Article
Mathematics, Applied
Yulian Yi, Yaozhong Hu, Jingjun Zhao
Summary: This paper proposes a class of explicit positivity preserving numerical methods for general stochastic differential equations with positive solutions. The convergence and convergence rate results for these methods are obtained under certain reasonable conditions. The main challenge lies in obtaining strong convergence and convergence rate for stochastic differential equations with coefficients of exponential growth. Numerical experiments are provided to illustrate the theoretical results for the proposed schemes.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Applied
Charles-Edouard Brehier, David Cohen
Summary: This paper analyzes the qualitative properties and order of convergence of a splitting scheme for a class of nonlinear stochastic Schrodinger equations. The numerical solution is shown to be symplectic and preserves the expected mass. Exponential moment bounds are proved for the exact and numerical solutions, enabling us to determine strong orders of convergence as well as orders of convergence in probability and almost surely. Extensive numerical experiments demonstrate the performance of the proposed numerical scheme.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Monika Eisenmann, Eskil Hansen
Summary: This paper introduces a sum splitting scheme for temporal approximation of nonlinear parabolic equations, with a straightforward parallelization strategy and convergence analysis in a variational framework. The focus is on illustrating the significant advantages of a variational framework for operator splittings and extending semigroup-based theory for this type of scheme.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Operations Research & Management Science
Truong Minh Tuyen, Ratthaprom Promkam, Pongsakorn Sunthrayuth
Summary: This paper studies the generalized monotone quasi-inclusion problem and proposes a forward-backward splitting method to solve the problem. By applying the Bregman distance function, the strong convergence of the algorithm is proven and applied to the variational inequality problem. Numerical examples demonstrate the performance of the algorithm.
Article
Mathematics, Applied
Jianqiang Xie, Muhammad Aamir Ali, Zhiyue Zhang
Summary: This paper focuses on the error estimation of a novel time second-order splitting conservative finite difference method for high-dimensional nonlinear fractional Schrodinger equation. The paper demonstrates the discrete preservation property and shows the accuracy of the method in terms of L2-norm. Numerical experiments are conducted to validate the accuracy and conservation property of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Andre Berg, David Cohen, Guillaume Dujardin
Summary: This article analyzes the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, proving that the strong order of the numerical approximation is 1/2 under globally Lipschitz nonlinear conditions. It demonstrates that the splitting scheme has a convergence order of 1/2 in probability and almost sure order 1/2- for cubic nonlinearities. Numerical experiments illustrate the results and efficiency of the scheme, while also investigating potential blowup of solutions for certain power-law nonlinearities.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jianbo Cui, Jialin Hong, Derui Sheng
Summary: This article studies the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. The existence and smoothness of the density function of the numerical solution are proven, along with the optimal strong convergence rate in every Malliavin-Sobolev norm. It is also shown that the convergence order of the density function of the numerical scheme coincides with its strong convergence order.
MATHEMATICS OF COMPUTATION
(2022)
Article
Mathematics, Interdisciplinary Applications
Charles-Edouard Brehier, David Cohen
Summary: We analyze a splitting integrator for the time discretization of the Schrodinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that this time integrator has first-order convergence in the pth mean sense, for any p greater than or equal to 1 in some Sobolev spaces. We demonstrate that the splitting schemes preserve the L-2-norm, which is essential for proving the strong convergence result. Finally, numerical experiments are conducted to illustrate the performance of the proposed numerical scheme.
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2022)
Article
Mathematics, Applied
Jingjun Zhao, Yulian Yi, Yang Xu
Summary: This paper proposes two projected Euler type schemes for stochastic differential equations with Markovian switching and super-linear coefficients, and investigates their convergence under polynomial growth and monotone conditions. Furthermore, the convergence rates of these schemes for highly nonlinear equations with small noise are discussed. Numerical experiments are conducted to validate the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Computer Science, Software Engineering
Meng Cai, Ruisheng Qi, Xiaojie Wang
Summary: In this paper, an explicit time-stepping scheme is proposed and analyzed for the spatial discretization of stochastic Cahn-Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space with a tamed exponential Euler method in time. The explicit scheme is easily implementable and significantly improves computational efficiency compared to implicit schemes. The paper presents the first result concerning an explicit scheme for the stochastic Cahn-Hilliard equation, and new arguments are developed to overcome the difficulties arising from the presence of an unbounded linear operator.
