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

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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

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 © 2021. 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 © 2020, 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 © 2020, The Authors. All rights reserved.

关键词： Quasicrystals

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

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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

Energy-Efficient Virtual Machine Placement in Data Centers Via Accelerated Genetic Algorithm with Improved Fitness computation

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SSRN

SSRN 2021年

作者： Hormozi, Elham Hu, Shuwen Ding, Zhe Tian, Yu-Chu Wang, You-Gan Yu, Zu-Guo Zhang, Weizhe School of Computer Science Queensland University of Technology GPO Box 2434 BrisbaneQLD4001 Australia School of Mathematical Sciences Queensland University of Technology GPO Box 2434 BrisbaneQLD4001 Australia Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China School of Cyberspace Science Harbin Institute of Technology Harbin China Cyberspace Security Research Center Peng Cheng Laboratory Shenzhen China Queensland University of Technology

Energy efficiency basically refers to how effectively energy can be utilised or consumed by any virtualised data centre giving cloud administrations.&#1.0;Since Genetic Algorithm (GA) is an acceptable method to solve ... 详细信息

Energy efficiency basically refers to how effectively energy can be utilised or consumed by any virtualised data centre giving cloud administrations. Since Genetic Algorithm (GA) is an acceptable method to solve the VM placement problem, this research formulates the Virtual Machine (VM) placement problem as a constrained optimisation problem. A considerable amount of residual resources is available for each new VM allocation, however they are distributed across multiple active PMs, rendering them inefficient. Our research focuses on improving a data centre’s energy and time efficiency by devising a technique to decrease PM’s residual resources and make them more valuable. As the most time consuming element of a GA is calculating the fitness function, this paper presents a new fitness function to simplify the fitness function calculation in GA. The results are compared to the standard GA and First Fit Decreasing (FFD) algorithm. The findings reveals a 8% energy savings for our GA compared to FFD and a 66% reduction in our GA execution time compared to the standard GA for virtualised data centres. The simulation experiments are conducted on small-, medium- and large-scale of the google data centre to demonstrate the effectiveness of our GA. © 2021. The Authors. All rights reserved.

关键词： Energy efficiency

Multicontinuum homogenization for coupled flow and transport equations

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Journal of computational and Applied Mathematics 2026年 471卷

作者： Dmitry Ammosov Jian Huang Wing Tat Leung Buzheng Shan Computational Technologies & Artificial Intelligence North-Eastern Federal University Yakutsk 677980 Russia School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan and Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan 411105 Hunan China Department of Mathematics City University of Hong Kong Hong Kong Department of Mathematics Texas A&M University College Station TX 77843 USA

In this paper, we present the derivation of a multicontinuum model for the coupled flow and transport equations by applying multicontinuum homogenization. We perform the multicontinuum expansion for both flow and transport solutions and formulate novel coupled constraint cell problems to capture the multiscale property, where oversampled regions are utilized to avoid boundary effects. Assuming the smoothness of macroscopic variables, we obtain a multicontinuum system composed of macroscopic elliptic equations and convection–diffusion–reaction equations with homogenized effective properties. Finally, we present numerical results for various coefficient fields and boundary conditions to validate our proposed algorithm.

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Multifractal temporally weighted detrended partial cross-correlation analysis to quantify intrinsic power-law cross-correlation of two non-stationary time series affected by common external factors

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arXiv 2020年

作者： Li, Bao-Gen Ling, Dian-Yi Yu, Zu-Guo Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan411105 China School of Electrical Engineering and Computer Science Queensland University of Technology GPO Box 2434 BrisbaneQ4001 Australia

When common factors strongly influence two cross-correlated time series recorded in complex natural and social systems, the results will be biased if we use multifractal detrended cross-correlation analysis (MF-DXA) without considering these common factors. Based on multifractal temporally weighted detrended cross-correlation analysis (MF-TWXDFA) proposed by our group and multifractal partial cross-correlation analysis (MF-DPXA) proposed by Qian et al., we propose a new method-multifractal temporally weighted detrended partial cross-correlation analysis (MF-TWDPCCA) to quantify intrinsic power-law cross-correlation of two non-stationary time series affected by common external factors in this paper. We use MF-TWDPCCA to characterize the intrinsic cross-correlations between the two simultaneously recorded time series by removing the effects of other potential time series. To test the performance of MF-TWDPCCA, we apply it, MF-TWXDFA and MF-DPXA on simulated series. Numerical tests on artificially simulated series demonstrate that MF-TWDPCCA can accurately detect the intrinsic cross-correlations for two simultaneously recorded series. To further show the utility of MF-TWDPCCA, we apply it on time series from stock markets and find that there exists significantly multifractal power-law cross-correlation between stock returns. A new partial cross-correlation coefficient is defined to quantify the level of intrinsic cross-correlation between two time series. Copyright © 2020, The Authors. All rights reserved.

关键词： Fractals