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On Krylov subspace methods for skew-symmetric and shifted skew-symmetric linear systems

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arXiv 2023年

作者： Du, Kui Fan, Jia-Jun Sun, Xiao-Hui Wang, Fang Zhang, Ya-Lan School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China School of Mathematical Sciences Xiamen University Xiamen361005 China School of Mathematics and Statistics Xiamen University of Technology Xiamen361024 China

Krylov subspace methods for solving linear systems of equations involving skew-symmetric matrices have gained recent attention. Numerical equivalences among Krylov subspace methods for nonsingular skew-symmetric linear systems have been given in Greif et al. [SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1071–1087]. In this work, we extend the results of Greif et al. to singular skew-symmetric linear systems. In addition, we systematically study three Krylov subspace methods (called S3CG, S3MR, and S3LQ) for solving shifted skew-symmetric linear systems. They all are based on Lanczos triangularization for skew-symmetric matrices, and correspond to CG, MINRES, and SYMMLQ for solving symmetric linear systems, respectively. To the best of our knowledge, this is the first work that studies S3LQ. We give some new theoretical results on S3CG, S3MR, and S3LQ. We also provide the relationship among the three methods and those based on Golub–Kahan bidiagonalization and Saunders–Simon–Yip tridiagonalization. Numerical examples are given to illustrate our theoretical findings. Copyright © 2023, The Authors. All rights reserved.

关键词： Numerical methods

An Unconditionally Energy-Stable and Orthonormality-Preserving Iterative Scheme for the Kohn-Sham Gradient Flow Based Model

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arXiv 2023年

作者： Wang, Xiuping Chen, Huangxin Kou, Jisheng Sun, Shuyu Saudi Arabia School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and HighPerformance Scientific Computing Xiamen University Fujian Xiamen China Key Laboratory of Rock Mechanics and Geohazards of Zhejiang Province Shaoxing University Shaoxing China School of Mathematics and Statistics Hubei Engineering University Hubei Xiaogan China

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The componentwise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. At the time step n, the orthogonality of the wave function being updated to other wave functions is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to all other wave functions known at the current time, while the normalization of this wave function is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to this wave function at tn+1/2. The unconditional energy stability is nontrivial, and it comes from a subtle treatment of the two-electron integral as well as a consistent treatment of the two projections. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature. Copyright © 2023, The Aut

关键词： Wave functions

Numerical analysis of a semi-implicit Euler scheme for the Keller-Segel model

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arXiv 2025年

作者： Huang, Xueling Goubet, Olivier Shen, Jie School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian Xiamen361005 China Laboratoire Paul Painlevé CNRS UMR 8524 et équipe projet INRIA PARADYSE Université de Lille Lille59000 France School of Mathematical Science Eastern Institute of Technology Zhejiang Ningbo315200 China

We study the properties of a semi-implicit Euler scheme that is widely used in time discretization of Keller-Segel equations both in the parabolic-elliptic form and the parabolic-parabolic form. We prove that this linear, decoupled, first-order scheme preserves unconditionally the important properties of Keller-Segel equations at the semi-discrete level, including the mass conservation and positivity preserving of the cell density, and the energy dissipation. We also establish optimal error estimates in Lp-norm (1 Copyright © 2025, The Authors. All rights reserved.

关键词： Euler equations

Benchmark modeling and 3D applications of solidification and macro-segregation based on an operator-splitting and fully decoupled scheme with term-wise matrix assembly

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arXiv 2023年

作者： Feng, Xiaoyu Chen, Huangxin Yu, Bo Sun, Shuyu Thuwal23955-6900 Saudi Arabia School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China School of Mechanical Engineering Beijing Institute of Petrochemical Technology Beijing102617 China

The solidification and macro-segregation problem involving unsteady multiphysics and multi-phase fields is typically a complex process with mass, momentum, heat, and species transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes an operator-splitting and matrix-based method to avoid non-linear systems. Also, the combination of vectorization and forward equation-based matrix assembly techniques enhances the implementability of extensions of 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and adequately capture the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect. Copyright © 2023, The Authors. All rights reserved.

关键词： Solidification

Benchmark modeling and 3d Applications of Solidification and Macro-Segregation Based on an Operator-Splitting and Fully Decoupled Scheme with Term-Wise Matrix Assembly

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SSRN

SSRN 2023年

The solidification and macro-segregation problem involving unsteady multi-physics and multi-phase fields is typically a complex process with mass, momentum, heat, and constituent transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes fully decoupled and iterative schemes based on operator-splitting and matrix-based ideas to avoid solving non-linear systems directly. Also, the combination of direct vectorization and equation-based forward matrix assembly techniques enhances the implementability and efficiency of extensions to 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and efficient computation. It adequately captures the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect. © 2023, The Authors. All rights reserved.

