hunan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan411105 China Space Engineering University Beijing101400 China

In this work, a subdiffusion equation with constant time delay τ is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at t = 0+ and its second-order time derivative shows singularity at both t = 0+ and τ+, while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Gronwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis. Copyright © 2.2., The Authors. All rights reserved.

关键词： Finite element method

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The Mean Time to Absorption on Horizontal Partitioned Sierpinski Gasket Networks

arXiv

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arXiv 2022年

作者： Zhang, Zhizhuo Wu, Bo Yu, Zuguo School of Mathematics Southeast University Nanjing210023 China School of Applied Mathematics Nanjing University of Finance and Economics Nanjing210023 China Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China

The random walk is one of the most basic dynamic properties of complex networks, which has gradually become a research hotspot in recent years due to its many applications in actual networks. An important characteristic of the random walk is the mean time to absorption, which plays an extremely important role in the study of topology, dynamics and practical application of complex networks. Analyzing the mean time to absorption on the regular iterative self-similar network models is an important way to explore the influence of self-similarity on the properties of random walks on the network. The existing literatures have proved that even local self-similar structures can greatly affect the properties of random walks on the global network, but they have failed to prove whether these effects are related to the scale of these self-similar structures. In this article, we construct and study a class of Horizontal Partitioned Sierpinski Gasket network model based on the classic Sierpinski gasket network, which is composed of local self-similar structures, and the scale of these structures will be controlled by the partition coefficient k. Then, the analytical expressions and approximate expressions of the mean time to absorption on the network model are obtained, which prove that the size of the self-similar structure in the network will directly restrict the influence of the self-similar structure on the properties of random walks on the network. Finally, we also analyzed the mean time to absorption of different absorption nodes on the network to find the location of the node with the highest absorption efficiency. Copyright © 2.2., The Authors. All rights reserved.

关键词： Complex networks

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NUMERICAL METHODS AND ANALYSIS OF COMPUTING QUASIPERIODIC SYSTEMS

arXiv

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arXiv 2022年

作者： Jiang, Kai Li, Shifeng Zhang, Pingwen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China

Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is of great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 2.6: 42., 2.14], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudo-spectral) method. Then we analyze the computational complexity of PM and QSM in calculating quasiperiodic systems. The PM can use fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of PM, QSM and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation. Copyright © 2.2., The Authors. All rights reserved.

关键词： Fast Fourier transforms

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A GENERAL ALTERNATING-DIRECTION IMPLICIT FRAMEWORK WITH GAUSSIAN PROCESS REGRESSION PARAMETER PREDICTION FOR LARGE SPARSE LINEAR SYSTEMS

arXiv

引用

arXiv 2021年

作者： Jiang, Kai Su, Xuehong Zhang, Juan Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper proposes an efficient general alternating-direction implicit (GADI) framework for solving large sparse linear systems. The convergence property of the GADI framework is discussed. Most of existing ADI methods can be unified in the developed framework. Meanwhile the GADI framework can derive new ADI methods. Moreover, as the algorithm efficiency is sensitive to the splitting parameters, we offer a data-driven approach, the Gaussian process regression (GPR) method based on the Bayesian inference, to predict the GADI framework’s relatively optimal parameters. The GPR method only requires a small training data set to learn the regression prediction mapping, can predict accurate splitting parameters, and has high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations. Finally, we use the three-dimensional convection-diffusion equation, two-dimensional parabolic equation, and continuous Sylvester matrix equation to examine the performance of our proposed methods. Numerical results demonstrate that the proposed framework is faster tens to thousands of times than the existing ADI methods, such as (inexact) Hermitian and skew-Hermitian splitting type methods in which the consumption of obtaining relatively optimal splitting parameters is ignored. As a result, our proposed methods can solve much larger linear systems which these existing ADI methods have not reached. Copyright © 2.2., The Authors. All rights reserved.

