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

Journal of computational Physics 2025年

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

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New Result on Interception of Stationary Targets at Arbitrary Time-Varying Velocity

arXiv

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arXiv 2021年

作者： Yuanhe, Liu Kebo, Li Yan’gang, Liang College of Aerospace Science and Engineering National University of Defense Technology Changsha410072 China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha410072 China

In this paper, some new results on time-varying missile against a stationary target using pure proportional navigation (PPN) are developed in the planar interception problem. First, the relative motion equation is established in arc-length domain based on the differential geometry theory, which eliminates the influence of time-varying missile speed. Then, the closed-form solution of time-varying speed missile intercepting stationary target with PPN is deduced, and the interception performance is analyzed. Additionally, considering the missile maneuvering acceleration limit, the capture region of time-varying speed missile is analyzed. Finally, the results derived in this paper are verified by numerical simulation analysis for various scenarios. © 2021, CC BY-NC-SA.

关键词： Equations of motion

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A unifying framework for n-dimensional quasi-conformal mappings

arXiv

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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 © 2021, The Authors. All rights reserved.

关键词： Numerical methods

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Spacecraft multi-impulse reachable domain

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Chinese Journal of Aeronautics 2024年

作者： Sai ZHANG Zhen YANG Yazhong LUO College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha 410073 China

The concept of the spacecraft Reachable Domain (RD) has garnered significant scholarly attention due to its crucial role in space situational awareness and on-orbit service applications. While the existing research has largely focused on single-impulse RD analysis, the challenge of Multi-Impulse RD (MIRD) remains a key area of interest. This study introduces a methodology for the precise calculation of spacecraft MIRD. The reachability constraints specific to MIRD are first formulated through coordinate transformations. Two restricted maneuvering strategies are examined. The derivation of two extremum conditions allows for determining the accessible orientation range and the nodes encompassing the MIRD. Subsequently, four nonlinear programming models are developed to address two types of MIRD by skillfully relaxing constraints using scale factors. Numerical results validate the robustness and effectiveness of the proposed approach, showing substantial agreement with Monte Carlo simulations and confirming its applicability to spacecraft on various elliptical orbits.

关键词： Orbital maneuver Reachable domain Multiple impulses Space situational awareness Nonlinear programing

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Non-Newtonian Fluid Flow in Micromachines: Magnetized Carreau Fluid Behavior with Slip and Viscous Dissipation at Inclined Stagnation Points

SSRN

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SSRN 2024年

作者： Liu, Hongjuan Khan, Shahzeb Dehraj, Sanaullah Haider, Ali Liu, Shengjun Ayub, Assad School of Mathematics and Statistics Central South University Hunan Changsha410083 China Department of Mathematics & Statistics Hazara University Manshera21300 Pakistan Department of Mathematics Government College Mansehra21300 Pakistan School of Mathematics and Statistics Xi’an Key Laboratory of Scientific Computation and Applied Statistics Northwestern Polytechnical University Xi’an710072 China Department of Mathematics and Statistics Quaid-e-Awam University of Engineering Science and Technology 67405 Sakrand Road Nawabshah Pakistan

Non-Newtonian fluid flow in micromachines has unique rheological properties and can be leveraged to enhance the performance and functionality of microscale devices. Viscosity of non-Newtonian fluid under different shear rates, can improve the efficiency of fluid transport, mixing, and control within micromachines. This paper presents the analysis of stagnation point flow of inclined magnetized Carreau fluid with slip constraints. This study employs a spectral relaxation method to analyze the flow and investigate the impacts of numerous parameters, such as the inclination angle and magnetic field strength. The study provides numerous applications in field of Chemical engineering and aerospace engineering. © 2024, The Authors. All rights reserved.

关键词： Porous materials

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

arXiv

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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 © 2020, The Authors. All rights reserved.

关键词： Finite element method

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Switching between singular points and exceptional-point-enhanced sensing in non-Hermitian photonic structures 13

Switching between singular points and exceptional-point-enha...

引用

Active Photonic Platforms XIII 2021

作者： Veronis, Georgios Huang, Yin Wang, Lanyan Shen, Yuecheng School of Electrical Engineering and Computer Science Louisiana State University Baton RougeLA70803 United States Center for Computation and Technology Louisiana State University Baton RougeLA70803 United States Department of Optoelectrics Information Science and Engineering School of Physics and Electronics Central South University Hunan Changsha410012 China State Key Laboratory of Optoelectronic Materials and Technologies School of Electronics and Information Technology Sun Yat-Sen University Guangdong Guangzhou510275 China

ISBN: (纸本)9781510644304

We investigate the exceptional points in a two-layer cylindrical waveguide structure consisting of absorbing and nonabsorbing dielectrics. We show that by tuning the parameters of the structure the complex effective indices of two waveguide modes can coalesce so that an exceptional point is formed. We show that the sensitivity of the effective index of the waveguide mode is enhanced at the exceptional point. We also investigate using phase-change materials in multilayer structures to switch between singular points. We show that in multilayer structures consisting of phasechange, lossless dielectric, lossy, and gain materials, absorbing or spectral singularities can be switched to exceptional points, and self-dual spectral singularities can be switched to unidirectional spectral singularities by switching the phasechange material from its crystalline to its amorphous phase. Our results could be important for developing new compact reconfigurable singularity-enhanced optical devices. © 2021 SPIE.

关键词： Physical optics

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

arXiv

引用

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 © 2020, The Authors. All rights reserved.

关键词： Quasicrystals

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An adaptive block Bregman proximal gradient method for computing stationary states of multicomponent phase-field crystal model

arXiv

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arXiv 2020年

作者： Bao, Chenglong Chen, Chang Jiang, Kai Yau Mathematical Sciences Center Tsinghua University Beijing100084 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem's block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods. Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

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