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A Two-level Overlapping Schwarz Method Using Energy Minimizing Multiscale Finite Element Functions 25th

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A Two-level Overlapping Schwarz Method Using Energy Minimizi...

2.th International Conference on Domain Decomposition Methods in science and engineering, DD 2.18

作者： Kim, Hyea Hyun Chung, Eric T. Wang, Junxian Department of Applied Mathematics and Institute of Natural Sciences Kyung Hee University Region Korea Republic of Department of Mathematics The Chinese University of Hong Kong SAR Hong Kong Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan China

In this paper, a two-level overlapping Schwarz algorithm is proposed for solving finite element discretization of the following model problem © 2.2., Springer Nature Switzerland AG.

ISBN: (纸本)9783030567491

In this paper, a two-level overlapping Schwarz algorithm is proposed for solving finite element discretization of the following model problem © 2.2., Springer Nature Switzerland AG.

关键词： Finite element method

AN ADAPTIVE FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL ELLIPTIC EQUATIONS WITH LINE DIRAC SOURCES

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arXiv 2021年

作者： Cao, Huihui Li, Hengguang Yi, Nianyu Yin, Peimeng Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Wayne State University Department of Mathematics DetroitMI48202 United States Multiscale Methods and Dynamics Group Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

In this paper, we propose a novel adaptive finite element method for an elliptic equation with line Dirac delta functions as a source term. We first study the well-posedness and global regularity of the solution in the whole domain. Instead of regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem with zero source term and nonzero flux jumps on line fractures. The transmission problem is defined in the same domain as the original problem excluding on line fractures, and the solution is therefore shown to be more regular. The estimator relies on meshes conforming to the line fractures and its edge jump residual essentially uses the flux jumps of the transmission problem on line fractures. The error estimator is proven to be both reliable and efficient, an adaptive finite element algorithm is proposed based on the error estimator and the bisection refinement method. Numerical tests show that quasi-optimal convergence rates are achieved even for high order approximations and the adaptive meshes are only locally refined at singular points. © 2.2., CC BY-NC-ND.

关键词： Fracture

Quantum circuits for the heat equation with physical boundary conditions via Schrödingerization

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Journal of computational Physics 2025年 538卷

作者： Shi Jin Nana Liu Yue Yu School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai 200240 China University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai 200240 China School of Mathematics and Computational Science Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 Hunan China

This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary dynamics, thus are not directly suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrödingerization method can be employed, which converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into one higher dimension. Despite advancements in Schrödingerization techniques, the explicit implementation of quantum circuits for solving general PDEs, especially with physical boundary conditions, remains underdeveloped. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. One approach utilizes Duhamel’s principle to express the solution in integral form and employs linear combination of unitaries (LCU) for coherent state preparation. Another method applies an augmentation to transform the inhomogeneous problem into a homogeneous one. We then apply the quantum simulation technique from [1] to transform the resulting non-autonomous system to an autonomous system in one higher dimension. We provide detailed implementations of these two methods and conduct a comprehensive complexity analysis in terms of queries to the time evolution input oracle.

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Developing Finite Element Methods for Simulating Transformation Optics Devices with Metamaterials

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Communications in computational Physics 2019年第1期25卷 135-154页

作者： Wei Yang Jichun Li Yungqing Huang Bin He Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105P.R.China Department of Mathematical Sciences University of Nevada Las VegasLas VegasNevada 89154-4020USA

In this paper,we first develop the mathematical modeling equations for wave propagation in several transformation optics devices,including electromagnetic concentrator,rotator and *** we propose the corresponding finite element time-domain methods for simulating wave propagation in these transformation optics *** implement the proposed algorithms and our numerical results demonstrate the effectiveness of our modeling *** our best knowledge,this is the first work on time-domain finite element simulation carried out for the electromagnetic concentrator,rotator and splitter.