关键词： Solidification

Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions

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arXiv 2023年

作者： Mao, Zhiping Meng, Xuhui School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen China Institute of Interdisciplinary Research for Mathematics and Applied Science School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China

We consider solving the forward and inverse PDEs which have sharp solutions using physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose adaptive sampling methods (ASMs) based on the residual and the gradient of the solution. We first present a residual only based ASM algorithm denoted by ASM I. In this approach, we first train the neural network by using a small number of residual points and divide the computational domain into a certain number of sub-domains, we then add new residual points in the sub-domain which has the largest mean absolute value of the residual, and those points which have largest absolute values of the residual in this sub-domain will be added as new residual points. We further develop a second type of ASM algorithm (denoted by ASM II) based on both the residual and the gradient of the solution due to the fact that only the residual may be not able to efficiently capture the sharpness of the solution. The procedure of ASM II is almost the same as that of ASM I except that in ASM II, we add new residual points which not only have large residual but also large gradient. To demonstrate the effectiveness of the present methods, we employ both ASM I and ASM II to solve a number of PDEs, including Burger equation, compressible Euler equation, Poisson equation over an L-shape domain as well as high-dimensional Poisson equation. It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASM I or ASM II algorithm, and both methods deliver much more accurate solution than original PINNs with the same number of residual points. Moreover, the ASM II algorithm has better performance in terms of accuracy, efficiency and stability compared with the ASM I algorithm. This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution. Furthermore

关键词： Poisson equation

THERMODYNAMIC FORMALISM FOR CORRESPONDENCES

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arXiv 2023年

作者： Li, Xiaoran Li, Zhiqiang Zhang, Yiwei School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Beijing International Center for Mathematical Research Peking University Beijing100871 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

In this work, we investigate the Variational Principle and develop a thermodynamic formalism for correspondences. We define the measure-theoretic entropy for transition probability kernels and topological pressure for correspondences. Based on these two notions, we establish the following results: The Variational Principle holds and equilibrium states exist for continuous potential functions, provided that the correspondence satisfies some expansion property called forward expansiveness. If, in addition, the correspondence satisfies the specification property and the potential function is Bowen summable, then the equilibrium state is unique. On the other hand, for a distance-expanding, open, strongly transitive correspondence and a Hölder continuous potential function, there exists a unique equilibrium state, and the backward orbits are equidistributed. Furthermore, we investigate the Variational Principle for general correspondences. In conformal dynamics, we establish the Variational Principle for the Lee-Lyubich-Markorov-Mazor-Mukherjee anti-holomorphic correspondences, which are matings of some anti-holomorphic rational maps with anti-Hecke groups and not forward expansive. We also show a Ruelle-Perron-Frobenius Theorem for a family of hyperbolic holomorphic correspondences of the form fc(z) = zq/p + *** Codes Primary: 37F05, Secondary: 54C60, 37D35, 37C30, 37D20 Copyright © 2023, The Authors. All rights reserved.

关键词： Thermodynamics

HIGH ORDER ASYMPTOTIC PRESERVING AND WELL-BALANCED SCHEMES FOR THE SHALLOW WATER EQUATIONS WITH SOURCE TERMS

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arXiv 2023年

作者： Huang, Guanlan Boscarino, Sebastiano Xiong, Tao School of Mathematical Sciences Xiamen University Fujian Xiamen361005 China Department of Mathematics and Computer Science University of Catania Catania95125 Italy School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this study, we investigate the Shallow Water Equations incorporating source terms accounting for Manning friction and a non-flat bottom topology. Our primary focus is on developing and validating numerical schemes that serve a dual purpose: firstly, preserving all steady states within the model, and secondly, maintaining the late-time asymptotic behavior of solutions, which is governed by a diffusion equation and coincides with a long time and stiff friction limit. Our proposed approach draws inspiration from a penalization technique adopted in [Boscarino et. al, SIAM Journal on scientific computing, 2014]. By employing an additive implicit-explicit Runge-Kutta method, the scheme can ensure a correct asymptotic behavior for the limiting diffusion equation, without suffering from a parabolic-type time step restriction which often afflicts multiscale problems in the diffusive limit. Numerical experiments are performed to illustrate high order accuracy, asymptotic preserving, and asymptotically accurate properties of the designed schemes. © 2023, CC BY.

关键词： Friction

The Onsager principle and structure preserving numerical schemes

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arXiv 2024年

作者： Chen, Huangxin Liu, Hailiang Xu, Xianmin School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian361005 China Iowa State University Department of Mathematics AmesIA50011 United States LSEC ICMSEC NCMIS Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We present a natural framework for constructing energy-stable time discretization schemes. By leveraging the Onsager principle, we demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems. Furthermore, this principle provides a robust basis for developing numerical schemes that uphold crucial physical properties. Within this framework, several widely used schemes emerge naturally, showing its versatility and *** Codes 65M12, 65M22, 76M30 Copyright © 2024, The Authors. All rights reserved.

关键词： Partial differential equations