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A highly accurate perfectly-matched-layer boundary integral equation solver for acoustic layered-medium problems

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arXiv 2022年

作者： Lu, Wangtao Xu, Liwei Yin, Tao Zhang, Lu School of Mathematical Sciences Zhejiang University Zhejiang Hangzhou310027 China School of Mathematical Sciences University of Electronic Science and Technology of China Sichuan Chengdu611731 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

Based on the perfectly matched layer (PML) technique, this paper develops a high-accuracy boundary integral equation (BIE) solver for acoustic scattering problems in locally defected layered media in both two and three dimensions. The original scattering problem is truncated onto a bounded domain by the PML. Assuming the vanishing of the scattered field on the PML boundary, we derive BIEs on local defects only in terms of using PML-transformed free-space Green’s function, and the four standard integral operators: single-layer, double-layer, transpose of double-layer, and hypersingular boundary integral operators. The hyper-singular integral operator is transformed into a combination of weakly-singular integral operators and tangential derivatives. We develop a high-order Chebyshev-based rectangular-polar singular-integration solver to discretize all weakly-singular integrals. Numerical experiments for both two- and three-dimensional problems are carried out to demonstrate the accuracy and efficiency of the proposed solver. Copyright © 2022, The Authors. All rights reserved.

关键词： Boundary integral equations

On the convergence of stochastic gradient descent with bandwidth-based step size

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arXiv 2021年

作者： Wang, Xiaoyu Yuan, Ya-Xiang Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences No.19A Yuquan Road Beijing100049 China 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 Beijing100190 China

We investigate the stochastic gradient descent (SGD) method where the step size lies within a banded region instead of being given by a fixed formula. The optimal convergence rate under mild conditions and large initial step size is proved. Our analysis provides comparable theoretical error bounds for SGD associated with a variety of step sizes. In addition, the convergence rates for some existing step size strategies, e.g., triangular policy and cosine-wave, can be revealed by our analytical framework under the boundary constraints. The bandwidth-based step size provides efficient and flexible step size selection in optimization. We also propose a 1/t up-down policy and give several non-monotonic step sizes. Numerical experiments demonstrate the efficiency and significant potential of the bandwidth-based step-size in many *** Codes 65K05, 65K10, 90C15 Copyright © 2021, The Authors. All rights reserved.

关键词： Bandwidth

Stochastic Three-Operator Splitting Algorithms for Nonconvex and Nonsmooth Optimization Arising from FLASH Radiotherapy

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arXiv 2023年

作者： Bian, Fengmiao Liu, Jiulong Zhang, Xiaoqun Gao, Hao Cai, Jian-Feng Department of Mathematics The Hong Kong University of Science and Technology Hong Kong Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing China School of Mathematical Sciences MOE-LSC Institute of Natural Sciences Shanghai Jiao Tong University Shanghai China Department of Radiation Oncology University of Kansas Medical Center Kansas CityKS United States HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute Futian Shenzhen China

Radiation therapy (RT) aims to deliver tumoricidal doses with minimal radiation-induced normal-tissue toxicity. Compared to conventional RT (of conventional dose rate), FLASH-RT (of ultra-high dose rate) can provide additional normal tissue sparing, which however has created a new nonconvex and nonsmooth optimization problem that is highly challenging to solve. In this paper, we propose a stochastic three-operator splitting (STOS) algorithm to address the FLASH optimization problem. We establish the convergence and convergence rates of the STOS algorithm under the nonconvex framework for both unbiased gradient estimators and variance-reduced gradient estimators. These stochastic gradient estimators include the most popular ones, such as SGD, SAGA, SARAH, and SVRG, among others. The effectiveness of the STOS algorithm is validated using FLASH radiotherapy planning for patients. Copyright © 2023, The Authors. All rights reserved.

