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A Deep Learning Method for computing Eigenvalues of the Fractional Schrödinger Operator

Journal of Systems Science & Complexity 2024年第2期37卷 391-412页

作者： GUO Yixiao MING Pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger *** proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the *** improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded *** authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger *** an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

关键词： Eigenvalue problem deep learning fractional Schrödinger operator isospectral problem

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A Derivative-Free Optimization Algorithm Combining Line-Search and Trust-Region Techniques

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Chinese Annals of mathematics,Series B 2023年第5期44卷 719-734页

作者： Pengcheng XIE Ya-xiang YUAN State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematicsand Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesUniversity of Chinese Academy of SciencesBeijing 100190China

The speeding-up and slowing-down(SUSD)direction is a novel direction,which is proved to converge to the gradient descent direction under some *** authors propose the derivative-free optimization algorithm SUSD-TR,which combines the SUSD direction based on the covariance matrix of interpolation points and the solution of the trust-region subproblem of the interpolation model function at the current iteration *** analyze the optimization dynamics and convergence of the algorithm *** of the trial step and structure step are *** results show their algorithm’s efficiency,and the comparison indicates that SUSD-TR greatly improves the method’s performance based on the method that only goes along the SUSD *** algorithm is competitive with state-of-the-art mathematical derivative-free optimization algorithms.

关键词： Nonlinear optimization Derivative-Free Quadratic model Line-Search Trust-Region

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STOCHASTIC TRUST-REGION METHODS WITH TRUST-REGION RADIUS DEPENDING ON PROBABILISTIC MODELS

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Journal of computational mathematics 2022年第2期40卷 294-334页

作者： Xiaoyu Wang Ya-xiang Yuan Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100049China State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic ***,we propose a specific algorithm termed STRME,in which the trust-region radius depends linearly on the gradient used to define the latest *** complexity results of the STRME method in nonconvex,convex and strongly convex settings are presented,which match those of the existing algorithms based on probabilistic *** addition,several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods.

关键词： Trust-region methods,Stochastic optimization Probabilistic models Trust-region radius Global convergence

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CONVERGENCE ANALYSIS OF SOME FINITE ELEMENT PARALLEL ALGORITHMS FOR THE STATIONARY INCOMPRESSIBLE MHD EQUATIONS

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Journal of computational mathematics 2024年第1期42卷 49-70页

作者： Xiaojing Dong Yinnian He 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 UniversityXiangtan 411105China Institute of Applied Physics and Computational Mathematics Beijing 100088China School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China

By combination of iteration methods with the partition of unity method(PUM),some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics(MHD)with different physical parameters are presented and *** algorithms are highly *** first,a global solution is obtained on a coarse grid for all approaches by one of the iteration *** parallelized residual schemes,local corrected solutions are calculated on finer meshes with overlapping *** subdomains can be achieved flexibly by a class of *** proposed algorithm is proved to be uniformly stable and ***,one numerical example is presented to confirm the theoretical findings.

关键词： Partition of unity method Local and parallel algorithm Finite element method Iteration methods Magnetohydrodynamics

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A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

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Communications on Applied mathematics and Computation 2025年第2期7卷 606-636页

作者： Qigang Liang Wei Wang Xuejun Xu School of Mathematical Science Tongji UniversityShanghai200092China Key Laboratory of Intelligent Computing and Applications(Tongji University) Ministry of EducationShanghai200092China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

Since the nonconforming finite elements(NFEs)play a significant role in approximating PDE eigenvalues from below,this paper develops a new and parallel two-level preconditioned Jacobi-Davidson(PJD)method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth(m=1.2)order elliptic eigenvalue *** a spectral projection on the coarse space and an overlapping domain decomposition(DD),a parallel preconditioned system can be solved in each iteration.A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and *** results supporting our theory are given.

关键词： PDE eigenvalue problems Nonconforming finite elements(NFEs) Preconditioned Jacobi-Davidson(PJD)method Overlapping domain decomposition(DD)

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Solving Time Dependent Fokker-Planck Equations via Temporal Normalizing Flow

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Communications in computational Physics 2022年第7期32卷 401-423页

作者： Xiaodong Feng Li Zeng Tao Zhou LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesBeijingChina

In this work,we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck(TFP)*** is well known that solutions of such equations are probability density functions,and thus our approach relies on modelling the target solutions with the temporal normalizing *** temporal normalizing flow is then trained based on the TFP loss function,without requiring any labeled *** a machine learning scheme,the proposed approach is mesh-free and can be easily applied to high dimensional *** present a variety of test problems to show the effectiveness of the learning approach.