关键词： Maxwell’s equations finite element time-domain methods edge elements transfor-mation optics

On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries

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arXiv 2025年

作者： Jin, Shi Liu, Nana Ma, Chuwen Yu, Yue School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

The Schr&#2.6;dingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schr&#2.6;dinger-type equations with unitary evolution. It does so via the so-ca... 详细信息

The Schrödingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schrödinger-type equation in one higher dimension [1,2.. We show that by employing a smooth initialization of the warped phase transform [3], Schrödingerization can in fact achieve optimal scaling in matrix queries. This paper presents the detailed implementation of three smooth initializations for the Schrödingerization method: (a) the cut-off function, (b) the higher-order polynomial interpolation, and (c) the Fourier transform methods, that achieve optimality for (a) and near-optimality for (b) and (c). A detailed analysis of key parameters affecting time complexity is conducted. Copyright © 2.2., The Authors. All rights reserved.

关键词： Ordinary differential equations

Schrödingerization based quantum algorithms for the fractional Poisson equation

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arXiv 2025年

作者： Jin, Shi Liu, Nana Yu, Yue School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan Xiangtan University Hunan Xiangtan411105 China

We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the d-dimensional fractional equation is reformulated as a local partial differential equation in d + 1 dimensions. We propose a quantum algorithm for the finite element discretization of this local problem, by capturing the steady-state of the corresponding differential equations using the Schrödingerization approach from [2., 39, 40]. The Schrödingerization technique transforms general linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum simulation. This is achieved through the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of the method and conduct a comprehensive complexity analysis, which can show up to exponential advantage – with respect to the inverse of the mesh size in high dimensions – compared to its classical counterpart. Specifically, while the classical method requires Oe(d1/2.3d/2.−d−2. operations, the quantum counterpart requires Oe(d33d/2.−2.5) queries to the block-encoding input models, with the quantum complexity being independent of the dimension d in terms of the inverse mesh size h−1. Numerical experiments are conducted to verify the validity of our formulation. © 2.2., CC BY-NC-ND.

关键词： Poisson equation

A unifying framework for n-dimensional quasi-conformal mappings

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arXiv 2021年

作者： Zhang, Daoping Choi, Gary P.T. Zhang, Jianping Lui, Lok Ming School of Mathematical Sciences Nankai University Tianjin300071 China Department of Mathematics Massachusetts Institute of Technology CambridgeMA02139 United States School of Mathematics and Computational Science Hunan National Center for Applied Mathematics Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China Department of Mathematics The Chinese University of Hong Kong Hong Kong

With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing n-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two- and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling. Copyright © 2.2., The Authors. All rights reserved.

关键词： Numerical methods

Local averaging type a posteriori error estimates for the nonlinear steady-state Poisson-Nernst-Planck equations

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arXiv 2020年

作者： Yang, Ying Shen, Ruigang Fang, Mingjuan Shu, Shi School of Mathematics and Computational Science Guangxi Colleges Universities Key Laboratory of Data Analysis and Computation Guilin University of Electronic Technology Guilin541004 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China School of Mathematics and Computational Science Guilin University of Electronic Technology GuilinGuangxi541004 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan411105 China

The a posteriori error estimates are studied for a class of nonlinear stead-state Poisson-Nernst-Planck equations, which are a coupled system consisting of the Nernst-Planck equation and the Poisson equation. Both the global upper bounds and the local lower bounds of the error estimators are obtained by using a local averaging operator. Numerical experiments are given to confirm the reliability and efficiency of the error estimators. Copyright © 2.2., The Authors. All rights reserved.

关键词： Finite element method

Efficient numerical methods for computing the stationary states of phase field crystal models

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arXiv 2020年

作者： Jiang, Kai Si, Wei Chen, Chang Bao, Chenglong School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan411105 China Yau Mathematical Sciences Center Tsinghua University Beijing100084 China

Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted for designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement of small effective time steps. In this paper, we discretize the energy functional and propose efficient numerical algorithms for solving the constrained non-convex minimization problem. A class of first order approaches, which is the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, is proposed and the convergence property is established without the global Lipschitz constant requirements. Moreover, we design a hybrid approach that applies an inexact Newton method to further accelerate the local convergence. One key feature of our algorithm is that the energy dissipation and mass conservation properties hold during the iteration process. Extensive numerical experiments, including two three dimensional periodic crystals in Landau-Brazovskii (LB) model and a two dimensional quasicrystal in Lifshitz-Petrich (LP) model, demonstrate that our approaches have adaptive time steps which lead to a significant acceleration over many existing methods when computing complex structures. Copyright © 2.2., The Authors. All rights reserved.

关键词： Quasicrystals