Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding pr...
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Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding preconditioned matrices. Numerical experiments show that these rotated block triangular pre- conditioners can be competitive to and even more efficient than the PMHSS preconditioner when they are used to accelerate Krylov subspeme iteration methods for solving block two-by-two linear systems with coefficient matrices possibly of nonsymmetric sub-blocks.
作者：
CUI LongMING PingBingLSEC
Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences
We study the effect of ＂ghost forces＂ for a quasicontinuum method in three dimension with a planar interface. ＂Ghost forces＂ are the inconsistency of the quasicontinuum method across the interface between the atom...
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We study the effect of ＂ghost forces＂ for a quasicontinuum method in three dimension with a planar interface. ＂Ghost forces＂ are the inconsistency of the quasicontinuum method across the interface between the atomistic region and the continuum region. Numerical results suggest that ＂ghost forces＂ may lead to a negilible error on the solution, while lead to a finite size error on the gradient of the solution. The error has a layer-like profile, and the interfacial layer width is of O（ε）. The error in certain component of the displacement gradient decays algebraically from O（1） to O（ε） away from the interface. A surrogate model is proposed and analyzed, which suggests the same scenario for the effect of ＂ghost forces＂. Our analysis is based on the explicit solution of the surrogate model.
The rise of multimodal generative AI is transforming the intersection of technology and art, offering deeper insights into large-scale artwork. Although its creative capabilities have been widely explored, its potenti...
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Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. It is mainly considered as a powerful tool with high flexibility to approximate functions. This paper pro...
Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. It is mainly considered as a powerful tool with high flexibility to approximate functions. This paper proposes a generalized NURBS based approach to solve nonlinear partial differential equations (PDEs) on arbitrary complex-geometry domains by using physics-informed neural networks (PINNs). Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network. An efficient and easy to implement algorithm is developed to obtain a posteriori error estimate for multiple goal functionals by employing the dual-weighted residual approach, which is followed by the computation of both primal and adjoint solutions using the neural network. The present study shows that such a data-driven model based learning has superior approximation of quantities of interest even with relatively less training data. Moreover, we illustrate the versatility of activation functions in achieving better learning capabilities and improving convergence rates, especially at the early training stage, and also in increasing solutions accuracies. The novel algorithmic developments are substantiated with several numerical test examples. It has been demonstrated that deep neural networks have distinct advantages over shallow neural networks, and the techniques for enhancing convergence have also been reviewed.
Professor Junzhi Cui was born on June 15, 1938 in Xinxiang, Henan Province in China. He graduated from the Department of mathematics and Mechanics, Northwestern Polytechnic University in 1962. Since then, he has been ...
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Professor Junzhi Cui was born on June 15, 1938 in Xinxiang, Henan Province in China. He graduated from the Department of mathematics and Mechanics, Northwestern Polytechnic University in 1962. Since then, he has been working in the institute of computing Technology （1962-1978）, the computing Center （1978-1995）,
In this paper, we consider the optimization method for monotone variational inequality problems on polyhedral sets. First, we consider the mixed complementarity problem based on the original problem. Then, a merit fun...
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In this paper, we consider the optimization method for monotone variational inequality problems on polyhedral sets. First, we consider the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed and some desirable properties of the merit function are obtained. Under certain assumptions: we show that any stationary point of the merit function is a solution of the original problem. A descent method for the optimization problem is proposed and the global convergence of the method is shown.
The four-roll mill has been traditionally viewed as a device generating simple extensional flow with a central stagnation point. Our systematic investigation using a two-relaxation-time regularized lattice Boltzmann (...
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This paper investigates the complex interplay between AI developers, regulators, users, and the media in fostering trustworthy AI systems. Using evolutionary game theory and large language models (LLMs), we model the ...
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