BIT NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Lin Chen, Siqing Gan, Xiaojie Wang
Summary: This paper proposes a novel explicit time-stepping scheme, called Lamperti smoothing truncation scheme, to approximate a stochastic SIS epidemic model. The scheme preserves the domain of the original SDEs and maintains a mean-square convergence rate of order one. Numerical examples are provided to confirm the theoretical findings.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Ruishu Liu, Xiaojie Wang
Summary: By combining a predictor-corrector method with a Lamperti-type transformation, we propose a higher-order, explicit, positivity preserving scheme for the stochastic susceptible-infected-susceptible (SIS) epidemic model that takes values in (0, N). The proposed scheme preserves the domain (0, N) of the original SIS model and allows for numerical approximations with exponential integrability. These findings help us recover the scheme's strong convergence rate of order 1.5. Furthermore, we investigate the dynamic behaviors of the numerical approximations, which show that the scheme can reproduce the extinction and persistence properties of the disease under certain assumptions. Lastly, numerical experiments are conducted to verify the theoretical findings. (c) 2023 Elsevier B.V. All rights reserved.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Chinedu Izuchukwu, Simeon Reich, Yekini Shehu, Adeolu Taiwo
Summary: In this paper, several strongly convergent versions of the forward-reflected-backward splitting method are proposed and studied for finding a zero of the sum of two monotone operators in a real Hilbert space. The proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration, which is a feature absent in many other available strongly convergent splitting methods. Inertial versions of the methods are also developed, and strong convergence results are obtained for these methods under certain conditions on the operators. Examples from image restorations and optimal control are discussed to demonstrate the effectiveness of the proposed methods compared to existing methods in the literature.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Operations Research & Management Science
Patrick R. Johnstone, Jonathan Eckstein, Thomas Flynn, Shinjae Yoo
Summary: This paper presents a new stochastic variant of the projective splitting algorithm for inclusion problems involving maximal monotone operators. The proposed method has significance in applications such as machine learning.
COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
(2023)
Article
Operations Research & Management Science
Van Dung Nguyen, Bang Cong Vu
Summary: We propose a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator and analyze its convergence. The algorithm only requires unbiased estimations of the Lipschitzian operator and achieves a convergence rate of O(log(n)/n) in expectation for the strongly monotone case, as well as almost sure convergence for the general case. Furthermore, in the context of application to convex-concave saddle point problems, we derive the convergence rate of the primal-dual gap, obtaining a convergence rate of O(1/n) in the deterministic setting.
OPTIMIZATION LETTERS
(2022)
Article
Mathematics, Applied
Chuchu Chen, Jianbo Cui, Jialin Hong, Derui Sheng
Summary: This paper investigates the numerical approximation of the density of the stochastic heat equation driven by space-time white noise using the accelerated exponential Euler scheme. The existence and smoothness of the density of the numerical solution are proved using Malliavin calculus. A test-function-independent weak convergence analysis is presented based on a priori estimates, which is crucial for showing the convergence of the density. The convergence order of the density is shown to be exactly 1/2 in the nonlinear drift case and nearly 1 in the affine drift case. To our knowledge, this is the first result on the existence and convergence of the density of the numerical solution to the stochastic partial differential equation.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Jianbo Cui, Luca Dieci, Haomin Zhou
Summary: This paper studies the discretizations of Hamiltonian systems on the probability density manifold with the L-2-Wasserstein metric. Based on discrete optimal transport theory, Hamiltonian systems on a graph with different weights are derived, which serve as spatial discretizations of the original systems. The consistency of these discretizations is proven. Moreover, by regularizing the system and obtaining an explicit lower bound for the density function, the use of symplectic schemes for time discretization is guaranteed. Desirable long time behavior of these symplectic schemes is shown and their performance is demonstrated through numerical examples. Finally, the present approach is compared with the standard viscosity methodology.
