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A BDDC method for three-dimensional advection-diffusion problems with an adaptive coarse space

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arXiv 2025年

作者： Peng, Jie Shu, Shi Wang, Junxian Zhong, Liuqiang College of Mathematics and Statistics Hengyang Normal University Hengyang421010 China School of Mathematical Sciences South China Normal University Guangzhou510631 China School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

The solution of nonsymmetric positive definite (NSPD) systems for advection-diffusion problems is an important research topic in science and engineering. The adaptive BDDC method is a significant class of non-overlapping domain decomposition methods, which is often used for solving symmetric positive definite problems. In this paper, we apply the adaptive BDDC method to solve the NSPD systems of te advection-diffusion problems. Moreover, by designing a class of edge generalized eigenvalue problems based on prior selected primal constraints, the number of primal unknowns is further reduced. Numerical experiments have verified the effectiveness of the proposed methods. Copyright © 2025, The Authors. All rights reserved.

关键词： Domain decomposition methods

PRISTA-Net: Deep Iterative Shrinkage Thresholding Network for Coded Diffraction Patterns Phase Retrieval

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arXiv 2023年

作者： Liu, Aoxu Fan, Xiaohong Yang, Yin 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 the Key Laboratory of Intelligent Computing & Information Processing the Ministry of Education Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University the National Center for Applied Mathematics in Hunan Xiangtan411105 China

The problem of phase retrieval (PR) involves recovering an unknown image from limited amplitude measurement data and is a challenge nonlinear inverse problem in computational imaging and image processing. However, many of the PR methods are based on black-box network models that lack interpretability and plug-and-play (PnP) frameworks that are computationally complex and require careful parameter tuning. To address this, we have developed PRISTA-Net, a deep unfolding network (DUN) based on the first-order iterative shrinkage thresholding algorithm (ISTA). This network utilizes a learnable nonlinear transformation to address the proximal-point mapping sub-problem associated with the sparse priors, and an attention mechanism to focus on phase information containing image edges, textures, and structures. Additionally, the fast Fourier transform (FFT) is used to learn global features to enhance local information, and the designed logarithmic-based loss function leads to significant improvements when the noise level is low. All parameters in the proposed PRISTA-Net framework, including the nonlinear transformation, threshold parameters, and step size, are learned end-to-end instead of being manually set. This method combines the interpretability of traditional methods with the fast inference ability of deep learning and is able to handle noise at each iteration during the unfolding stage, thus improving recovery quality. Experiments on Coded Diffraction Patterns (CDPs) measurements demonstrate that our approach outperforms the existing state-of-the-art methods in terms of qualitative and quantitative evaluations. Our source codes are available at https://***/liuaxou/PRISTA-Net. Copyright © 2023, The Authors. All rights reserved.

关键词： Mapping

Robust approximation of chance constrained optimization with polynomial perturbation

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arXiv 2022年

作者： Rao, Bo Yang, Liu Zhong, Suhan Zhou, Guangming School of Mathematics and Computational Sciences Xiangtan University Hunan China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan China Department of Mathematics Texas A&M University College StationTX77843 United States

This paper proposes a robust approximation method for solving chance constrained optimization (CCO) of polynomials. Assume the CCO is defined with an individual chance constraint that is affine in the decision variables. We construct a robust approximation by replacing the chance constraint with a robust constraint over an uncertainty set. When the objective function is linear or SOS-convex, the robust approximation can be equivalently transformed into linear conic optimization. Semidefinite relaxation algorithms are proposed to solve these linear conic transformations globally and their convergent properties are studied. We also introduce a heuristic method to find efficient uncertainty sets such that optimizers of the robust approximation are feasible to the original problem. Numerical experiments are given to show the efficiency of our method. Copyright © 2022, The Authors. All rights reserved.

关键词： Heuristic methods

An adaptive high-order surface finite element method for polymeric self-consistent field theory on general curved surfaces

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arXiv 2021年

作者： Jiang, Kai Wang, Xin Liu, Jianggang Wei, Huayi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new Log marking strategy is proposed. Compared with the traditional strategy, the Log marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes. Copyright © 2021, The Authors. All rights reserved.

关键词： Molecular orbitals

SOLVING TIME-DEPENDENT PARAMETRIC PDES BY MULTICLASS CLASSIFICATION-BASED REDUCED ORDER MODEL

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arXiv 2021年

作者： Cui, Chen Jiang, Kai Shu, Shi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In this paper, we propose a network model, the multiclass classification-based reduced order model (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) [14] to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based on this fact, we classify the dataset according to the magnitude of the solutions and construct corresponding subnets dependent on different types of data. Then we train a classifier to integrate different subnets together to obtain the MC-ROM. When subsets have the same architecture, we can use transfer learning techniques to accelerate offline training. Numerical experiments show that the MC-ROM improves the generalization ability of the DL-ROM both for diffusion- and convection-dominant problems, and maintains the DL-ROM’s advantage of good approximation ability. We also compare the approximation accuracy and computational efficiency of the proper orthogonal decomposition (POD) which is not suitable for convection-dominant problems. For diffusion-dominant problems, the MC-ROM has better approximation accuracy than the POD in a small dimensionality reduction space, and its computational performance is more efficient than the POD’s. Copyright © 2021, The Authors. All rights reserved.

