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A lowest-degree strictly conservative finite element scheme for incompressible stokes problem on general triangulations

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arXiv 2021年

作者： Liu, Wenjia Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, we propose a finite element pair for incompressible Stokes problem. The pair uses a slightly enriched piecewise linear polynomial space for velocity and piecewise constant space for pressure, and is illustrated to be a lowest-degree conservative stable pair for the Stokes problem on general triangulations. Copyright © 2021, The Authors. All rights reserved.

关键词： Navier Stokes equations

A monotone finite volume method for time fractional Fokker-Planck equations

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Science China mathematics 2019年第4期62卷 783-794页

作者： Yingjun Jiang Xuejun Xu School of Mathematics and Statistics Changsha University of Science and Technology Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences Tongji University

We develop a monotone finite volume method for the time fractional Fokker-Planck equations and theoretically prove its unconditional stability. We show that the convergence rate of this method is of order 1 in the space and if the space grid becomes sufficiently fine, the convergence rate can be improved to order *** results are given to support our theoretical findings. One characteristic of our method is that it has monotone property such that it keeps the nonnegativity of some physical variables such as density, concentration,etc.

关键词： time fractional Fokker-Planck equations finite volume methods monotone convergence

Optimality conditions for homogeneous polynomial optimization on the unit sphere

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arXiv 2021年

作者： Huang, Lei Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing China

In this note, we prove that for homogeneous polynomial optimization on the sphere, if the objective f is generic in the input space, all feasible points satisfying the first order and second order necessary optimality conditions are local minimizers, which addresses an issue raised in the recent work by Lasserre (Optimization Letters, 2021). As a corollary, this implies that Lasserre’s hierarchy has finite convergence when f is generic. Copyright © 2021, The Authors. All rights reserved.

关键词： Spheres

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

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

A FOURTH-ORDER UNFITTED CHARACTERISTIC FINITE ELEMENT METHOD FOR SOLVING THE ADVECTION-DIFFUSION EQUATION ON TIME-VARYING DOMAINS

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arXiv 2021年

作者： Ma, Chuwen Zhang, Qinghai Zheng, Weiying School of Mathematical Science University of Chinese Academy of Sciences China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences Zhejiang University 38 Zheda Road Zhejiang Province Hangzhou310027 China LSEC NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Numerical experiments are performed on a rotating domain and a severely deformed domain to verify our theoretical results and to demonstrate the optimal convergence of the proposed method. Copyright © 2021, The Authors. All rights reserved.

关键词： Finite element method

Efficient spectral methods for quasi-equilibrium closure approximations of symmetric problems on unit circle and sphere

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arXiv 2021年

作者： Jiang, Shan Yu, Haijun School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China NCMIS LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China

MSC Codes 65M70, 65D40, ***-equilibrium approximation is a widely used closure approximation approach for model reduction with applications in complex fluids, materials science, etc. It is based on the maximum entropy principle and leads to thermodynamically consistent coarse-grain models. However, its high computational cost is a known barrier for fast and accurate applications. Despite its good mathematical properties, there are very few works on the fast and efficient implementations of quasi-equilibrium approximations. In this paper, we give efficient implementations of quasi-equilibrium approximations for antipodally symmetric problems on unit circle and unit sphere using polynomial and piecewise polynomial approximations. Comparing to the existing methods using linear or cubic interpolations, our approach achieves high accuracy (double precision) with much less storage cost. The methods proposed in this paper can be directly extended to handle other moment closure approximation problems. © 2021, CC BY-NC-SA.

关键词： Spectroscopy

Verifiable blind quantum computation with identity authentication for different types of clients

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arXiv 2022年

作者： Quan, Junyu Li, Qin Li, Lvzhou School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Computer Science Xiangtan University Xiangtan411105 China Institute of Quantum Computing and Computer Theory School of Computer Science and Engineering Sun Yat-sen University Guangzhou510006 China

Quantum computing has considerable advantages in solving some problems over its classical counterpart. Currently various physical systems are developed to construct quantum computers but it is still challenging and the first use of quantum computers may adopt the cloud style. Blind quantum computing (BQC) provides a solution for clients with limited quantum capabilities to delegate their quantum computation to remote quantum servers while keeping input, output, and even algorithm private. In this paper, we propose three multi-party verifiable blind quantum computing (VBQC) protocols with identity authentication to handle clients with varying quantum capabilities in quantum networks, such as those who can just make measurements, prepare single qubits, or perform a few single-qubit gates. They are client-friendly and flexible since the clients can achieve BQC depending on their own quantum devices and resist both insider outsider attacks in quantum networks. Furthermore, all the three proposed protocols are verifiable, namely that the clients can verify the correctness of their calculations. Copyright © 2022, The Authors. All rights reserved.

关键词： Qubits

Machine learning for elliptic PDEs: Fast rate generalization bound, neural scaling law and minimax optimality

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arXiv 2021年

作者： Lu, Yiping Chen, Haoxuan Lu, Jianfeng Ying, Lexing Blanchet, Jose Institute for Computational and Mathematical Engineering Stanford University StanfordCA United States Department of Computing and Mathematical Sciences Caltech Mathematics Department Duke University United States Department of Mathematics Institute for Computational and Mathematical Engineering Stanford University StanfordCA United States Department of Management Science & Engineering Stanford University StanfordCA United States

In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To simplify the problem, we focus on a prototype elliptic PDE: the Schrödinger equation on a hypercube with zero Dirichlet boundary condition, which has wide application in the quantum-mechanical systems. We establish upper and lower bounds for both methods, which improves upon concurrently developed upper bounds for this problem via a fast rate generalization bound. We discover that the current Deep Ritz Methods is sub-optimal and propose a modified version of it. We also prove that PINN and the modified version of DRM can achieve minimax optimal bounds over Sobolev spaces. Empirically, following recent work which has shown that the deep model accuracy will improve with growing training sets according to a power law, we supply computational experiments to show a similar behavior of dimension dependent power law for deep PDE solvers. © 2021, CC BY.

关键词： Optimization