检索结果-内蒙古大学图书馆

arXiv 2021年

作者： Pu, Wenqiang Liu, Ya-Feng Luo, Zhi-Quan Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen518172 China 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

An important preliminary procedure in multi-sensor data fusion is sensor registration, and the key step in this procedure is to estimate sensor biases from their noisy measurements. There are generally two difficulties in this bias estimation problem: one is the unknown target states which serve as the nuisance variables in the estimation problem, and the other is the highly nonlinear coordinate transformation between the local and global coordinate systems of the sensors. In this paper, we focus on the 3-dimensional asynchronous multi-sensor scenario and propose a weighted nonlinear least squares (NLS) formulation by assuming that there is a target moving with a nearly constant velocity. We propose two possible choices of the weighting matrix in the NLS formulation, which correspond to classical and weighted NLS estimation and maximum likelihood (ML) estimation, respectively. To address the intrinsic nonlinearity, we propose a block coordinate descent (BCD) algorithm for solving the formulated problem, which alternately updates different kinds of bias estimates. Specifically, the proposed BCD algorithm involves solving linear LS problems and nonconvex quadratically constrained quadratic program (QCQP) problems with special structures. Instead of adopting the semidefinite relaxation technique, we develop a much more computationally efficient algorithm based on the alternating direction method of multipliers (ADMM) to solve the nonconvex QCQP subproblems. The convergence of the ADMM to the global solution of the QCQP subproblems is established under mild conditions. The effectiveness and efficiency of the proposed BCD algorithm are demonstrated via numerical simulations. Copyright © 2021, The Authors. All rights reserved.

关键词： Maximum likelihood estimation

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Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations

arXiv

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arXiv 2022年

作者： Liao, Hong-Lin Tang, Tao Zhou, Tao School of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing211106 China MIIT Nanjing211106 China Division of Science and Technology BNU-HKBU United International College Guangdong Province Zhuhai China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see e.g. [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the L2 norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical *** Codes 65M06, 65M12 © 2022, CC BY-NC-SA.

关键词： Partial differential equations

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Disentangling quantum neural networks for unified estimation of quantum entropies and distance measures

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Physical Review A 2024年第6期110卷 062418-062418页

作者： Myeongjin Shin Seungwoo Lee Junseo Lee Mingyu Lee Donghwa Ji Hyeonjun Yeo Kabgyun Jeong School of Computing KAIST Daejeon 34141 South Korea Quantum AI Team Norma Inc. Seoul 04799 South Korea Department of Computer Science and Engineering Seoul National University Seoul 08826 South Korea College of Liberal Studies Seoul National University Seoul 08826 South Korea Department of Physics and Astronomy Seoul National University Seoul 08826 South Korea Research Institute of Mathematics Seoul National University Seoul 08826 South Korea School of Computational Sciences Korea Institute for Advanced Study Seoul 02455 South Korea

The estimation of quantum entropies and distance measures, such as von Neumann entropy, Rényi entropy, Tsallis entropy, trace distance, and fidelity-induced distances such as the Bures distance, has been a key area of research in quantum information science. In our study, we introduce the disentangling quantum neural network (DEQNN), designed to efficiently estimate various physical quantities in quantum information. Estimation algorithms for these quantities are generally tied to the size of the Hilbert space of the quantum state to be estimated. Our proposed DEQNN offers a unified dimensionality reduction methodology that can significantly reduce the size of the Hilbert space while preserving the values of diverse physical quantities. We provide an in-depth discussion of the physical scenarios and limitations in which our algorithm is applicable, as well as the learnability of the proposed quantum neural network.

关键词： Hilbert spaces

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CONVERGENCE OF THE PLANEWAVE APPROXIMATIONS FOR QUANTUM INCOMMENSURATE SYSTEMS

arXiv

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arXiv 2022年

作者： Wang, Ting Chen, Huajie Zhou, Aihui Zhou, Yuzhi Massatt, Daniel School of Mathematical Sciences Beijing Normal University Beijing100875 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Mathematics Department Louisiana State University LA70802 United States

Incommensurate structures come from stacking the single layers of low-dimensional materials on top of one another with misalignment such as a twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. This paper studies the physical observables of continuum Schrödinger operators for incommensurate systems. We characterize the density of states and local density of states in real and reciprocal spaces, and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states and local density of states via the operator kernel characterization;and (ii) propose efficient numerical schemes based on planewave approximation and trapezoidal rule in reciprocal space. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical *** Codes 65Z05, 81-08, 47G30 Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods

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PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units

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数学研究 2020年第2期53卷 159-191页

作者： Bo Li Shanshan Tang Haijun Yu NCMIS&LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing 100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China China Justice Big Data Institute Beijing 100043 China

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations only when smoothness is not required. In this paper, we construct deep neural networks with rectified power units (RePU), which can give better approximations for smooth functions. Optimal algorithms are proposed to explicitly build neural networks with sparsely connected RePUs, which we call PowerNets, to represent polynomials with no approximation error. For general smooth functions, we first project the function to their polynomial approximations, then use the proposed algorithms to construct cor-responding PowerNets. Thus, the error of best polynomial approximation provides an upper bound of the best RePU network approximation error. For smooth functions in higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product grid and sparse grid discretization to get polynomial approximations. Our construc-tive algorithms show clearly a close connection between spectral methods and deep neural networks: PowerNets with n hidden layers can exactly represent polynomials up to degree sn, where s is the power of RePUs. The proposed PowerNets have po-tential applications in the situations where high-accuracy is desired or smoothness is required.