关键词： Partial differential equations

Phase behavior of symmetric diblock copolymers under 3D soft confinement

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arXiv 2024年

作者： He, Zhijuan Huang, Jin Jiang, Kai Shi, An-Chang School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education 411105 China Laboratory of Mathematics and Complex Systems Ministry of Education School of Mathematical Sciences Beijing Normal University Beijing100875 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

The phase behavior of symmetric diblock copolymers under three-dimensional (3D) soft confinement is investigated using the self-consistent field theory. The soft confinement is realized in binary blends composed AB diblock copolymers and C homopolymers, where the copolymers self-assemble to form a droplet embedded in the homopolymer matrix. The phase behavior of the confined block copolymers is regulated by the degree of confinement and the selectivity of the homopolymers, resulting in a rich variety of novel structures. When the C homopolymers are neutral to the A- and B-blocks, stacked lamellae (SL) are formed where the number of layers increases with the droplet volume, resulting in a morphological transition sequence from Janus particle to square SL. When the C homopolymers are strongly selective to the B-blocks, a series of non-lamellar morphologies, including onion-, hamburger-, cross-, ring-, and cookie-like structures, are observed. A detailed free energy analysis reveals a first-order reversible transformation between SL and onion-like (OL) structures when the selectivity of the homopolymers is changed. Our results provide a comprehensive understanding of how various factors, such as the copolymer concentration, homopolymer chain length, degree of confinement, homopolymer selectivity, affect the self-assembled structures of diblock copolymers under soft 3D confinement. Copyright © 2024, The Authors. All rights reserved.

关键词： Free energy

Multicontinuum Homogenization for Coupled Flow and Transport Equations

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arXiv 2024年

作者： Ammosov, Dmitry Leung, W.T. Shan, Buzheng Huang, Jian Computational Technologies & Artificial Intelligence North-Eastern Federal University Yakutsk677980 Russia Department of Mathematics City University of Hong Kong Hong Kong Department of Mathematics Texas A&M University College StationTX77843 United States School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan Xiangtan411105 China

In this paper, we present the derivation of a multicontinuum model for the coupled flow and transport equations by applying multicontinuum homogenization. We perform the multicontinuum expansion for both flow and transport solutions and formulate novel coupled constraint cell problems to capture the multiscale property, where oversampled regions are utilized to avoid boundary effects. Assuming the smoothness of macroscopic variables, we obtain a multicontinuum system composed of macroscopic elliptic equations and convection-diffusion-reaction equations with homogenized effective properties. Finally, we present numerical results for various coefficient fields and boundary conditions to validate our proposed algorithm. Copyright © 2024, The Authors. All rights reserved.

关键词：

VIRTUAL ELEMENT METHODS WITHOUT EXTRINSIC STABILIZATION

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arXiv 2022年

作者： Chen, Chunyu Huang, Xuehai Wei, Huayi Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Mathematics Shanghai University of Finance and Economics Shanghai200433 China

Virtual element methods (VEMs) without extrinsic stabilization in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local H(div)-conforming macro finite element spaces such that the associated L2 projection of the gradient of virtual element functions is computable, and the L2 projector has a uniform lower bound on the gradient of virtual element function spaces in L2 norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic *** Codes 65N12, 65N22, 65N30 Copyright © 2022, The Authors. All rights reserved.

关键词： Stabilization

GEOMETRIC DECOMPOSITION AND EFFICIENT IMPLEMENTATION OF HIGH ORDER FACE AND EDGE ELEMENTS

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arXiv 2023年

作者： Chen, Chunyu Chen, Long Huang, Xuehai Wei, Huayi School of Mathematics and Computational Science Xiangtan University National Center of Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China Department of Mathematics University of California at Irvine IrvineCA92697 United States School of Mathematics Shanghai University of Finance and Economics Shanghai200433 China

This study investigates high-order face and edge elements in finite element methods, with a focus on their geometric attributes, indexing management, and practical application. The exposition begins by a geometric decomposition of Lagrange finite elements, setting the foundation for further analysis. The discussion then extends to H(div)-conforming and H(curl)-conforming finite element spaces, adopting variable frames across differing sub-simplices. The imposition of tangential or normal continuity is achieved through the strategic selection of corresponding bases. The paper concludes with a focus on efficient indexing management strategies for degrees of freedom, offering practical guidance to researchers and engineers. It serves as a comprehensive resource that bridges the gap between theory and *** Codes 65N30, 35Q60 Copyright © 2023, The Authors. All rights reserved.

关键词： Degrees of freedom (mechanics)