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Identifying source term in the subdiffusion equation with L2-TV regularization

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arXiv 2021年

作者： Fan, Bin Xu, Chuanju School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China

In this paper, we consider the inverse source problem for the time-fractional diffusion equation, which has been known to be an ill-posed problem. To deal with the ill-posedness of the problem, we propose to transform the problem into a regularized problem with L2 and total variational (TV) regularization terms. Differing from the classical Tikhonov regularization with L2 penalty terms, the TV regularization is beneficial for reconstructing discontinuous or piecewise constant solutions. The regularized problem is then approximated by a fully discrete scheme. Our theoretical results include: estimate of the error order between the discrete problem and the continuous direct problem;the convergence rate of the discrete regularized solution to the target source term;and the convergence of the regularized solution with respect to the noise level. Then we propose an accelerated primal-dual iterative algorithm based on an equivalent saddle-point reformulation of the discrete regularized model. Finally, a series of numerical tests are carried out to demonstrate the efficiency and accuracy of the algorithm. Copyright © 2021, The Authors. All rights reserved.

关键词： Partial differential equations

AN EFFICIENT NUMERICAL APPROACH FOR STOCHASTIC EVOLUTION PDES DRIVEN BY RANDOM DIFFUSION COEFFICIENTS AND MULTIPLICATIVE NOISE∗

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arXiv 2022年

作者： Xiao, Q.I. Azaiez, Mejdi Huang, Can Chuanju, X.U. School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China Bordeaux INP Laboratoire I2M UMR 5295 Pessac33607 France

In this paper, we investigate the stochastic evolution equations (SEEs) driven by log-Whittle-Matérn (W-M) random diffusion coefficient field and Q-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result. Copyright © 2022, The Authors. All rights reserved.

关键词： Diffusion

ON SHRINKING TARGETS FOR LINEAR EXPANDING AND HYPERBOLIC TORAL ENDOMORPHISMS

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arXiv 2024年

作者： Hu, Zhang-Nan Persson, Tomas Wu, Wanlou Zhang, Yiwei College of Science China University of Petroleum Beijing102249 China Centre for Mathematical Sciences Lund University Box 118 Lund221 00 Sweden School of Mathematics and Statistics Jiangsu Normal University Xuzhou221116 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Let A be an invertible d × d matrix with integer elements. Then A determines a self-map T of the d-dimensional torus Td = Rd/Zd. Given a real number τ > 0, and a sequence {zn} of points in Td, let Wτ be the set of points x ∈ Td such that Tn(x) ∈ B(zn, e−nτ) for infinitely many n ∈ . The Hausdorff dimension of Wτ has previously been studied by Hill-Velani and Li-Liao-Velani-Zorin. We provide complete results on the Hausdorff dimension of Wτ for any expanding matrix. For hyperbolic matrices, we compute the dimension of Wτ only when A is a 2 × 2 matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension *** Codes 37C45, 37E05 Copyright © 2024, The Authors. All rights reserved.

关键词： Matrix algebra

A Progressive Image Restoration Network for High-order Degradation Imaging in Remote Sensing

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arXiv 2024年

作者： Feng, Yujie Yang, Yin Fan, Xiaohong Zhang, Zhengpeng Bu, Lijing Zhang, Jianping School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory for Intelligent Computing and Information Processing of the Ministry of Education Xiangtan411105 China School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science Xiangtan411105 China College of Mathematical Medicine Zhejiang Normal University Jinhua321004 China School of Automation and Electronic Information Xiangtan University Xiangtan411105 China

Recently, deep learning methods have gained remarkable achievements in the field of image restoration for remote sensing (RS). However, most existing RS image restoration methods focus mainly on conventional first-order degradation models, which may not effectively capture the imaging mechanisms of remote sensing images. Furthermore, many RS image restoration approaches that use deep learning are often criticized for their lacks of architecture transparency and model interpretability. To address these problems, we propose a novel progressive restoration network for high-order degradation imaging (HDI-PRNet), to progressively restore different image degradation. HDI-PRNet is developed based on the theoretical framework of degradation imaging, offering the benefit of mathematical interpretability within the unfolding network. The framework is composed of three main components: a module for image denoising that relies on proximal mapping prior learning, a module for image deblurring that integrates Neumann series expansion with dual-domain degradation learning, and a module for super-resolution. Extensive experiments demonstrate that our method achieves superior performance on both synthetic and real remote sensing images. Copyright © 2024, The Authors. All rights reserved.