关键词： Linear systems

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Nucleation and phase transition of decagonal quasicrystals

arXiv

引用

arXiv 2024年

作者： Zhou, Tiejun Zhang, Lei Zhang, Pingwen Shi, An-Chang Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Beijing International Center for Mathematical Research Peking University Beijing100871 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China Department of Physics and Astronomy McMaster University HamiltonL8S 4M1 Canada

In this work, we study the nucleation of quasicrystals from liquid or periodic crystals by developing an efficient order-order phase transition algorithm, namely the nullspace-preserving saddle search method. Specifically, we focus on nucleation and phase transitions of the decagonal quasicrystal (DQC) based on the Lifshitz-Petrich model. We present the nucleation path of DQC from the liquid and demonstrate one- and two-stage transition paths between DQC and periodic crystals. We provide a perspective of the group-subgroup phase transition and nucleation rates to understand the nucleation and phase transition mechanisms involving DQC. These results reveal the one-step and stepwise modes of symmetry breaking or recovery in the phase transition from DQC, where the stepwise modes are more probable. Copyright © 2.2., The Authors. All rights reserved.

关键词： Quasicrystals

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Nest-DGIL: Nesterov-optimized Deep Geometric Incremental Learning for CS Image Reconstruction

arXiv

引用

arXiv 2023年

作者： Fan, Xiaohong Yang, Yin Chen, Ke Feng, Yujie Zhang, Jianping The School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center Xiangtan411105 China The Centre for Mathematical ImagingTechniques Department of Mathematical Sciences The University of Liverpool Department of Mathematics and Statistics University of Strathclyde Glasgow United Kingdom The School of Mathematics and Computational Science Xiangtan University The Key Laboratory of Intelligent Computing & Information Processing The Ministry of Education Xiangtan411105 China

Proximal gradient-based optimization is one of the most common strategies to solve inverse problem of images, and it is easy to implement. However, these techniques often generate heavy artifacts in image reconstruction. One of the most popular refinement methods is to fine-tune the regularization parameter to alleviate such artifacts, but it may not always be sufficient or applicable due to increased computational costs. In this work, we propose a deep geometric incremental learning framework based on the second Nesterov proximal gradient optimization. The proposed end-to-end network not only has the powerful learning ability for high-/low-frequency image features, but also can theoretically guarantee that geometric texture details will be reconstructed from preliminary linear reconstruction. Furthermore, it can avoid the risk of intermediate reconstruction results falling outside the geometric decomposition domains and achieve fast convergence. Our reconstruction framework is decomposed into four modules including general linear reconstruction, cascade geometric incremental restoration, Nesterov acceleration, and post-processing. In the image restoration step, a cascade geometric incremental learning module is designed to compensate for missing texture information from different geometric spectral decomposition domains. Inspired by the overlap-tile strategy, we also develop a post-processing module to remove the block effect in patch-wise-based natural image reconstruction. All parameters in the proposed model are learnable, an adaptive initialization technique of physical parameters is also employed to make model flexibility and ensure converging smoothly. We compare the reconstruction performance of the proposed method with existing state-of-the-art methods to demonstrate its superiority. Our source codes are available at https://***/fanxiaohong/Nest-DGIL. Copyright © 2.2., The Authors. All rights reserved.

关键词： Image reconstruction

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Theory of polygonal phases self-assembled from T-shaped liquid crystalline molecules

arXiv

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arXiv 2023年

作者： He, Zhijuan Wang, Xin Zhang, Pingwen Shi, An-Chang Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

Extensive experimental studies have shown that numerous ordered phases can be formed via the self-assembly of T-shaped liquid crystalline molecules (TLCMs) composed of a rigid backbone, two flexible end chains and a flexible side chain. However, a comprehensive understanding of the stability and formation mechanisms of these intricately nanostructured phases remains incomplete. Here we fill this gap by carrying out a theoretical study of the phase behaviour of TLCMs. Specifically, we construct phase diagrams of TLCMs by computing the free energy of different ordered phases of the system. Our results reveal that the number of polygonal edges increases as the length of side chain or interaction strength increases, consistent with experimental observations. The theoretical study not only reproduces the experimentally observed phases and phase transition sequences, but also systematically analyzes the stability mechanism of the polygonal phases. Copyright © 2.2., The Authors. All rights reserved.

关键词： Free energy

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