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arXiv 2024年

作者： Jin, Shi Liu, Nana Ma, Chuwen School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China

We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerisation, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerised equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the (constant and variable coefficient) backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialisation for the Schrödingerised equation which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost log log 1/Ε where Ε is the desired precision. This optimizes the complexity of the Schrödingerisation based quantum algorithms for any non-unitary dynamical system introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603]. Copyright © 2024, The Authors. All rights reserved.

关键词： Hamiltonians

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The Macroscopic Finite-Difference Scheme and Modified Equations of the General Propagation Multiple-Relaxation-Time Lattice Boltzmann Model

SSRN

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SSRN 2023年

作者： Chen, Ying Chai, Zhenhua Liu, Xi Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, two benchmark problems, Gauss hill problem and Poiseuille flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results are not only in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis. © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations

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Schrödingerization based Quantum Circuits for Maxwell’s Equation with time-dependent source terms

arXiv

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arXiv 2024年

作者： Ma, Chuwen Jin, Shi Liu, Nana Wang, Kezhen Zhang, Lei School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China

The Schrödingerisation method combined with the autonomozation technique in [10] converts general non-autonomous linear differential equations with non-unitary dynamics into systems of autonomous Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into two higher dimension. Despite the success of Schrödingerisation techniques, they typically require the black box of the sparse Hamiltonian simulation, suitable for continuous-variable based analog quantum simulation. For qubit-based general quantum computing one needs to design the quantum circuits for practical implementation. This paper explicitly constructs a quantum circuit for Maxwell’s equations with perfect electric conductor (PEC) boundary conditions and time-dependent source terms, based on Schrödingerization and autonomozation, with corresponding computational complexity analysis. Through initial value smoothing and high-order approximation to the delta function, the increase in qubits from the extra dimensions only requires minor rise in computational complexity, almost log log 1/Ε where Ε is the desired precision. Our analysis demonstrates that quantum algorithms constructed using Schrödingerisation exhibit polynomial acceleration in computational complexity compared to the classical Finite Difference Time Domain (FDTD) format. Copyright © 2024, The Authors. All rights reserved.

关键词： Delta functions

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POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS II: HIGH DIMENSIONAL CASE

arXiv

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arXiv 2024年

作者： Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan Institute of Applied Physics and Computational Mathematics Beijing100088 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Sun Yat-sen University Guangdong Zhuhai519082 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, for any odd n and any integer m ≥ 1 with n > 4m, we study the fundamental solution of the higher order Schrödinger equation i∂tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ R, x ∈ Rn, where V is a real-valued C n+21 −2m potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m+V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying n(m−1) |K(t, x, y)| ≤ C(1 + |t|)−(2nm −σ)(1 + |t|− 2nm ) (1 + |t|− 21m |x − y|)− 2m−1 , t , 0, x, y ∈ Rn, where σ = 2 if 0 is an eigenvalue of H, and σ = 0 otherwise. A similar result for smoothing operators H 2αm e−itHPac(H) is also given. The regularity condition V ∈ C n+21 −2m is optimal in the second order case, and it also seems optimal when m > 1. © 2024, CC0.

关键词： Schrodinger equation

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Analog quantum simulation of parabolic partial differential equations using Jaynes-Cummings-like models

arXiv

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arXiv 2024年

作者： Jin, Shi Liu, Nana Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China

We present a simplified analog quantum simulation protocol for preparing quantum states that embed solutions of parabolic partial differential equations, including the heat, Black-Scholes and Fokker-Planck equations. The key idea is to approximate the heat equations by a system of hyperbolic heat equations that involve only first-order differential operators. This scheme requires relatively simple interaction terms in the Hamiltonian, which are the electric and magnetic dipole moment-like interaction terms that would be present in a Jaynes-Cummings-like model. For a ddimensional problem, we show that it is much more appropriate to use a single d-level quantum system - a qudit - instead of its qubit counterpart, and d + 1 qumodes. The total resource cost is efficient in d and precision error, and has potential for realisability for instance in cavity and circuit QED systems. © 2024, CC BY.

关键词： Fokker Planck equation

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Phase-field-based lattice Boltzmann method for the transport of insoluble surfactant in two-phase flows

arXiv

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arXiv 2024年

作者： Zhan, Chengjie Liang, Hong Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Physics Hangzhou Dianzi University Hangzhou310018 China

In this work, we present a general second-order phase-field model for the transport of insoluble surfactant in incompressible two-phase flows. In this model, the second-order local Allen-Cahn equation is applied for interface capturing, a general form of the simple scalar transport equation [S. S. Jain, J. Comput. Phys. 515, 113277 (2024)] is adopted for interface-confined surfactant, and the consistent and conservative Navier-Stokes equations with the Marangoni force is used for fluid flows. To solve this model, we further developed a mesoscopic lattice Boltzmann (LB) method, in which the LB model for surfactant transport equation is proposed under the general LB framework for the convection-diffusion type equation, and it can correctly recover the governing equation for surfactant transport. The accuracy of the present LB method is tested by several benchmark problems, and the numerical results show it has a good performance for the transport of the insoluble surfactant in two-phase flows. © 2024, CC BY-NC-ND.