关键词： Image reconstruction

A high-order discrete energy decay and maximum-principle preserving scheme for time fractional Allen-Cahn equation

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arXiv 2023年

作者： Zhang, Guoyu Huang, Chengming Alikhanov, Anatoly A. Yin, Baoli School of Mathematical Sciences Inner Mongolia University Hohhot010021 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China North-Caucasus Center for Mathematical Research North-Caucasus Federal University Stavropol355017 Russia

The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen-Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for the first time. Based on these properties, we rigorously demonstrate the discrete energy decay property and maximum-principle preservation for the scheme. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. Comparison with the classic fractional BDF2 and L2-1σ method further validates the superiority of SFTR in solving the tFAC equation. Experiments concerning both discrete energy decay and discrete maximum-principle also verify the correctness of the theoretical results. © 2023, CC BY-NC-SA.

关键词： Maximum principle

HIGH ORDER ASYMPTOTIC PRESERVING WELL-BALANCED FINITE DIFFERENCE WENO SCHEMES FOR ALL MACH FULL EULER EQUATIONS WITH GRAVITY

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arXiv 2022年

作者： Huang, Guanlan Xing, Yulong Xiong, Tao School of Mathematical Sciences Xiamen University Fujian Xiamen361005 China Department of Mathematics The Ohio State University ColumbusOH43210 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, we propose a high order semi-implicit well-balanced finite difference scheme for all Mach Euler equations with a gravitational source term. To obtain the asymptotic preserving property, we start from the conservative form of full compressible Euler equations and add the evolution equation of the perturbation of potential temperature. The resulting system is then split into a (non-stiff) nonlinear low dynamic material wave to be treated explicitly, and (stiff) fast acoustic and gravity waves to be treated implicitly. With the aid of explicit time evolution for the perturbation of potential temperature, we design a novel well-balanced finite difference WENO scheme for the conservative variables, which can be proven to be both asymptotic preserving and asymptotically accurate in the incompressible limit. Extensive numerical experiments were provided to validate these properties. © 2022, CC BY.

关键词： Euler equations

Moment-based multi-resolution HWENO scheme for hyperbolic conservation laws

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arXiv 2022年

作者： Li, Jiayin Shu, Chi-Wang Qiu, Jianxian School Of Mathematical Sciences Xiamen University Fujian Xiamen361005 China Division Of Applied Mathematics Brown University ProvidenceRI02912 United States School Of Mathematical Sciences Fujian Provincial Key Laboratory Of Mathematical Modeling And High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J. Comput. Phys., 446 (2021) 110653], in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries. However, in this paper, only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments. In addition, an extra modification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolution near discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights can also be any positive numbers as long as their sum equals one and the CFL number can still be 0.6 whether for the one or two dimensional case. Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO *** Codes 65M60, 35L65 © 2022, CC BY.

关键词： Physical properties

A Jacobi Collocation Method for the Fractional Ginzburg-Landau Differential Equation

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Advances in Applied Mathematics and Mechanics 2020年第1期12卷 57-86页

作者： Yin Yang Jianyong Tao Shangyou Zhang Petr V.Sivtsev Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan 411105China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA International Scientific and Research Laboratory of Multiscale Model Reduction and High Performance Computing Ammosov North Eastern UniversityKulakovskogo677013YakutskRussia

In this paper,we design a collocation method to solve the fractional Ginzburg-Landau equation.A Jacobi collocation method is developed and implemented in two ***,we space-discretize the equation by the Jacobi-Gauss-Lobatto collocation(JGLC)method in one-and two-dimensional *** equation is then converted to a system of ordinary differential equations(ODEs)with the time variable based on *** second step applies the Jacobi-Gauss-Radau collocation(JGRC)method for the time ***,we give a theoretical proof of convergence of this Jacobi collocation method and some numerical results showing the proposed scheme is an effective and high-precision algorithm.

关键词： The fractional Ginzburg-Landau equation Jacobi collocation method convergence

ASYMPTOTIC ERROR DISTRIBUTION OF ACCELERATED EXPONENTIAL EULER METHOD FOR PARABOLIC SPDES

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arXiv 2024年

作者： Hong, Jialin Jin, Diancong Wang, Xu Yang, Guanlin LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

The asymptotic error distribution of numerical methods applied to stochastic ordinary differential equations has been well studied, which characterizes the evolution pattern of the error distribution in the small step-size regime. It is still open for stochastic partial differential equations whether the normalized error process of numerical methods admits a nontrivial limit distribution. We answer this question by presenting the asymptotic error distribution of the temporal accelerated exponential Euler (AEE) method when applied to parabolic stochastic partial differential equations. In order to overcome the difficulty caused by the infinite-dimensional setting, we establish a uniform approximation theorem for convergence in distribution. Based on it, we derive the limit distribution of the normalized error process of the AEE method by studying the limit distribution of its certain appropriate finite-dimensional approximation process. As applications of our main result, the asymptotic error distribution of a fully discrete AEE method for the original equation and that of the AEE method for a stochastic ordinary differential equation are also obtained. Copyright © 2024, The Authors. All rights reserved.

关键词： Ordinary differential equations