In this paper, we propose a numerical method to solve the classic $L^2$-optimal transport problem. Our algorithm is based on use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the
transport problem. We exploit the viewpoint of Wasserstein Hamiltonian flow with initial and target densities, and our method is designed to retain the underlying Hamiltonian structure. Several numerical examples are presented to illustrate the performance of the method.
In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schrodinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic
Schrodinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of (RSlogS) equations. In addition, temporal Holder regularity estimates and uniform estimates in energy space $mathbb H^1(mathcal O)$ and weighted Sobolev space $L^2_{alpha}(mathcal O)$ of the solutions for both SlogS equation and RSlogS equation are also obtained.
We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the
optimal transport problem, Schrodinger equation and Schrodinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.
We study discretizations of Hamiltonian systems on the probability density manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on graph (lattice) with different weights are deri
ved, which can be viewed as spatial discretizations to the original Hamiltonian systems. We prove the consistency and provide the approximate orders for those discretizations. By regularizing the system using Fisher information, we deduce an explicit lower bound for the density function, which guarantees that symplectic schemes can be used to discretize in time. Moreover, we show desirable long time behavior of these schemes, and demonstrate their performance on several numerical examples.
In this article, we develop and analyze a full discretization, based on the spatial spectral Galerkin method and the temporal drift implicit Euler scheme, for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise. By
introducing an appropriate decomposition of the numerical approximation, we first use the factorization method to deduce the a priori estimate and regularity estimate of the proposed full discretization. With the help of the variation approach, we then obtain the sharp spatial and temporal convergence rate in negative Sobolev space in mean square sense. Furthermore, the sharp mean square convergence rates in both time and space are derived via the Sobolev interpolation inequality and semigroup theory. To the best of our knowledge, this is the first result on the convergence rate of temporally and fully discrete numerical methods for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise.
In this article, we consider the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise with diffusion coefficient of sublinear growth. By introducing the spectral Galerkin method, we first obtain the well-posedness of the
approximated equation in finite dimension. Then with the help of the semigroup theory and the factorization method, the approximation processes is shown to possess many desirable properties. Further, we show that the approximation process is strongly convergent in certain Banach space via the interpolation inequality and variational approach. Finally, the global existence and regularity estimate of the unique solution process are proven by means of the strong convergence of the approximation process.
In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. To deal with the non-globally monotone coefficient in the considered equation, we first
present the exponential integrability properties of the exact and numerical solutions. Then we show the existence and smoothness of the density function of the numerical solution by proving its uniform non-degeneracy in Malliavin sense. In order to analyze the approximate error between the density function of the exact solution and that of the numerical solution, we derive the optimal strong convergence rate in every Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus. Combining the approximation result of Donskers delta function and the smoothness of the density functions, we prove that the convergence rate in density coincides with the optimal strong convergence rate of the numerical scheme.
This article presents explicit exponential integrators for stochastic Maxwells equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numeric
al approximation is $frac 12$ for general multiplicative noise. Combing a proper decomposition with the stochastic Fubinis theorem, the strong order of the proposed scheme is shown to be $1$ for additive noise. Moreover, for linear stochastic Maxwells equation with additive noise, the proposed time integrator is shown to preserve exactly the symplectic structure, the evolution of the energy as well as the evolution of the divergence in the sense of expectation. Several numerical experiments are presented in order to verify our theoretical findings.
In this article, we consider the stochastic Cahn--Hilliard equation driven by space-time white noise. We discretize this equation by using a spatial spectral Galerkin method and a temporal accelerated implicit Euler method. The optimal regularity pro
perties and uniform moment bounds of the exact and numerical solutions are shown. Then we prove that the proposed numerical method is strongly convergent with the sharp convergence rate in a negative Sobolev space. By using an interpolation approach, we deduce the spatial optimal convergence rate and the temporal super-convergence rate of the proposed numerical method in strong convergence sense. To the best of our knowledge, this is the first result on the strong convergence rates of numerical methods for the stochastic Cahn--Hilliard equation driven by space-time white noise. This interpolation approach is also applied to the general noise and high dimension cases, and strong convergence rate results of the proposed scheme are given.
Approximating the invariant measure and the expectation of the functionals for parabolic stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients is an active research area and is far from being well understood. In t
his article, we study such problem in terms of a full discretization based on the spectral Galerkin method and the temporal implicit Euler scheme. By deriving the a priori estimates and regularity estimates of the numerical solution via a variational approach and Malliavin calculus, we establish the sharp weak convergence rate of the full discretization. When the SPDE admits a unique $V$-uniformly ergodic invariant measure, we prove that the invariant measure can be approximated by the full discretization. The key ingredients lie on the time-independent weak convergence analysis and time-independent regularity estimates of the corresponding Kolmogorov equation. Finally, numerical experiments confirm the theoretical findings.
