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Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

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Science China mathematics 2022年第8期65卷 1731-1748页

作者： Waixiang Cao Dongfang Li Zhimin Zhang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

In this paper,we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave *** error estimates in time and superconvergent error estimates in space are established without certain time-step *** key is to estimate directly the solution bounds in the H-norm for both the nonlinear wave equation and the corresponding fully discrete scheme,while the previous investigations rely on the temporal-spatial error splitting *** examples are presented to confirm energy-conserving properties,unconditional convergence and optimal error estimates,respectively,of the proposed fully discrete schemes.

关键词： scalar auxiliary variable wave equations stability error estimate superconvergence

Metrically regular mappings and its application to convergence analysis of a confined Newton-type method for nonsmooth generalized equations

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Science China mathematics 2020年第1期63卷 39-60页

作者： Mohammed Harunor Rashid Ya-xiang Yuan Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Department of Mathematics Faculty of ScienceUniversity of Rajshahi 6205Bangladesh

Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:U→Y is a nonsmooth function and F:X■Y is a set-valued mapping with closed *** introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this ***,under the point-based approximation of f on U and metrically regular property of f+F,we present quadratic rate of convergence of this ***,superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f+F is metrically *** example of nonsmooth functions that have p-point-based approximation is ***,a numerical experiment is given which illustrates the theoretical result.

关键词： set-valued mappings generalized equations metrically regular mapping semilocal convergence point-based approximation

NEW C0 INTERIOR PENALTY METHOD FOR MONGE-AMPÈRE EQUATIONS

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arXiv 2024年

作者： Chu, Tianyang Guo, Hailong Zhang, Zhimin LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematics and Statistics The University of Melbourne ParkvilleVIC3010 Australia Department of Mathematics Wayne State University DetroitMI48202 United States

Monge-Ampère equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the C0 interior penalty method to approximation the viscosity solution of the Monge-Ampère equation. The new methods is inspired from the discrete Miranda-Talenti estimate. Based on the vanishing moment representation, we approximate the Monge-Ampère equation by the fourth order semi-linear equation with some additional boundary conditions. We will use the discrete Miranda-Talenti estimates to ensure the well-posedness of the numerical scheme and derive the error estimates. © 2024, CC BY.

关键词： Partial differential equations

Inverse design of all-dielectric metasurfaces with accidental bound states in the continuum

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Nanophotonics 2023年第19期12卷 3767-3779页

作者： Gladyshev, Sergei Karamanos, Theodosios D. Kuhn, Lina Beutel, Dominik Weiss, Thomas Rockstuhl, Carsten Bogdanov, Andrey Institute of Physics University of Graz Universitätsplatz 5 Graz8010 Austria Institut Langevin ESPCI Paris Université PSL CNRS Paris75005 France Institute of Theoretical Solid State Physics Karlsruhe Institute of Technology Karlsruhe76131 Germany Steinbuch Centre for Computing - Scientific Computing & Mathematics Karlsruhe Institute of Technology Karlsruhe Germany Institute of Nanotechnology Karlsruhe Institute of Technology Eggenstein-Leopoldshafen76344 Germany Qingdao Innovation and Development Center of Harbin Engineering University Shandong Qingdao266000 China 4th Physics Institute and SCoPE University of Stuttgart Pfaffenwaldring 57 StuttgartD-70569 Germany Harbin Engineering University Harbin150001 China

Metasurfaces with bound states in the continuum (BICs) have proven to be a powerful platform for drastically enhancing light-matter interactions, improving biosensing, and precisely manipulating near- and far-fields. However, engineering metasurfaces to provide an on-demand spectral and angular position for a BIC remains a prime challenge. A conventional solution involves a fine adjustment of geometrical parameters, requiring multiple time-consuming calculations. In this work, to circumvent such tedious processes, we develop a physics-inspired, inverse design method on all-dielectric metasurfaces for an on-demand spectral and angular position of a BIC. Our suggested method predicts the core-shell particles that constitute the unit cell of the metasurface, while considering practical limitations on geometry and available materials. Our method is based on a smart combination of a semi-analytical solution, for predicting the required dipolar Mie coefficients of the meta-atom, and a machine learning algorithm, for finding a practical design of the meta-atom that provides these Mie coefficients. Although our approach is exemplified in designing a metasurface sustaining a BIC, it can, also, be applied to many more objective functions. With that, we pave the way toward a general framework for the inverse design of metasurfaces in specific and nanophotonic structures in general. © 2023 the author(s), published by De Gruyter, Berlin/Boston.

关键词： Machine learning

Derivative-Free Optimization with Transformed Objective Functions (DFOTO) and the Algorithm Based on the Least Frobenius Norm Updating Quadratic Model

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arXiv 2023年

作者： Xie, Peng-Cheng Yuan, Ya-Xiang State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Zhong Guan Cun Donglu 55 Beijing100190 China University of Chinese Academy of Sciences Zhong Guan Cun Donglu 55 Beijing100190 China