关键词： Temporal normalizing flow Fokker-Planck equations adaptive density approximation

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MC-Nonlocal-PINNs:Handling Nonlocal Operators in PINNs Via Monte Carlo Sampling

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Numerical mathematics(Theory,Methods and Applications) 2023年第3期16卷 769-791页

作者： Xiaodong Feng Yue Qian Wanfang Shen Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina Shandong Key Laboratory of Blockchain Finance Shandong University of Finance and EconomicsJinan 250014China

We propose Monte Carlo Nonlocal physics-informed neural networks(MC-Nonlocal-PINNs),which are a generalization of MC-fPINNs in *** et al.(*** ***.400(2022),115523)for solving general nonlocal models such as integral equations and nonlocal *** to MC-fPINNs,our MC-Nonlocal-PINNs handle nonlocal operators in a Monte Carlo way,resulting in a very stable approach for high dimensional *** present a variety of test problems,including high dimensional Volterra type integral equations,hypersingular integral equations and nonlocal PDEs,to demonstrate the effectiveness of our approach.

关键词： Nonlocal models PINNs Monte Carlo sampling deep neural networks

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Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

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Numerical mathematics(Theory,Methods and Applications) 2023年第1期16卷 1-25页

作者： Xiaoying Dai Liwei Zhang Aihui Zhou LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure *** this paper,we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model,with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given *** addition,we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.

关键词： Gradient flow based model density functional theory orthogonality preserving scheme convergence temporal discretization

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A two-dimensional double layer-averaged model of hyperconcentrated turbidity currents with non-Newtonian rheology

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International Journal of Sediment Research 2023年第6期38卷 794-810页

作者： Yining Sun Ji Li Zhixian Cao Alistair George Liam Borthwick State Key Laboratory of Water Resources Engineering and Management Wuhan University Zienkiewicz Centre for Computational Engineering Faculty of Science and EngineeringSwansea University Institute for Infrastructure and Environment University of Edinburgh School of Engineering Computing and MathematicsUniversity of Plymouth

Hyperconcentrated turbidity currents typically display non-Newtonian characteristics that influence sediment transport and morphological evolution in alluvial ***,hydro-sedimentmorphological processes involving hyperconcentrated turbidity currents are poorly understood,with little known about the effect of the non-Newtonian *** current paper extends a recent twodimensional double layer-averaged model to incorporate non-Newtonian constitutive *** extended model is benchmarked against experimental and numerical data for cases including subaerial mud flow,subaqueous debris flow,and reservoir turbidity *** computational results agree well with observations for the subaerial mud flow and independent numerical simulations of subaqueous debris *** between the non-Newtonian and Newtonian model results become more pronounced in terms of propagation distance and sediment transport rate as sediment concentration *** model is then applied to turbidity currents in the Guxian Reservoir planned for middle Yellow River,China,which connects to a tributary featuring hyperconcentrated sediment-laden *** non-Newtonian model predicts slower propagation of turbidity currents and more significant bed aggradation at the confluence between the tributary Wuding River and the Yellow River in the reservoir than its Newtonian *** difference in model performance could be of considerable importance when optimizing reservoir operation schemes.

关键词： Double layer-averaged model Non-Newtonian rheology Mud flow Reservoir turbidity current Yellow River

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A Stochastic Gradient Descent Method for computational Design of Random Rough Surfaces in Solar Cells

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Communications in computational Physics 2023年第10期34卷 1361-1390页

作者： Qiang Li Gang Bao Yanzhao Cao Junshan Lin Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Department of Mathematics Zhejiang UniversityChina Department of Mathematics and Statistics Auburn UniversityAuburnAL 36849USA

In this work,we develop a stochastic gradient descent method for the computational optimal design of random rough surfaces in thin-film solar *** formulate the design problems as random PDE-constrained optimization problems and seek the optimal statistical parameters for the random *** optimizations at fixed frequency as well as at multiple frequencies and multiple incident angles are *** evaluate the gradient of the objective function,we derive the shape derivatives for the interfaces and apply the adjoint state method to perform the *** stochastic gradient descent method evaluates the gradient of the objective function only at a few samples for each iteration,which reduces the computational cost *** numerical experiments are conducted to illustrate the efficiency of the method and significant increases of the absorptance for the optimal random *** also examine the convergence of the stochastic gradient descent algorithm theoretically and prove that the numerical method is convergent under certain assumptions for the random interfaces.

关键词： Optimal design random rough surface solar cell Helmholtz equation stochastic gradient descent method

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