MATHEMATICS OF COMPUTATION
(2022)
Article
Automation & Control Systems
Xu Liu, Wei Peng, Zhiqiang Gong, Weien Zhou, Wen Yao
Summary: Temperature field inversion of heat-source systems (TFI-HSS) with limited observations is essential for monitoring system health. This study proposes a physics-informed neural network-based temperature field inversion method and a coefficient matrix condition number-based position selection of observations method. The results demonstrate that the PINN-TFI method can significantly improve prediction precisions and the CMCN-PSO method can find good positions to improve the robustness of the PINN-TFI method.
ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE
(2022)
Article
Mathematics, Applied
Jialin Hong, Zhihui Liu, Derui Sheng
Summary: This study investigates the optimal Holder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet. It proves the well-posedness and Holder continuity of the solutions using stochastic calculus for fractional Brownian motion. The study also obtains the optimal Holder exponents, which is the first result on the optimal Holder continuity of the systems driven by a rough fractional Brownian sheet in space. Based on this sharp regularity, lower and upper bounds of hitting probabilities are obtained using Bessel-Riesz capacity and Hausdorff measure.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Jianbo Cui, Jialin Hong, Derui Sheng
Summary: This article studies the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. The existence and smoothness of the density function of the numerical solution are proven, along with the optimal strong convergence rate in every Malliavin-Sobolev norm. It is also shown that the convergence order of the density function of the numerical scheme coincides with its strong convergence order.
MATHEMATICS OF COMPUTATION
(2022)
Article
Mathematics, Applied
Jianbo Cui, Luca Dieci, Haomin Zhou
Summary: This paper proposes a numerical method for solving the classic L-2-optimal transport problem. The algorithm is based on multiple shooting and a continuation procedure to solve the associated boundary value problem. By considering the viewpoint of Wasserstein Hamiltonian flow, the algorithm reflects the Hamiltonian structure of the problem and utilizes it in the numerical discretization. Several numerical examples are provided to demonstrate the performance of the method.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Jianbo Cui, Jialin Hong
Summary: This article investigates the approximation problem of the one-dimensional stochastic Cahn-Hilliard equation. The well-posedness of the approximated equation in finite dimension is obtained using the spectral Galerkin method. The desirable properties and explicit convergence rate of the approximation processes are shown through the semigroup theory and factorization method. Additionally, the global existence and regularity estimates of the unique solution process are proven by means of the strong convergence of the approximation process, filling a gap in the global existence of the mild solution for the stochastic Cahn-Hilliard equation.
STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS
(2023)
Article
Engineering, Multidisciplinary
Zeyu Zhang, Wen Yao, Yu Li, Weien Zhou, Xiaoqian Chen
Summary: With the rapid development of artificial intelligence (AI) technology, scientific research has entered a new era of AI. The cross development between topology optimization (TO) and AI technology has been receiving continuous attention. This paper introduces the concept of Implicit Neural Representations from AI into the TO field and establishes a novel TO framework called TOINR.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Jianbo Cui, Shu Liu, Haomin Zhou
Summary: In this paper, the stochastic Hamiltonian flow in Wasserstein manifold is studied via the Wong-Zakai approximation. It is shown that the stochastic Euler-Lagrange equation can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold, regardless of its derivation from either the variational principle or particle dynamics. A novel variational formulation is proposed to derive more general stochastic Wasserstein Hamiltonian flows, and its applicability to various systems including the stochastic Schrodinger equation, Schrodinger equation with random dispersion, and Schrodinger bridge problem with common noise is demonstrated.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Jianbo Cui, Shu Liu, Haomin Zhou
Summary: This article studies the Wasserstein Hamiltonian flow with common noise on the density manifold of a finite graph. Under the framework of the stochastic variational principle, the formulation of stochastic Wasserstein Hamiltonian flow is developed, and the local existence of a unique solution is shown. A sufficient condition for the global existence of the solution is also established. Consequently, the global well-posedness for the nonlinear Schroedinger equations with common noise on a graph is obtained. Additionally, the existence of the minimizer for an optimal control problem with common noise is proved using Wong-Zakai approximation, and it is shown that the minimizer satisfies the stochastic Wasserstein Hamiltonian flow on a graph as well.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2023)
Article
Thermodynamics
Xianqi Chen, Wen Yao, Weien Zhou, Zeyu Zhang, Yu Li
Summary: In this work, a general and differentiable heat source layout optimization framework based on parameterized level set functions is proposed. The framework incorporates Heaviside projection for an analytical description of the heat source intensity function and automatic differentiation technique for sensitivity analysis. An adaptive multiresolution FEA method is introduced to eliminate gradient oscillations caused by finite element discretization. Numerical experiments demonstrate the positive effects of the adaptive multiresolution strategy and the effectiveness of the proposed approach in heat conduction problems.
