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作者： Ng, Michael K. Bai, Zhong-Zhi Department of Mathematics University of Hong Kong Pokfulam Road Hong Kong Hong Kong State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of Sciences P.O. Box 2719 Beijing 100080 China

The symmetric Sinc-Galerkin method applied to a sparable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations(Ψx⊗Dy+Dx⊗Ψ y)u=g,where⊗ is the Kronecker product symbol, Ψx and Ψy are Toeplitz-plus-diagonal matrices, and Dx and Dy are diagonal matrices. The main contribution of this paper is to present and analyze a two-step preconditioning strategy based on the banded matrix approximation (BMA) and the alternating direction implicit (ADI) iteration for these Sinc-Galerkin systems. In particular, we show that the two-step preconditioner is symmetric positive definite, and the condition number of the preconditioned matrix is bounded by the convergence factor of the involved ADI iteration. Numerical examples show that the new preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc-Galerkin linear system. © 2003 Elsevier Science Inc.

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Message from the IWPAPS 2008 workshop chair

Proceedings - 10th IEEE International Conference on High Per...

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Proceedings - 10th IEEE International Conference on High Performance computing and Communications, HPCC 2008 2008年 xxiv页

作者： Linbo, Zhang State Key Laboratory of Scientific and Engineering Computing CAS China

No abstract available

ISBN: (纸本)9780769533520

No abstract available

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Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems

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Communications in Computational Physics 2023年第2期33卷 596-627页

作者： Sidi Wu Aiqing Zhu Yifa Tang Benzhuo Lu State Key Laboratory of Scientific and Engineering Computing National Center for Mathematics and Interdisciplinary SciencesAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China.

With the remarkable empirical success of neural networks across diverse scientific disciplines,rigorous error and convergence analysis are also being developed and ***,there has been little theoretical work focusing on neural networks in solving interface *** this paper,we perform a convergence analysis of physics-informed neural networks(PINNs)for solving second-order elliptic interface ***,we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump *** is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in H2 as the number of samples *** experiments are provided to demonstrate our theoretical analysis.

关键词： Elliptic interface problems generalization errors convergence analysis neural networks.

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On the worst-case complexity of nonlinear stepsize control algorithms for convex unconstrained optimization

On the worst-case complexity of nonlinear stepsize control a...

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作者： Grapiglia, G.N. Yuan, J. Yuan, Y. Department of Mathematics Federal University of Paraná Curitiba Brazil State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

A Nonlinear Stepsize Control (NSC) framework has been proposed by Toint [Nonlinear stepsize control, trust regions and regularizations for unconstrained optimization, *** Softw. 28 (2013), pp. 82-95] for unconstrained optimization, generalizing many trust-region and regularization algorithms. More recently, worst-case complexity bounds for the generic NSC framework were proved by Grapiglia et al. [On the convergence and worst-case complexity of trust-region and regularization methods for unconstrained optimization, Math. Program. 152 (2015), pp. 491-520] in the context of non-convex problems. In this paper, improved complexity bounds are obtained for convex and strongly convex objectives. © 2015 Taylor & Francis.

关键词： Optimization

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A new reconstruction algorithm in tomography with geometric feature-preserving regularization

A new reconstruction algorithm in tomography with geometric ...

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International Conference on BioMedical engineering and Informatics (BMEI)

作者： Chong Chen Guoliang Xu Institute of Computational Mathematics and Scientific Engineering Computing State Key Laboratory of Scientific and Engineering Computing Chinese Academy of Sciences Beijing China Institute of Computational Mathematics and Scientific Engineering Computing State Key Laboratory of Scientific and Engineering Computing Chinese Academy and Sciences Beijing China

We present a new optimization-based reconstruction model with geometric feature-preserving regularization. The previous construction approach of optimization-based requires generally solving a tremendously large discrete linear system. In addition, the analytic algorithms, such as, back-projection related methods, Fourier transform based reconstruction algorithms, are sensitive to noise. These disadvantages motivate us naturally to present a new optimization-based regularization algorithm, as distinct from traditionally constructing a discrete linear system, that is based upon a continuous energy functional model like analytic one. Some theoretical derivations and numerical computing of our model are discussed. Experimental results illustrate the desirable performance of the algorithm under various noiseless data situations. And the geometric feature-preserving regularizer is used in numerical simulations to demonstrate the robustness and effectiveness of the algorithm under various contaminated data scenarios.

关键词： Image reconstruction Noise Distributed databases Mathematical model Computational modeling Numerical models Computed tomography

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A Brief Introduction to Manifold Optimization

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Journal of the Operations Research Society of China 2020年第2期8卷 199-248页

作者： Jiang Hu Xin Liu Zai-Wen Wen Ya-Xiang Yuan Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100190China

Manifold optimization is ubiquitous in computational and appliedmathematics,statistics,engineering,machine learning,physics,chemistry,*** of the main challenges usually is the non-convexity of the manifold *** utilizing the geometry of manifold,a large class of constrained optimization problems can be viewed as unconstrained optimization problems on *** this perspective,intrinsic structures,optimality conditions and numerical algorithms for manifold optimization are *** recent progress on the theoretical results of manifold optimization is also presented.

