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A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

Guang-hua Gao, Zhi-zhong Sun (2012)

Open Mathematics

Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers...

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional brownian motion

A. Deya, A. Neuenkirch, S. Tindel (2012)

Annales de l'I.H.P. Probabilités et statistiques

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second-order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds...

A stochastic phase-field model determined from molecular dynamics

Erik von Schwerin, Anders Szepessy (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The dynamics of dendritic growth of a crystal in an undercooled melt is determined by macroscopic diffusion-convection of heat and by capillary forces acting on the nanometer scale of the solid-liquid interface width. Its modelling is useful for instance in processing techniques based on casting. The phase-field method is widely used to study evolution of such microstructural phase transformations on a continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landau equation...

A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems

M. Billaud-Friess, A. Nouy, O. Zahm (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be designed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take...

Approximation of stochastic advection diffusion equations with stochastic alternating direction explicit methods

Ali R. Soheili, Mahdieh Arezoomandan (2013)

Applications of Mathematics

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed....

Asymptotics of a Time-Splitting Scheme for the Random Schrödinger Equation with Long-Range Correlations

Christophe Gomez, Olivier Pinaud (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This work is concerned with the asymptotic analysis of a time-splitting scheme for the Schrödinger equation with a random potential having weak amplitude, fast oscillations in time and space, and long-range correlations. Such a problem arises for instance in the simulation of waves propagating in random media in the paraxial approximation. The high-frequency limit of the Schrödinger equation leads to different regimes depending on the distance of propagation, the oscillation pattern of the initial...

Behavior of the Euler scheme with decreasing step in a degenerate situation

Vincent Lemaire (2007)

ESAIM: Probability and Statistics

The aim of this short note is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant measures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously.

Convergence rates for the full gaussian rough paths

Peter Friz, Sebastian Riedel (2014)

Annales de l'I.H.P. Probabilités et statistiques

Under the key assumption of finite ρ -variation, ρ [ 1 , 2 ) , of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), ρ = 1 resp. ρ = 1 / ( 2 H ) , we recover and extend the respective results of (Trans. Amer. Math. Soc.361 (2009) 2689–2718) and (Ann. Inst. Henri Poincasé Probab. Stat.48(2012) 518–550). In particular, we establish an a.s. rate k - ( 1 / ρ - 1 / 2 - ε ) , any ε g t ; 0 , for Wong–Zakai and Milstein-type...

Discrete approximations of generalized RBSDE with random terminal time

Katarzyna Jańczak-Borkowska (2012)

Discussiones Mathematicae Probability and Statistics

The convergence of discrete approximations of generalized reflected backward stochastic differential equations with random terminal time in a general convex domain is studied. Applications to investigation obstacle elliptic problem with Neumann boundary condition for partial differential equations are given.

Discrete Approximations of Strong Solutions of Reflecting SDEs with Discontinuous Coefficients

Alina Semrau (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We study L p convergence for the Euler scheme for stochastic differential equations reflecting on the boundary of a general convex domain D ⊆ ℝd. We assume that the equation has the pathwise uniqueness property and its coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. In the case D=[0,∞) new sufficient conditions ensuring pathwise uniqueness for equations with possibly discontinuous coefficients are given.

Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification

Kamaljit Chowdhary, Paul Dupuis (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all.The main tool used is the duality between risk-sensitive integrals and relative entropy, and we...

Elements of uncertainty modeling

Chleboun, Jan (2010)

Programs and Algorithms of Numerical Mathematics

The goal of this contribution is to introduce some approaches to uncertainty modeling in a way accessible to non-specialists. Elements of the Monte Carlo method, polynomial chaos method, Dempster-Shafer approach, fuzzy set theory, and the worst (case) scenario method are presented.

Elliptic equations of higher stochastic order

Sergey V. Lototsky, Boris L. Rozovskii, Xiaoliang Wan (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which...

Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere

Ľubomír Baňas, Zdzisław Brzeźniak, Mikhail Neklyudov, Martin Ondreját, Andreas Prohl (2015)

Czechoslovak Mathematical Journal

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also...

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