Numerical models for groundwater flow and transport on a regional scale require the assignment of hydraulic parameters to grid blocks of a size larger than typical scales of available field data. Most natural formations are heterogeneous with hydraulic properties varying on a multiplicity of scales. This difference in scales is often handled by using effective (upscaled) parameters. Here we inv...

On Asymptotic Preserving Schemes for a Class of Stochastic Differential Equations in Averaging and Diffusion Approximation Regimes

This paper investigates a non-autonomous slow–fast system, which is generalized by stochastic differential equations with locally Lipschitz coefficients, subjected to standard Brownian motion and fractional Hurst parameter 1/2<H<1. The pathwise approach the Itô calculus are combined technique of stopping time establish averaging principle. Then, we conclude that slow component original system c...

[1] Nemirovski and Yudin. Problems Complexity and Method Efficiency in Optimization. Wiley-Interscience series in discrete mathematics. Wiley, 1983. [2] W. Krichene, A. Bayen and P. Bartlett. Accelerated Mirror Descent in Continuous and Discrete Time. NIPS 2015. [3] W. Su, S. Boyd and E. Candes. A differential equation for modeling Nesterov's accelerated gradient method: theory and insights. NI...

We consider the minimization of a function defined on a Riemannian manifold M accessible only through unbiased estimates of its gradients. We develop a geometric framework to transform a sequence of slowly converging iterates generated from stochastic gradient descent (SGD) on M to an averaged iterate sequence with a robust and fast O(1/n) convergence rate. We then present an application of our...

We demonstrate deterministic control over the photon-number distribution by interfering two coherent beams within a disordered photonic lattice. By sweeping a relative phase between two equal-amplitude coherent fields with Poissonian statistics that excite adjacent sites in a lattice endowed with disorder-immune chiral symmetry, we measure an output photon-number distribution that changes perio...

We investigate stochastic averaging on the infinite time interval for a class of continuous-time nonlinear systems with stochastic perturbation and remove or weaken several restrictions present in existing results: global Lipschitzness of the nonlinear vector field, equilibrium preservation under the stochastic perturbation, global exponential stability of the average system, and compactness of...

We obtain new transport-entropy inequalities and, as a by-product, new deviation estimates for the laws of two kinds of discrete stochastic approximation schemes. The first one refers to the law of an Euler like discretization scheme of a diffusion process at a fixed deterministic date and the second one concerns the law of a stochastic approximation algorithm at a given time-step. Our results ...