In this paper we discuss parametrized partial differential equations (PDEs) for parameters that describe the geometry of the underlying problem. One can think of applications in control theory and optimization which depend on time-consuming parameter-studies of such problems. Therefore, we want to reduce the order of complexity of the numerical simulations for such PDEs. Reduced Basis (RB) meth...

Article history: Received 29 December 2010 Revised 19 January 2011 Available online 4 February 2011 We prove an abstract version of the striking diffusion phenomenon that offers a strong connection between the asymptotic behavior of abstract parabolic and dissipative hyperbolic equations. An important aspect of our approach is that we use in a natural way spectral analysis without involving com...

We consider the parabolic type equation with a source-sink term and construct the Monte Carlo estimator for it. The procedure is based on the Hopf-Cole transformation and a Monte Carlo estimator for the correspondent Burgers equation. Monte Carlo estimators for the heat and diffusion equations are very well known since the first steps of the method. The algorithms are based usually on the intim...

Title of dissertation: ASYMPTOTIC PROBLEMS FOR STOCHASTIC PROCESSES AND CORRESPONDING PARTIAL DIFFERENTIAL EQUATIONS Lucas Tcheuko, Doctor of Philosophy, 2014 Dissertation directed by: Professor Mark Freidlin & Professor Leonid Koralov Department of Mathematics We consider asymptotic problems for diffusion processes that rely on large deviations. In Chapter 2, we study the long time behavior (a...

Fick’s law expresses the proportionality of solute flux with respect to concentration gradient. Similar relations are Darcy’s law for the fluid flow in porous media, Ohm’s law for the electric flux and Fourier’s law for heat transfers. When introduced in the corresponding balance equations, these flux laws yield diffusion equations of parabolic character. Different attempts have been made to ob...

My research work centers around two broad topics: The analysis of nonlinear Partial Differential Equations (PDE) of elliptic and parabolic type (mainly free boundary problems and reaction-diffusion equations) and the study of transport phenomena (particularly the connection between kinetic and hydrodynamic models). My main contributions are described below (preprints are available at http://www...

In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain Ω× [0, T ] which is decomposed into an overlapping collection of cylindrical subregions of the form Ωl ×[0, T ], fo...

A procedure is described for defining a generalized solution for stochastic differential equations using the Cameron-Martin version of the Wiener Chaos expansion. Existence and uniqueness of this Wiener Chaos solution is established for parabolic stochastic PDEs such that both the drift and the diffusion operators are of the second order.

We propose a new Complex Diffusion Monte Carlo (CDMC) method for the simulation of quantum systems with complex wave function. In CDMC the modulus and phase of wave function are simulated both in contrast to other methods. We successfully test CDMC by the simulation of the ground state for 2D electron in magnetic field and 2D fermionsanyons in parabolic well.

We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Hölder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.