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER
(2023)
Article
Automation & Control Systems
Jianbo Cui, Shu Liu, Haomin Zhou
Summary: We study the optimal control formulation for the stochastic nonlinear Schrodinger equation (SNLSE) on a finite graph. By treating the SNLSE as a stochastic Wasserstein Hamiltonian flow on the density manifold, we prove the global existence of a unique strong solution for SNLSE with a linear drift control or a linear diffusion control on the graph. Additionally, we provide the gradient formula, the existence of the optimal control, and a description of the optimal condition through the forward and backward stochastic differential equations.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2023)
Article
Mathematics, Applied
Jianbo Cui, Liying Sun
Summary: This paper proves the global existence and uniqueness of the solution of the stochastic logarithmic Schrodinger equation driven by either additive noise or multiplicative noise. The key lies in the regularized logarithmic Schrodinger equation with regularized energy and the strong convergence analysis of the solutions. In addition, temporal Holder regularity estimates and uniform estimates in energy space and weighted Sobolev space are obtained for the solutions of both the original and the regularized equation.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2023)
Article
Thermodynamics
Xiaoyu Zhao, Xiaoqian Chen, Zhiqiang Gong, Weien Zhou, Wen Yao, Yunyang Zhang
Summary: The perception of the full state is crucial for monitoring, analyzing, and designing physical systems. This paper introduces a novel end-to-end physical field reconstruction method called RecFNO, which learns the mapping from sparse observations to flow and heat fields in infinite-dimensional space. The proposed method achieves excellent performance and mesh transferability. Experimental results demonstrate its superiority over existing POD-based and CNN-based methods in most cases, and its ability to achieve zero-shot super-resolution.