关键词： Deep neural network rectified linear unit rectified power unit sparse grid Power-Net

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HOMOGENIZATION FOR POLYNOMIAL OPTIMIZATION WITH UNBOUNDED SETS

arXiv

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arXiv 2021年

作者： Huang, Lei Nie, Jiawang Yuan, Ya-Xiang Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100190 China Department of Mathematics University of California 9500 Gilman Drive La Jolla San DiegoCA92093 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work. Polynomial optimization and Homogenization and Moment-SOS relaxations and Optimality conditions. Copyright © 2021, The Authors. All rights reserved.

关键词： Polynomials

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An Adaptive Finite Element DtN Method for the Acoustic-Elastic Interaction Problem in Periodic Structures

arXiv

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arXiv 2025年

作者： Lin, Lei Lv, Junliang School of Mathematics Jilin University Qianjin Street Jilin Province Changchun130012 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, while the elastic body is assumed to be isotropic and linear. By introducing the Dirichlet-to-Neumann (DtN) operators for acoustic and elastic waves simultaneously, the model is formulated as an acoustic-elastic interaction problem in periodic structures. Based on a duality argument, an a posteriori error estimate is derived for the associated truncated finite element approximation. The a posteriori error estimate consists of the finite element approximation error and the truncation error of two different DtN operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem in periodic structures. Numerical experiments are presented to demonstrate the effectiveness of the proposed *** Codes 65N12, 65N15, 65N30 © 2025, CC BY-NC-ND.

关键词： Acoustic noise

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CONVERGENCE RATE OF GRADIENT DESCENT METHOD FOR MULTI-OBJECTIVE OPTIMIZATION

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Journal of computational mathematics 2019年第5期37卷 689-703页

作者： Liaoyuan Zeng Yuhong Dai Yakui Huang Institute of Computational Mathematics and Scientific/Engineering Computing State Key Labomtory of Scientific and Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Science Hebei University of TechnologyTianjin 300401 China

The convergence rate of the gradient descent method is considered for unconstrained multi-objective optimization problems (MOP). Under standard assumptions, we prove that the gradient descent method with constant stepsizes converges sublinearly when the objective functions are convex and the convergence rate can be strengthened to be linear if the objective functions are strongly convex. The results are also extended to the gradient descent method with the Armijo line search. Hence, we see that the gradient descent method for MOP enjoys the same convergence properties as those for scalar optimization.

关键词： Multi-objective optimization Gradient descent Convergence rate.

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ERROR ESTIMATE OF MULTISCALE FINITE ELEMENT METHOD FOR PERIODIC MEDIA REVISITED

arXiv

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arXiv 2022年

作者： Ming, Pingbing Song, Siqi LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences No. 55 East Road Zhong-Guan-Cun Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We derive the optimal energy error estimate for multiscale finite element method with oversampling technique applying to elliptic systems with rapidly oscillating periodic coefficients that are bounded measurable, which may admit rough microstructures. As a by-product of the energy error estimate, we derive the rate of convergence in Ld/(d−1)−norm with d the dimensionality. © 2022, CC BY.

关键词： Finite element method

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An Energy-Stable Finite Element Method for Nonlinear Maxwell's Equations

SSRN

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SSRN 2022年

作者： Li, Lingxiao Lyu, Maohui Zheng, Weiying School of Mathematics and Statistics Henan University Kaifeng475004 China Center for Applied Mathematics of Henan Province Henan University Zhengzhou450046 China LSEC NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Science University of Chinese Academy of Sciences Beijing100049 China

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell’s equations in an optical medium characterized by the linear Lorentz dispersion, the nonlinear Kerr effect, and the delayed Raman scattering. By introducing auxiliary variables, we propose a fully-discrete and energy-stable finite element method for the nonlinear problem. The discrete scheme is unconditionally stable and admits second-order convergence. To solve the nonlinear discrete problem, we propose a Newton-Krylov iterative method, where the linearized algebraic problems are solved with a preconditioned GMRES method based on the auxiliary space preconditioning for Maxwell’s equations. Numerical experiment for a spatial soliton shows agreements with the two-dimensional numerical simulations in the literature. Moreover, we also demonstrate the efficiency and parallel scalability of the finite element method numerically on 1–512 computer processors. © 2022, The Authors. All rights reserved.

关键词： Finite element method

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