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Time Complexity Analysis of Quantum Algorithms Via Linear Representations For Nonlinear Ordinary and Partial Differential Equations

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SSRN

SSRN 2022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 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 construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrodinger framework to solve the Liouville equation for the HJE, since it can be recast as the semiclassical limit of the Wigner transform of the Schrodinger equation. Comparsion between the Schrodinger and the Liouville framework will also be made. © 2022, The Authors. All rights reserved.

关键词： Approximation algorithms

Time Complexity Analysis of Quantum Difference Methods for Linear High Dimensional and Multiscale Partial Differential Equations

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SSRN

SSRN 2022年

We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the ''quantum difference methods''). Our detailed analyses show that for the heat and linear hyperbolic equations the quantum difference methods provide exponential speedup over the classical difference method with respect to the spatial dimension. For the multiscale problem, the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as $O(1/\varepsilon)$, where $\varepsilon$ is the scaling parameter, while the scaling for the Asymptotic-Preserving (AP) schemes does not depend on $\varepsilon$. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing. © 2022, The Authors. All rights reserved.

关键词： Partial differential equations

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

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arXiv 2022年

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrödinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schrödinger equation. Comparsion between the Schrödinger and the Liouville framework will also be made. © 2022, CC BY-NC-ND.

关键词： Ordinary differential equations

Mean square stability of stochastic theta method for stochastic differential equations driven by fractional Brownian motion

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arXiv 2021年

作者： Li, Min Hu, Yaozhong Huang, Chengming Wang, Xiong School of Mathematics and Physics Center for Mathematical Sciences China University of Geosciences Wuhan China Department of Mathematical and Statistical Sciences University of Alberta Edmonton Canada School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modelling and Scientific Computing Huazhong University of Science and Technology Wuhan China

In this paper, we study the mean square stability of the solution and its stochastic theta scheme for the following stochastic differential equations drive by fractional Brownian motion with Hurst parameter H ∈ (1/2,1): dX(t) = f(t,X(t))dt+g(t,X(t))dBH(t). Firstly, we consider the special case when f(t,X) = −λκtκ-1X and g(t,X) = µX. The solution is explicit and is mean square stable when κ ≥ 2H. It is proved that if the parameter 2H ≤ κ ≤ 3/2 and √3/2·e/√3/2·e+1 (≈ 0.77) ≤ θ ≤ 1 or κ > 3/2 and 1/2 MSC Codes 65C30, 65L20, 60G22, 60H10, 33C15 Copyright © 2021, The Authors. All rights reserved.

关键词： Stochastic systems

GLOBAL SOLVABILITY AND ASYMPTOTICAL BEHAVIOR IN A TWO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION

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arXiv 2021年

作者： Ren, Guoqiang Xiang, Tian School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Institute for Mathematical Sciences Renmin University of China Bejing100872 China

In this work, we study global existence, eventual smoothness and asymptotical behavior of positive solutions for the following two-species chemotaxis consumption model: ut = ∆u - χ1∇ · (u∇w), x ∈ Ω, t > 0, {{{{{{{ vt = ∆v - χ2∇ · (u∇w), x ∈ Ω, t > 0, wt = ∆w - (αu + βv)w, x ∈ Ω, t > 0, in a bounded smooth but not necessarily convex domain Ω ⊂ Rn(n = 2, 3, 4, 5) with nonnegative initial data u0, v0, w0 and homogeneous Neumann boundary data. Here, the parameters χ1, χ2 are positive and α, β are nonnegative. Under a smallness condition max{χ1, χ2}kw0kL∞ MSC Codes 35K59, 35K57 © 2021, CC BY.

关键词： Biochemistry

GLOBAL SOLVABILITY IN A TWO-SPECIES CHEMOTAXIS SYSTEM WITH SIGNAL PRODUCTION

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arXiv 2021年

In this paper, we study the following prototypical two-species chemotaxis system with Lotka-Volterra competition and signal production: (Equation presented). We show that if (Equation presented). the associated Neumann initial-boundary value problem for (∗) admits a global generalized solutions in a bounded and smooth domain Ω ⊂ N (N ≥ 2) under merely integrable initial *** Codes 35A01, 35K57, 35Q92, 92C17 © 2021, CC BY.

