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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

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(2.-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 2.-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

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 © 2.2., 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 [CJL2.] 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. © 2.2., The Authors. All rights reserved.

关键词： Boundary conditions

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 © 2.2., 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 [CJL2.] 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. © 2.2., CC BY-NC-ND.

关键词： Ordinary differential equations

Design and Characterization of Low-Carbon Supersulfated Cement Prepared Using As-Received Electrolysis Manganese Dioxide Residue

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SSRN

SSRN 2023年

作者： Ma, Wenbo Bai, Tianwen Zhu, Wei Guo, Shuaicheng School of Mechanical Engineering and Mechanics Xiangtan University Xiangtan411105 China College of Civil Engineering Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China Key Laboratory of Building Safety and Energy Efficiency of the Ministry of Education Hunan University Changsha410082 China International Science Innovation Collaboration Base for Green & Advanced Civil Engineering Materials of Hunan Province Hunan University Changsha410082 China Key Laboratory for Green & Advanced Civil Engineering Materials and Application Technology of Hunan Province College of Civil Engineering Hunan University Changsha410082 China

Utilizing electrolysis manganese dioxide residue (EMDR) to prepare the supersulfated cement can achieve effective waste recycling, and its application can reduce the carbon emission caused by Portland cement. This study aims to quantitatively build the correlation between the mechanical performance and cementitious matrix composition of EMDR-based supersulfated cement. The supersulfated cement containing clinker, slag and as received EMDR is designed and prepared based on the simplex centroid design method. The compressive strength, tensile strength and flexural strength of the EMDR based supersulfated cement paste were measured at 3-day, 7-day and 2.-day ages. The compressive strength of the electrolysis manganese dioxide residue (EMDR) based supersulfated cement first increases and then decreases with the mass ratio between slag and EMDR. The added cement clinker can increase the compressive strength at 3-day and 7-day ages, but the 2.-day age strength first decrease and then increase with the cement clinker content. Meanwhile, the flexural and tensile performances of the EMDR based supersulfated cement generally increase with the clinker content. The measured elastic modulus results indicated the added clinker could enhance the stiffness of the stiffness development. Then the strength development mechanisms of EMDR-based supersulfated cement are analyzed based on phase and microstructure characterization. It is found the clinker can accelerate the reaction rate between the slag and EMDR, and the consumption rate of the gypsum contained in EMDR. The calcium silicate hydrate (CSH) gel phase's contribution to tensile strength is more significant than that of ettringite phase. Based on the mechanism analysis, the index of [SO3]/[Al2.3] molar ratio is proposed for the matrix design of EMDR based SSC. This study can support the design and application of EMDR-based supersulfated cement. © 2.2., The Authors. All rights reserved.

关键词： Aluminum oxide

A Progressive Image Restoration Network for High-order Degradation Imaging in Remote Sensing

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arXiv 2024年

作者： Feng, Yujie Yang, Yin Fan, Xiaohong Zhang, Zhengpeng Bu, Lijing Zhang, Jianping School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory for Intelligent Computing and Information Processing of the Ministry of Education Xiangtan411105 China School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science Xiangtan411105 China College of Mathematical Medicine Zhejiang Normal University Jinhua321004 China School of Automation and Electronic Information Xiangtan University Xiangtan411105 China

Recently, deep learning methods have gained remarkable achievements in the field of image restoration for remote sensing (RS). However, most existing RS image restoration methods focus mainly on conventional first-order degradation models, which may not effectively capture the imaging mechanisms of remote sensing images. Furthermore, many RS image restoration approaches that use deep learning are often criticized for their lacks of architecture transparency and model interpretability. To address these problems, we propose a novel progressive restoration network for high-order degradation imaging (HDI-PRNet), to progressively restore different image degradation. HDI-PRNet is developed based on the theoretical framework of degradation imaging, offering the benefit of mathematical interpretability within the unfolding network. The framework is composed of three main components: a module for image denoising that relies on proximal mapping prior learning, a module for image deblurring that integrates Neumann series expansion with dual-domain degradation learning, and a module for super-resolution. Extensive experiments demonstrate that our method achieves superior performance on both synthetic and real remote sensing images. Copyright © 2.2., The Authors. All rights reserved.

关键词： Image reconstruction

MVMS-RCN: A Dual-Domain Unified CT Reconstruction with Multi-sparse-view and Multi-scale Refinement-correction

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arXiv 2024年

作者： Fan, Xiaohong Chen, Ke Yi, Huaming 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 Centre for Mathematical Imaging Techniques Department of Mathematical Sciences The University of Liverpool LiverpoolL6972L United Kingdom The School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Xiangtan411105 China

X-ray Computed Tomography (CT) is one of the most important diagnostic imaging techniques in clinical applications. Sparse-view CT imaging reduces the number of projection views to a lower radiation dose and alleviates the potential risk of radiation exposure. Most existing deep learning (DL) and deep unfolding sparse-view CT reconstruction methods: 1) do not fully use the projection data;2. do not always link their architecture designs to a mathematical theory;3) do not flexibly deal with multi-sparse-view reconstruction assignments. This paper aims to use mathematical ideas and design optimal DL imaging algorithms for sparse-view CT reconstructions. We propose a novel dual-domain unified framework that offers a great deal of flexibility for multi-sparse-view CT reconstruction through a single model. This framework combines the theoretical advantages of model-based methods with the superior reconstruction performance of DL-based methods, resulting in the expected generalizability of DL. We propose a refinement module that utilizes unfolding projection domain to refine full-sparse-view projection errors, as well as an image domain correction module that distills multi-scale geometric error corrections to reconstruct sparse-view CT. This provides us with a new way to explore the potential of projection information and a new perspective on designing network architectures. The multi-scale geometric correction module is end-to-end learnable, and our method could function as a plug-and-play reconstruction technique, adaptable to various applications. Extensive experiments demonstrate that our framework is superior to other existing state-of-the-art methods. Our source codes are available at https://***/fanxiaohong/MVMS-RCN. Copyright © 2.2., The Authors. All rights reserved.

关键词： Geometry