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Quantum simulation of the Fokker-Planck equation via Schrödingerization

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arXiv 2024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 Xiangtan University Hunan Xiangtan411105 China

This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrödingerization method - it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrödinger-type equations. It does so via the so-called warped phase transformation that maps the equation into one higher dimension. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrödingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrödingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrödingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique, and give explicitly its corresponding quantum circuit. We discuss the relationship between operator splitting in the Schrödingerization method and its application directly to the original problem, illustrating how the Schrödingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibit

关键词： Hamiltonians

Nonlinear Constitutive Model of Hydroxyl Terminated Polybutadiene Propellant Based on Parallel Rheological Frameworks

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SSRN

SSRN 2024年

作者： Zhou, Shiming Xu, Yihang Li, Daokui College of Aerospace Science and Engineering National University of Defense Technology Changsha410073 China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha410073 China

In this paper, the mechanical response of HTPB propellant is established based on the parallel rheological framework, and a widely applicable nonlinear constitutive model is constructed. Firstly, the mechanical response of HTPB propellant is decoupled into equilibrium response and viscous response by cyclic tensile-relaxation experiment;then, the molecular chain elongation response in the equilibrium response is constructed based on the non-affine eight-chain model, the cross-linking response in the molecular chain network is described based on the tube model, and the viscous response is predicted based on the relaxation rheology of molecular chain;in the end, the propellant equilibrium response and viscous response are integrated to form the total propellant response based on the parallel rheological framework. Finally, the propellant equilibrium response and viscous response are integrated through a parallel rheological framework, constituting the total mechanical response of the propellant. Comparison with the propellant cyclic tensile-relaxation test, and tensile and relaxation test data shows that the established constitutive model can accurately capture the nonlinear mechanical behaviour of HTPB propellant materials. The results of this paper lay a good foundation for accurately describing propellant mechanical properties and provide a method for rapid prediction and evaluation of propellant mechanical response. © 2024, The Authors. All rights reserved.

关键词： Polybutadienes

A fast finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction-diffusion equation

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International Journal of Modeling, simulation, and Scientific Computing 2020年第2期11卷 115-132页

作者： Yaping Zhang Jiliang Cao Weiping Bu Aiguo Xiao School of Mathematics and Computational Science and Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University 411105 HunanP.R.China College of Science Shaoyang University 422000 ShaoyangP.R.China

In this work,we develop a finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction-diffusion equation(2D-DOTSFRDE)with low regularity solution at the initial time.A fast evaluation of the distributedorder time fractional derivative based on graded time mesh is obtained by substituting the weak singular kernel for the *** stability and convergence of the developed semi-discrete scheme to 2D-DOTSFRDE are *** the spatial approximation,the finite element method is *** convergence of the corresponding fully discrete scheme is ***,some numerical tests are given to verify the obtained theoretical results and to demonstrate the effectiveness of the method.

关键词： Distributed-order fractional derivative fractional reaction-diffusion equation fast evaluation graded mesh finite element method error estimate

Analytical Delta-V Approximation for Nonlinear Programming of Multi-target Rendezvous and Flyby Trajectories

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arXiv 2024年

作者： Huang, An-Yi Li, Heng-Nian Luo, Ya-Zhong State Key Laboratory of Astronautic Dynamics Xi’an710043 China National University of Defense Technology Changsha410073 China College of Aerospace Science and Engineering Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions China

This study proposes an analytical Delta-V approximation of short-timetransfers based on the linear relative motion and a gradient-based nonlinear programming model of multi-target rendezvous and fly by trajectories. In previous studies, the Lambert's solution is commonly used to evaluate Delta-Vof short-duration transfers. In this study, to avoid the iteration process forobtaining the Lambert's solution and its gradient, the linear relative motion equations are applied to form an analytical two-point boundary value model for the near-circular orbit rendezvous problems. Although the relative motion equations are usually applicable when the two orbits are close enough, and the position and velocity errors would become more significant as the orbital differences increase, the errors of the velocity increments were proved acceptable in our simulations. Moreover, the analytical formula facilitates the calculation of the gradients to the start epoch and flight time, which are used to establish a nonlinear programming model for sequence optimization that gradient-based algorithms can easily solve. simulation results demonstrated that the analytical Delta-V approximation requires much less calculation than the Lambert's solution, and the proposed gradient-based nonlinear programming algorithms can obtain similar results in less time than previous methods. Copyright © 2024, The Authors. All rights reserved.