关键词： Initial value problems

Data-driven method to learn the most probable transition pathway and stochastic differential equations

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arXiv 2021年

作者： Chen, Xiaoli Duan, Jinqiao Hu, Jianyu Li, Dongfang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Department of Mathematics National University of Singapore 119077 Singapore Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics College of Computing Illinois Institute of Technology ChicagoIL60616 United States

Transition phenomena between metastable states play an important role in complex systems due to noisy fluctuations. We introduce the Onsager-Machlup theory and the Freidlin-Wentzell theory to quantify rare events in stochastic differential equations. By the variational principle, the most probable transition pathway is the minimizer of the action functional, which is governed by the Euler-Lagrange equation. In this paper, the physics-informed neural networks (PINNs) are presented to compute the most probable transition pathway through computing the Euler-Lagrange equation. The convergence result for the empirical loss is obtained for the forward problem. Then we investigate the inverse problem to extract the stochastic differential equation from the most probable transition pathway data. Finally, several numerical experiments are presented to verify the effectiveness of our methods. Copyright © 2021, The Authors. All rights reserved.

关键词： Inverse problems

Convergence rate of inertial forward-backward splitting algorithms based on the local error bound condition

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arXiv 2020年

作者： Liu, Hongwei Wang, Ting Liu, Zexian School Of Mathematics And Statistics Xidian University Xi'An710126 China State Key Laboratory Of Scientific And Engineering Computing Institute Of Computational Mathematics And Scientific/Engineering Computing AMSS Chinese Academy Of Sciences Beijing100190 China School Of Mathematics And Statistics Guizhou University Guiyang550025 China

The " Inertial Forward-Backward algorithm " (IFB) is a powerful tool for convex nonsmooth minimization problems, and under the local error bound condition, the R-linear convergence rates for the sequences of objective values and iterates have been proved if the inertial parameter γksatisfies supkγkkγk= 1 is not know. In this paper, based on the local error bound condition, we exploit a new assumption condition for the important parameter tkin IFB, which implies that limk→∞γk= 1, and establish the convergence rate of function values and strong convergence of the iterates generated by the IFB algorithms with six tksatisfying the above assumption condition in Hilbert space. It is remarkable that, under the local error bound condition, we show that the IFB algorithms with some tkcan achieve sublinear convergence rate of o(1/kp) for any positive integer p > 1. In addition, we propose a class of Inertial Forward-Backward algorithm with adaptive modification and show it has same convergence results as IFB under the error bound condition. Some numerical experiments are conducted to illustrate our results. Copyright © 2020, The Authors. All rights reserved.

关键词： Parameter estimation

ANALYSIS ON THE STEADY EULER FLOWS WITH STAGNATION POINTS IN AN INFINITELY LONG NOZZLE

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arXiv 2022年

作者： Li, Congming Lü, Yingshu Shahgholian, Henrik Xie, Chunjing School of Mathematical Sciences CMA-Shanghai Shanghai Jiao Tong University Shanghai200240 China Institute of Natural Sciences CMA-Shanghai Shanghai Jiao Tong University Shanghai200240 China Department of Mathematics Royal Institute of Technology Stockholm Sweden School of Mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University Shanghai200240 China

A recent prominent result asserts that steady incompressible Euler flows strictly away from stagnation in a two-dimensional infinitely long strip must be shear flows. On the other hand, flows with stagnation points, very challenging in analysis, are interesting and important phenomenon in fluids. In this paper, we not only prove the uniqueness and existence of steady flows with stagnation points, but also obtain the regularity of the boundary of stagnation set, which is a class of obstacle type free boundary. First, we prove a global uniqueness theorem for steady Euler system with Poiseuille flows as upstream far field state in an infinitely long strip. Due to the appearance of stagnation points, the nonlinearity of the semilinear equation for the stream function becomes non- Lipschitz. This creates a challenging analysis problem since many classical analysis methods do not apply directly. Second, the existence of steady incompressible Euler flows, tending to Poiseuille flows in the upstream, are established in an infinitely long nozzle via variational approach. A very interesting phenomenon is the regularity of the boundary of non-stagnant region, which can be regarded as an obstacle type free boundary and is proved to be globally C1. Finally, the existence of stagnation region is proved as long as the nozzle is wider than the width of the nozzle at upstream where the flows tend to Poiseuille flows. Copyright © 2022, The Authors. All rights reserved.

关键词： Nozzles