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We study the nonlinear Fokker-Planck equation on graphs, which is the gradient flow in the space of probability measures supported on the nodes with respect to the discrete Wasserstein metric. The energy functional driving the gradient flow consists of a Boltzmann entropy, a linear potential and a quadratic interaction energy. We show that the solution converges to the Gibbs measures exponentially fast with a rate that can be given analytically. The continuous analog of this asymptotic rate is related to the Yano's formula.

This paper is concerned with the inverse medium scattering problem with a stochastic source, the reconstruction of the refractive index of an inhomogeneous medium from the boundary measurements of the scattered field. As an inverse problem, there are two major difficulties in addition to being highly nonlinear: the ill-posedness and the presence of many local minima. To overcome these difficulties, a stable and efficient recursive linearization method has been recently developed for solving the inverse medium scattering problem with a deterministic source. Compared to classical inverse problems, stochastic inverse problems, referred to as inverse problems involving uncertainties, have substantially more difficulties due to randomness and uncertainties. Based on the Wiener chaos expansion, the stochastic problem is converted into a set of decoupled deterministic problems. The strategy developed is a new hybrid method combining the WCE with the recursive linearization method for solving the inverse medium problem with a stochastic source. Numerical experiments are reported to demonstrate the effectiveness of the proposed approach.

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Casopis pro pSstovani matematiky, rofc. 111 (1986), Praha

First, we propose a new stochastic model for a spatially incoherent source in optical phenomena. The model naturally incorporates the incoherent property into the electromagnetic wave equation through a random source term. Then we propose a new numerical method based on Wiener chaos expansion (WCE) and apply it to solve the resulting stochastic wave equation. The main advantage of the WCE method is that it separates random and deterministic effects and allows the random effects to be factored out of the primary partial differential equation (PDE) very effectively. Therefore, the stochastic PDE is reduced to a set of deterministic PDEs for the coefficients of the WCE method which can be solved by conventional numerical algorithms. We solve these secondary deterministic PDEs by a finite-difference time domain (FDTD) method and demonstrate that the numerical computations based on the WCE method are considerably more efficient than the brute-force simulations. Moreover, the WCE approach does not require generation of random numbers and results in less computational errors compared to Monte Carlo simulations.