关键词： Navier Stokes equations

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On Schrödingerization based quantum algorithms for linear dynamical systems with inhomogeneous terms

arXiv

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arXiv 2024年

We analyze the Schrödingerisation method for quantum simulation of a general class of non-unitary dynamics with inhomogeneous source terms. The Schrödingerisation technique, introduced in [26, 28], transforms any linear ordinary and partial differential equations with non-unitary dynamics into a system under unitary dynamics via a warped phase transition that maps the equations into a higher dimension, making them suitable for quantum simulation. This technique can also be applied to these equations with inhomogeneous terms modeling source or forcing terms or boundary and interface conditions, and discrete dynamical systems such as iterative methods in numerical linear algebra, through extra equations in the system. Difficulty airses with the presense of inhomogeneous terms since it can change the stability of the original system. In this paper, we systematically study–both theoretically and numerically–the important issue of recovering the original variables from the Schrödingerized equations, even when the evolution operator contains unstable modes. We show that even with unstable modes, one can still construct a stable scheme, yet to recover the original variable one needs to use suitable data in the extended space. We analyze and compare both the discrete and continuous Fourier transforms used in the extended dimension, and derive corresponding error estimates, which allows one to use the more appropriate transform for specific equations. We also provide a smoother initialization for the Schrodödingerized system to gain higher order accuracy in the extended space. We homogenize the inhomogeneous terms with a stretch transformation, making it easier to recover the original variable. Our recovering technique also provides a simple and generic framework to solve general ill-posed problems in a computationally stable way. Copyright © 2024, The Authors. All rights reserved.

关键词： Dynamics

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POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS I: LOW DIMENSIONAL CASE

arXiv

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arXiv 2024年

In this paper, for any odd n and any integer m ≥ 1 with n tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ , x ∈ n, where V is a real-valued potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m + V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying (Equation presented), where the constants C, h > 0, and h can be specified by m, n and the zero energy resonances of H. A similar result for smoothing operators like (Equation presented) is also given. One of the main novelties lies in a new and unified approach to obtain the asymptotic expansions of perturbed resolvents under various types of zero energy resonances. © 2024, CC0.

关键词： Schrodinger equation

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Differentially Private Energy Sharing Among Smart Grid-Powered Base Stations

Differentially Private Energy Sharing Among Smart Grid-Power...

引用

IEEE Conference on Vehicular Technology (VTC)

作者： Liwan Qi Bochun Wu Fei Wang Kai Li Wei Ni Abbas Jamalipour State Key Laboratory of Integrated Chips and Systems Fudan University Shanghai China College of Physics and Electromechanical Engineering Hexi University Zhangye China University of Cambridge Cambridge U.K. Real-Time and Embedded Computing Systems Research Centre (CISTER) Porto Portugal Commonwealth Scientific and Industrial Research Organisation (CSIRO) Sydney NSW Australia School of Electrical and Information Engineering University of Sydney NSW Australia

ISBN: (数字)9798331517786

ISBN: (纸本)9798331517793

Allowing for energy sharing among base stations (BSs), we investigate a distributed BS system equipped with renewable power units and energy storage batteries. While offering significant benefits, this raises concerns about privacy protection. By leveraging the Laplace mechanism, our differential privacy (DP)-based method safeguards energy consumption data related to processing data tasks from different BSs. To address a long-term average cost minimization problem, we propose a distributed online algorithm for efficient energy sharing. We demonstrate that the boundary constraints of energy storage batteries can still be satisfied by choosing parameters appropriately. We provide a theoretical bound for the optimality gap and validate the effectiveness of our theoretical results. Numerical results indicate our algorithm can potentially reduce the average costs of BSs by up to 34%.

关键词： Renewable energy sources Privacy Energy consumption Base stations Vehicular and wireless technologies Costs Batteries Smart grids Protection Optimization

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Force-based gradient descent method for ab initio atomic structure relaxation

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Physical Review B 2022年第10期106卷 104101-104101页

作者： Yukuan Hu Xingyu Gao Yafan Zhao Xin Liu Haifeng Song State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China University of Chinese Academy of Sciences Beijing 100049 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing 100088 China CAEP Software Center for High Performance Numerical Simulation Beijing 100088 China

Force-based algorithms for ab initio atomic structure relaxation, such as conjugate gradient methods, usually get stuck in the line minimization processes along search directions, where expensive ab initio calculations are triggered frequently to test trial positions before locating the next iterate. We present a force-based gradient descent method, WANBB, that circumvents the deficiency. At each iteration, WANBB enters the line minimization process with a trial step size capturing the local curvature of the energy surface. The exit is controlled by a nonrestrictive criterion that tends to accept early trials. These two ingredients streamline the line minimization process in WANBB. The numerical simulations on nearly 80 systems with good universality demonstrate the considerable compression of WANBB on the cost for the unaccepted trials compared with conjugate gradient methods. We also observe across the board significant and universal speedups as well as the superior robustness of WANBB over several widely used methods. The latter point is theoretically established. The implementation of WANBB is pretty simple, in that no a priori physical knowledge is required and only three parameters are present without tuning.

关键词： Crystal structure Structural properties Density functional theory development

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