关键词： Convergence First-order-type algorithms Manifold optimization Retraction Second-order-type algorithms

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Stochastic Multi-Symplectic Integrator for Stochastic Nonlinear Schrodinger Equation

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Communications in Computational Physics 2013年第7期14卷 393-411页

作者： Shanshan Jiang Lijin Wang Jialin Hong College of Science Beijing University of Chemical TechnologyBeijing 100029P.R.China School of Mathematical Sciences Graduate University of Chinese Academy of SciencesBeijing 100049P.R.China State Key Laboratory of Scientific and Engineering Computing Institute of ComputationalMathematics and Scientific/Engineering ComputingAcademy of Mathematics and System ScienceChinese Academy of SciencesP.O.Box 2719Beijing 100190P.R.China

In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations,and develop a stochastic multisymplectic method for numerically solving a kind of stochastic nonlinear Schrodinger *** is shown that the stochasticmulti-symplecticmethod preserves themultisymplectic structure,the discrete charge conservation law,and deduces the recurrence relation of the discrete *** experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision.

关键词： Stochastic nonlinear Schrodinger equations stochasticmulti-symplectic Hamiltonian systems multi-symplectic integrators

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Dynamic Load Balancing on Adaptive Unstructured Meshes

Dynamic Load Balancing on Adaptive Unstructured Meshes

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IEEE International Conference on High Performance computing and Communications (HPCC)

作者： Hui Liu State Key Laboratory of Scientific and Engineering Computing CAS Beijing China

In this paper the work on implementing two mesh partitioning algorithms, the refinement-tree based partitioning algorithm and the space-filling curve partitioning algorithm, in the parallel adaptive finite element toolbox PHG (parallel hierarchical grid) is presented. These algorithms are used for both initial mesh partitioning and mesh repartitioning for dynamical load balancing in adaptive finite element computations. In the implementations improved algorithms are designed. Partitioning time and quality of our code are compared with existing publicly available mesh or graph partitioners, including ParMETIS and Zoltan, through some numerical examples.

关键词： Finite element methods Partitioning algorithms Indexes Face Complexity theory Algorithm design and analysis Classification algorithms

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Impacts of prior parameter distributions on Bayesian evaluation of groundwater model complexity

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Water Science and engineering 2018年第2期11卷 89-100页

作者： Saeideh Samani Ming Ye Fan Zhang Yong-zhen Pei Guo-ping Tang Ahmed Elshall Asghar A.Moghaddam Department of Earth Sciences University of Tabriz Department of Scientific Computing Florida State University School of Computer Science and Software Engineering Tianjin Polytechnic University Department of Earth Ocean and Atmospheric Science Florida State University Key Laboratory of Tibetan Environment Changes and Land Surface Processes Institute of Tibetan Plateau Research Chinese Academy of Sciences Environmental Sciences Division Oak Ridge National Laboratory

This study used the marginal likelihood and Bayesian posterior model probability for evaluation of model complexity in order to avoid using over-complex models for numerical simulations. It focused on investigation of the impacts of prior parameter distributions(involved in calculating the marginal likelihood) on the evaluation of model complexity. We argue that prior parameter distributions should define the parameter space in which numerical simulations are made. New perspectives on the prior parameter distribution and posterior model probability were demonstrated in an example of groundwater solute transport modeling with four models, each simulating four column experiments. The models had different levels of complexity in terms of their model structures and numbers of calibrated parameters. The posterior model probability was evaluated for four cases with different prior parameter distributions. While the distributions substantially impacted model ranking, the model ranking in each case was reasonable for the specific circumstances in which numerical simulations were made. For evaluation of model complexity, it is thus necessary to determine the parameter spaces for modeling, which can be done by conducting numerical simulation and usineg engineering judgment based on understanding of the system being studied.

关键词： Marginal likelihood Posterior model probability Advection-dispersion equation Mobile-immobile model Groundwater model

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Self-consistent phonon approaches for the hydrogen bond chain

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Physical Review E 1997年第4期55卷 4531-4531页

作者： Yong-li Zhang Wei-Mou Zheng and State Key Laboratory for Scientific and Engineering Computing CAS P.O. Box 2719 Beijing 100080 China

The hydrogen bonded ammonia chain model is studied by means of the standard self-consistent phonon approach and two modified versions. The effective crystal constant, force constant, free energy, and ratio of the first-order free energy to the zero order as a function of temperature are numerically obtained with the three approaches, and then compared. The standard approach gives the lowest free energy. Violation of the convexity is found in one of the modified approaches near the temperature which is regarded as the melting temperature.

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