关键词： Radiotherapy

Self-propulsion dynamics of small droplets on general surfaces with curvature gradient

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arXiv 2021年

作者： Chen, Yujuan Xu, Xianmin LSEC Institute of Computational Mathematics and Scientific/Engineering Computing NCMIS AMSS Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We study theoretically the self-propulsion dynamics of a small droplet on general curved surfaces by a variational approach. A new reduced model is derived based on careful computations for the capillary energy and the viscous dissipation in the system. The model describes quantitatively the spontaneous motion of a liquid droplet on general surfaces. In particular, it recovers previous models for droplet motion on the outside surface of a cone. In this case, we derive a scaling law of the displacement s ∼ t1/3 of a droplet with respect to time t by asymptotic analysis. Theoretical results are in good agreement with experiments in previous literature without adjusting the friction coefficient in the model. Copyright © 2021, The Authors. All rights reserved.

关键词： Asymptotic analysis

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

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Communications in mathematical Research 2020年第2期36卷 128-153页

作者： Ling Guo Akil Narayan Yongle Liu Tao Zhou Department of Mathematics Shanghai Normal UniversityShanghai 200234China Department of Mathematics and Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUT 84112USA Department of Mathematics Southern University of Science and TechnologyShenzhen 518055China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China.

One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random *** challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random *** address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ *** first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random *** second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward *** enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability *** demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the *** demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.

关键词： Uncertainty quantification data-driven polynomial chaos expansions sparse approximation equilibrium measure induced measure

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

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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

Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise

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arXiv 2021年

作者： Hong, Jialin Hou, Baohui Sun, Liying 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

In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Pad'e approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the averaged energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings. © 2021, CC BY.

关键词： Finite difference method

TAYLOR-HOOD LIKE FINITE ELEMENTS FOR NEARLY INCOMPRESSIBLE STRAIN GRADIENT ELASTICITY PROBLEMS

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arXiv 2021年

作者： Liao, Yulei Ming, Pingbing Xu, Yun 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 Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China

We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73-100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with 1/2 order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical *** Codes 65N15, 65N30 (Primary), 74K20 (Secondary) Copyright © 2021, The Authors. All rights reserved.

关键词： Elasticity

Stochastic Higher-Order Three-Scale Strength Prediction Model for Composite Structures with Micromechanical Analysis

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SSRN

SSRN 2022年

作者： Dong, Hao Yang, Zihao Guan, Xiaofei Cui, Junzhi School of Mathematics and Statistics Xidian University Xi’an710071 China School of Mathematics and Statistics Northwestern Polytechnical University Xi’an710129 China School of Mathematical Sciences Tongji University Shanghai China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Stochastic multiscale modeling and analysis for the strength prediction of composite structures with complex multiscale random configurations remains a challenging problem. This is mainly due to the facts of high-dimensional physical properties, non-linear and non-Gaussian features, which many repeated evaluations of the corresponding stochastic multiscale model are often required. In this paper, we develop a stochastic higher-order three-scale strength prediction (SHTSP) model for composite structures, which is designed to overcome limitations of prohibitive computation involving the microscale, the mesoscale and the macroscale. By virtue of asymptotic homogenization theory and micromechanical analysis, the SHTSP model is derived from the detailed stochastic higher-order three-scale homogenization analysis implemented with analytic solutions of composite structures subjected to tension, bending and twist loads. The SHTSP model represents strength anisotropy through different strength criterions for evaluating the yield state of the different component materials of composite structures, which are induced by aggregate-matrix interface deboning or matrix cracking in multiple scales. Moreover, these are constructed and calibrated from the high-accuracy mechanical analysis with help of two classes of mesoscopic and microscopic auxiliary cell functions, respectively. The corresponding numerical algorithm of SHTSP model and a preprocessing multilevel strategy is designed to improve the computational efficiency. Finally, the numerical experiments in 3D cases illustrate the outstanding performance of the proposed SHTSP model, and the proposed method can significantly reduce the computational time. © 2022, The Authors. All rights reserved.

关键词： Stochastic systems