Derivative-free optimization (DFO) problems are optimization problems where the derivative information is unavailable. The least Frobenius norm updating quadratic interpolation model function is one of the essential under-determined model functions for model-based derivative-free trust-region methods. This article proposes derivative-free optimization with transformed objective functions (DFOTO) and gives a model-based trust-region method with the least Frobenius norm model. The model updating formula is based on Powell’s formula and can be easily implemented. The method shares the same framework with those for problems without transformations, and its query scheme is given. We propose the definitions related to optimality-preserving transformations to understand the interpolation model in our method when minimizing transformed objective functions. We prove the existence of model optimality-preserving transformations beyond translation transformations. The necessary and sufficient condition for such transformations is given. An interesting discovery is that, as a fundamental transformation, the affine transformation with a (non-trivial) positive multiplication coefficient is not model optimality-preserving. We also analyze the corresponding least Frobenius norm updating model and its interpolation error when the objective function is affinely transformed. The convergence property of a provable algorithmic framework containing the least Frobenius norm updating quadratic model for minimizing transformed objective functions is given. Numerical results show that our method can successfully solve most test problems with objective optimality-preserving transformations, even though some of such transformations will change the optimality of the model function. To our best knowledge, this is the first work providing the model-based derivative-free algorithm and analysis for transformed problems with the function evaluation oracle. This article also proposes the "moving-targe

关键词： Interpolation

A New Two-dimensional Model-based Subspace Method for Large-scale Unconstrained Derivative-free Optimization: 2D-MoSub

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arXiv 2023年

作者： Xie, Pengcheng Yuan, Ya-Xiang State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Zhong Guan Cun Donglu No.55 Beijing China University of Chinese Academy of Sciences Zhong Guan Cun Donglu No.55 Beijing China

This paper proposes the method 2D-MoSub (2-dimensional model-based subspace method), which is a novel derivative-free optimization (DFO) method based on the subspace method for general unconstrained optimization and especially aims to solve large-scale DFO problems. Our method combines 2-dimensional quadratic interpolation models and trust-region techniques to iteratively update the points and explore the 2-dimensional subspace. Its framework includes initialization, constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. We introduce the framework and computational details of 2D-MoSub, and discuss the poisedness and quality of the interpolation set in the corresponding 2-dimensional subspace. We also analyze some properties of our method, including the model’s approximation error with projection property and the algorithm’s convergence. Numerical results demonstrate the effectiveness and efficiency of 2D-MoSub for solving a variety of unconstrained optimization *** Codes 90C56, 90C30, 65K05, 90C90 Copyright © 2023, The Authors. All rights reserved.

关键词： Interpolation

Mathematical and numerical study of an inverse source problem for the biharmonic wave equation

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arXiv 2023年

作者： Chang, Yan Guo, Yukun Yin, Tao Zhao, Yue School of Mathematics Harbin Institute of Technology Harbin China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematics and Statistics Central China Normal University Wuhan China

In this paper, we study the inverse source problem for the biharmonic wave equation. Mathematically, we characterize the radiating sources and non-radiating sources at a fixed wavenumber. We show that a general source can be decomposed into a radiating source and a non-radiating source. The radiating source can be uniquely determined by Dirichlet boundary measurements at a fixed wavenumber. Moreover, we derive a Lipschitz stability estimate for determining the radiating source. On the other hand, the non-radiating source does not produce any scattered fields outside the support of the source function. Numerically, we propose a novel source reconstruction method based on Fourier series expansion by multi-wavenumber boundary measurements. Numerical experiments are presented to verify the accuracy and efficiency of the proposed method. Copyright © 2023, The Authors. All rights reserved.

关键词： Wave equations

Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

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Science China mathematics 2021年第5期64卷 887-902页

作者： Hong-Lin Liao Xuehua Song Tao Tang Tao Zhou Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211101China Department of Mathematics and International Center for Mathematics Southern University of Science and TechnologyShenzhen 518055China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China NCMIS&LSEC Institute of Computational Mathematics and Scientific Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope *** first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step *** we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r_(k):=τ_(k)/τ_(k-1)<***,with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms,the L^(2)norm stability and rigorous error estimates are established,under the same step-ratio constraint that ensures the energy stability,i.e.,0

关键词： molecular beam epitaxial growth variable-step BDF2 scheme discrete orthogonal convolution kernels energy stability convergence analysis

QoS-based Beamforming and Compression Design for Cooperative Cellular Networks via Lagrangian Duality

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arXiv 2023年

作者： Fan, Xilai Liu, Ya-Feng Liu, Liang Chang, Tsung-Hui State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Electronic and Information Engineering The Hong Kong Polytechnic University Hong Kong School of Science and Engineering The Chinese University of Hong Kong Shenzhen Research Institute of Big Data Shenzhen Shenzhen China

This paper considers the quality-of-service (QoS)based joint beamforming and compression design problem in the downlink cooperative cellular network, where multiple relay-like base stations (BSs), connected to the central processor via rate-limited fronthaul links, cooperatively transmit messages to the users. The problem of interest is formulated as the minimization of the total transmit power of the BSs, subject to all users’ signal-to-interference-plus-noise ratio (SINR) constraints and all BSs’ fronthaul rate constraints. In this paper, we first show that there is no duality gap between the considered joint optimization problem and its Lagrangian dual by showing the tightness of its semidefinite relaxation (SDR). Then, we propose an efficient algorithm based on the above duality result for solving the considered problem. The proposed algorithm judiciously exploits the special structure of an enhanced Karush-Kuhn-Tucker (KKT) conditions of the considered problem and finds the solution that satisfies the enhanced KKT conditions via two fixed point iterations. Two key features of the proposed algorithm are: (1) it is able to detect whether the considered problem is feasible or not and find its globally optimal solution when it is feasible;(2) it is highly efficient because both of the fixed point iterations in the proposed algorithm are linearly convergent and evaluating the functions in the fixed point iterations are computationally cheap. Numerical results show the global optimality and efficiency of the proposed algorithm. Copyright © 2023, The Authors. All rights reserved.

关键词： Quality of service