INTERNATIONAL JOURNAL OF THERMAL SCIENCES
(2024)
Article
Mathematics, Applied
Jianbo Cui, Jialin Hong, Liying Sun
Summary: In this article, the stochastic Cahn-Hilliard equation is discretized using the spectral Galerkin method in space and a temporally accelerated implicit Euler method. The proposed numerical method is proven to have strong convergence with a sharp convergence rate in a negative Sobolev space. By utilizing semigroup theory and interpolation inequality, the spatial optimal convergence rate and temporal superconvergence rate of the numerical method in the strong sense are deduced.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics
Daniele Cassani, Zhisu Liu, Giulio Romani
Summary: This article investigates the strongly coupled nonlinear Schrodinger equation and Poisson equation in two dimensions. The existence of solutions is proved using a variational approximating procedure, and qualitative properties of the solutions are established through the moving planes technique.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Giovanni Alessandrini, Romina Gaburro, Eva Sincich
Summary: This paper considers the inverse problem of determining the conductivity of a possibly anisotropic body Ω, subset of R-n, by means of the local Neumann-to-Dirichlet map on a curved portion Σ of its boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities, the paper provides a Hölder stability estimate on Σ when the conductivity is a priori known to be a constant matrix near Σ.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nuno Costa Dias, Cristina Jorge, Joao Nuno Prata
Summary: This article studies the time dependent Euler-Bernoulli beam equation with discontinuous and singular coefficients, and obtains an explicit formulation of the differential problem using an extension of the Hormander product of distributions. The dynamics of the Euler-Bernoulli beam model with discontinuous flexural stiffness and structural cracks are further explored, and the relationship between the characteristic frequencies of the beam and the singularities in the flexural stiffness is investigated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Baoquan Zhou, Hao Wang, Tianxu Wang, Daqing Jiang
Summary: This paper is Part I of a two-part series that presents a mathematical framework for approximating the invariant probability measures and density functions of stochastic generalized Kolmogorov systems with small diffusion. It introduces two new approximation methods and demonstrates their utility in various applications.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Yun Li, Danhua Jiang, Zhi-Cheng Wang
Summary: In this study, a nonlocal reaction-diffusion equation is used to model the growth of phytoplankton species in a vertical water column with changing-sign advection. The species relies solely on light for metabolism. The paper primarily focuses on the concentration phenomenon of phytoplankton under conditions of large advection amplitude and small diffusion rate. The findings show that the phytoplankton tends to concentrate at certain critical points or the surface of the water column under these conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Monica Conti, Stefania Gatti, Alain Miranville
Summary: The aim of this paper is to study a perturbation of the Cahn-Hilliard equation with nonlinear terms of logarithmic type. By proving the existence, regularity and uniqueness of solutions, as well as the (strong) separation properties of the solutions from the pure states, we finally demonstrate the convergence to the Cahn-Hilliard equation on finite time intervals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Xiaolong He
Summary: This article employs the CWB method to construct quasi-periodic solutions for nonlinear delayed perturbation equations, and combines the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, thus extending the applicability of the CWB method. As an application, it studies the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nicolas Camps, Louise Gassot, Slim Ibrahim
Summary: In this paper, we consider the probabilistic local well-posedness problem for the Schrodinger half-wave equation with a cubic nonlinearity in quasilinear regimes. Due to the lack of probabilistic smoothing in the Picard's iterations caused by high-low-low nonlinear interactions, we need to use a refined ansatz. The proof is an adaptation of Bringmann's method on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, ill-posedness results for this equation are discussed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Elie Abdo, Mihaela Ignatova
Summary: In this study, we investigate the Nernst-Planck-Navier-Stokes system with periodic boundary conditions and prove the exponential nonlinear stability of constant steady states without constraints on the spatial dimension. We also demonstrate the exponential stability from arbitrary large data in the case of two spatial dimensions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Peter De Maesschalck, Joan Torregrosa
Summary: This paper provides the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. The key idea is the perturbation of a vector field with many cusp equilibria, which is constructed using elements of catastrophe theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Leyi Jiang, Taishan Yi, Xiao-Qiang Zhao
Summary: This paper studies the propagation dynamics of a class of integro-difference equations with a shifting habitat. By transforming the equation using moving coordinates and establishing the spreading properties of solutions and the existence of nontrivial forced waves, the paper contributes to the understanding of the propagation properties of the original equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Mckenzie Black, Changhui Tan
Summary: This article investigates a family of nonlinear velocity alignments in the compressible Euler system and shows the asymptotic emergent phenomena of alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are studied, resulting in a variety of different asymptotic behaviors.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Lorenzo Cavallina
Summary: In this paper, the concept of variational free boundary problem is introduced, and a unified functional-analytical framework is provided for constructing families of solutions. The notion of nondegeneracy of a critical point is extended to this setting.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Ying-Chieh Lin, Kuan-Hsiang Wang, Tsung-Fang Wu
Summary: In this study, we investigate a linearly coupled Schrodinger system and establish the existence of positive ground states under suitable assumptions and by using variational methods. We also relax some of the conditions and provide some results on the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