关键词： Nonlinear programming

PRISTA-Net: Deep Iterative Shrinkage Thresholding Network for Coded Diffraction Patterns Phase Retrieval

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arXiv 2023年

作者： Liu, Aoxu Fan, Xiaohong Yang, Yin Zhang, Jianping The School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University the Key Laboratory of Intelligent Computing & Information Processing the Ministry of Education Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University the National Center for Applied Mathematics in Hunan Xiangtan411105 China

The problem of phase retrieval (PR) involves recovering an unknown image from limited amplitude measurement data and is a challenge nonlinear inverse problem in computational imaging and image processing. However, many of the PR methods are based on black-box network models that lack interpretability and plug-and-play (PnP) frameworks that are computationally complex and require careful parameter tuning. To address this, we have developed PRISTA-Net, a deep unfolding network (DUN) based on the first-order iterative shrinkage thresholding algorithm (ISTA). This network utilizes a learnable nonlinear transformation to address the proximal-point mapping sub-problem associated with the sparse priors, and an attention mechanism to focus on phase information containing image edges, textures, and structures. Additionally, the fast Fourier transform (FFT) is used to learn global features to enhance local information, and the designed logarithmic-based loss function leads to significant improvements when the noise level is low. All parameters in the proposed PRISTA-Net framework, including the nonlinear transformation, threshold parameters, and step size, are learned end-to-end instead of being manually set. This method combines the interpretability of traditional methods with the fast inference ability of deep learning and is able to handle noise at each iteration during the unfolding stage, thus improving recovery quality. Experiments on Coded Diffraction Patterns (CDPs) measurements demonstrate that our approach outperforms the existing state-of-the-art methods in terms of qualitative and quantitative evaluations. Our source codes are available at https://***/liuaxou/PRISTA-Net. Copyright © 2023, The Authors. All rights reserved.

关键词： Mapping

Quantum Circuits for the Heat Equation with Physical Boundary Conditions Via Schrödingerisation

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SSRN

SSRN 2024年

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 suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrödingerisation 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 Schrödingerisation 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 [CJL23] 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. © 2024, The Authors. All rights reserved.

关键词： Boundary conditions

The Control Design of Spaee Flying Net Robot System Based on RBF Network

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The Control Design of Spaee Flying Net Robot System Based on...

2021 China Automation Congress, CAC 2021

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

ISBN: (纸本)9781665426473

Space Flying Net Robot System (SFNRS) is a new capture method of space debris. In this paper, the SFNRS dynamic equations are established by using the lumped mass method. The previous studies about controller design assumed that the mass of the maneuvering unit in SFNRS is known and a constant while it is unknown in fact. In addition, the upper boundary of tension in the space net is often assumed as a known constant, but it is difficult to get the value. In this paper, the radial basis function (RBF) network is used to design a controller in order to overcome the shortcomings above Finally, the simulation is carried out to verify the effectiveness and the robustness of the proposed method. © 2021 IEEE

关键词： Controllers

A BDDC method for three-dimensional advection-diffusion problems with an adaptive coarse space

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arXiv 2025年

作者： Peng, Jie Shu, Shi Wang, Junxian Zhong, Liuqiang College of Mathematics and Statistics Hengyang Normal University Hengyang421010 China School of Mathematical Sciences South China Normal University Guangzhou510631 China School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

The solution of nonsymmetric positive definite (NSPD) systems for advection-diffusion problems is an important research topic in science and engineering. The adaptive BDDC method is a significant class of non-overlapping domain decomposition methods, which is often used for solving symmetric positive definite problems. In this paper, we apply the adaptive BDDC method to solve the NSPD systems of te advection-diffusion problems. Moreover, by designing a class of edge generalized eigenvalue problems based on prior selected primal constraints, the number of primal unknowns is further reduced. Numerical experiments have verified the effectiveness of the proposed methods. Copyright © 2025, The Authors. All rights reserved.

关键词： Domain decomposition methods

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

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arXiv 2024年

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 suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrödingerisation 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 Schrödingerisation 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 [CJL23] 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. © 2024, CC BY-NC-ND.

关键词： Ordinary differential equations