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Transfer operators, atomic decomposition and the BestiaryMar 16 2019Arbieto and S. recently used atomic decomposition to study transfer operators. We give a long list of old and new expanding dynamical systems for which those results can be applied, obtaining the quasi-compactness of transfer operator acting on Besov ... More
Transfer operators and atomic decompositionMar 16 2019Apr 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More
Transfer operators and atomic decompositionMar 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More
Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearitiesMar 14 2019The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing ... More
Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noiseJan 16 2019In numerical analysis for stochastic partial differential equations one distinguishes between weak and strong convergence rates. Often the weak convergence rate is twice the strong convergence rate. However, there is no standard way to prove this: to ... More
Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noiseDec 18 2018We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence ... More
Optimal real-time detection of a drifting Brownian coordinateDec 18 2018Consider the motion of a Brownian particle in three dimensions, whose two spatial coordinates are standard Brownian motions with zero drift, and the remaining (unknown) spatial coordinate is a standard Brownian motion with a non-zero drift. Given that ... More
Strong convergence rate of a full discretization for stochastic Cahn--Hilliard equation driven by space-time white noiseDec 15 2018Jan 02 2019In 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 properties ... More
Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equationsNov 01 2018Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete ... More
Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic TermsOct 15 2018This paper introduces a statistical treatment of inverse problems constrained by models with stochastic terms. The solution of the forward problem is given by a distribution represented numerically by an ensemble of simulations. The goal is to formulate ... More
Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic TermsOct 15 2018Apr 16 2019This paper introduces a statistical treatment of inverse problems constrained by models with stochastic terms. The solution of the forward problem is given by a distribution represented numerically by an ensemble of simulations. The goal is to formulate ... More
Numerical approximation for non-colliding particle systemsJul 24 2018We apply the semi-discrete method, c.f. \emph{N. Halidias and I.S. Stamatiou (2016), On the numerical solution of some non-linear stochastic differential equations using the semi-discrete method, Computational Methods in Applied Mathematics, 16(1)}, to ... More
Uncertainty quantification for an optical grating coupler with an adjoint-based Leja adaptive collocation methodJul 19 2018This paper addresses uncertainties arising in the nano-scale fabrication of optical devices. The stochastic collocation method is used to propagate uncertainties in material and geometry to the scattering parameters of the system. A dimension-adaptive ... More
Rapid covariance-based sampling of linear SPDE approximations in the multilevel Monte Carlo methodJun 29 2018Oct 02 2018The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equations (SPDE) with additive Gaussian noise at a fixed time is considered. A Galerkin finite element method is employed ... More
Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficientsMay 28 2018Mar 01 2019We present strongly convergent explicit and semi-implicit numerical schemes for systems of stiff stochastic differential equations (SDEs) where both the drift and diffusion coefficients are non-globally Lipschitz continuous. This stiffness may originate ... More
Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficientsMay 28 2018We present strongly convergent explicit and semi-implicit numerical schemes for systems of stiff stochastic differential equations (SDEs) where both the drift and diffusion coefficients are non-globally Lipschitz continuous. This stiffness may originate ... More
Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noiseApr 30 2018May 02 2018Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive trace-class noise in space dimension $d \le 3$. The full discretization ... More
Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulationApr 19 2018We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic ... More
Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulationApr 19 2018Apr 02 2019We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic ... More
Weak convergence rates of splitting schemes for the stochastic Allen-Cahn equationApr 11 2018Apr 18 2018This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work, for the temporal discretization of the stochastic Allen-Cahn equation driven by space-time white noise. The schemes are based ... More
Numerical methods for conservation laws with rough fluxFeb 02 2018Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with a rough path dependent flux function. For a convex flux, it is demonstrated that rough path oscillations may lead to "cancellations" ... More
Scalable hierarchical PDE sampler for generating spatially correlated random fields using non-matching meshesDec 19 2017This work describes a domain embedding technique between two non-matching meshes used for generating realizations of spatially correlated random fields with applications to large-scale sampling-based uncertainty quantification. The goal is to apply the ... More
Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputsNov 30 2017Nov 02 2018In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems ... More
Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equationsNov 07 2017The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence ... More
Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEsSep 27 2017This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form $dX(t)=AX(t)dt+F(X(t))dt+dW(t)$, driven by space-time white noise. In previous results, test functions ... More
A note on pathwise stability and positivity of nonlinear stochastic differential equationsAug 25 2017We use the semi-discrete method, originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6), to reproduce qualitative properties of ... More
A boundary preserving numerical scheme for the Wright-Fisher modelApr 13 2017Jun 27 2017We are interested in the numerical approximation of non-linear stochastic differential equations (SDEs) with solution in a certain domain. Our goal is to construct explicit numerical schemes that preserve that structure. We generalize the semi-discrete ... More
A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random FieldsMar 24 2017We propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen-Lo\`{e}ve (KL) ... More
Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensionsFeb 24 2017Jul 07 2017The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and ... More
On some nonlinear equation from theory of the flows on networksJan 19 2017Here we study the nonlinear hyperbolic equations of the type of equations from theory of flows on networks, for which we prove the solvability theorem under the appropriate conditions and also investigate the behaviour of the solution.
Stochastic Least-Squares Petrov-Galerkin Method for Parameterized Linear SystemsJan 05 2017We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated ... More
Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equationDec 30 2016Jan 02 2018We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension $d\le 3$. We discretize the equation using a standard finite element method in space and a fully implicit backward ... More
Numerical methods for the deterministic second moment equation of parabolic stochastic PDEsNov 07 2016Apr 19 2018Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the first and second moments, as well as the covariance. ... More
A dynamical polynomial chaos approach for long-time evolution of SPDEsMay 15 2016Dec 14 2016We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously ... More
A dynamical polynomial chaos approach for long-time evolution of SPDEsMay 15 2016We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously ... More
Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximationsMay 14 2016This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater ... More
An accelerated exponential time integrator for semi-linear stochastic strongly damped wave equation with additive noiseFeb 19 2016This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed by a spectral Galerkin method, we introduce a kind of accelerated ... More
The Eddy Current--LLG Equations: FEM-BEM Coupling and A Priori Error EstimatesFeb 01 2016Feb 05 2017We analyze a numerical method for the coupled system of the eddy current equations in $\mathbb{R}^3$ with the Landau-Lifshitz-Gilbert equation in a bounded domain. The unbounded domain is discretized by means of finite-element/boundary-element coupling. ... More
The Eddy Current-LLG Equations-Part II: A Priori Error EstimatesFeb 01 2016A numerical method for the coupled system of the eddy current equations in $\mathbb{R}^3$ with the Landau-Lifshitz-Gilbert equation in a bounded domain has been analysed in the first part of this paper. The method involves a symmetric finite-element/boundary-element ... More
The Eddy Current-LLG Equations-Part I: FEM-BEM CouplingFeb 01 2016We analyse a numerical method for the coupled system of the eddy current equations in $\mathbb{R}^3$ with the Landau-Lifshitz-Gilbert equation in a bounded domain. The unbounded domain is discretised by means of finite-element/boundary-element coupling. ... More
Exponential integrators for stochastic Schrödinger equations driven by Ito noiseJan 25 2016We study an explicit exponential scheme for the time discretisation of stochastic Schr\"odinger equations driven by additive or multiplicative Ito noise. The numerical scheme is shown to converge with strong order $1$ if the noise is additive and with ... More
Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equationsJan 21 2016This article proposes and analyzes explicit and easily implementable temporal numerical approximation schemes for additive noise-driven stochastic partial differential equations (SPDEs) with polynomial nonlinearities such as, e.g., stochastic Ginzburg-Landau ... More
Numerical analysis of lognormal diffusions on the sphereJan 11 2016Numerical solutions of stationary diffusion equations on $\mathbb{S}^2$ with isotropic lognormal diffusion coefficients are considered. H\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of ... More
Numerical analysis of lognormal diffusions on the sphereJan 11 2016Nov 01 2016Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity ... More
Approximating Stochastic Evolution Equations with Additive White and Rough NoisesJan 09 2016Jan 14 2016We analyze Galerkin schemes for stochastic evolution equations (SEEs) driven by an additive Gaussian noise which is temporally white and spatially fractional with Hurst index less than or equal to $\frac{1}{2}$. We first regularize the noise by Wong-Zakai ... More
On the discretisation in time of the stochastic Allen-Cahn equationOct 13 2015Mar 13 2016We consider the stochastic Allen--Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$, and study the semidiscretisation in time of the equation by an Euler type split-step method. We ... More
On the discretisation in time of the stochastic Allen-Cahn equationOct 13 2015Jan 02 2018We consider the stochastic Allen--Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$, and study the semidiscretisation in time of the equation by an Euler type split-step method. We ... More
Error estimates of finite element method for semi-linear stochastic strongly damped wave equationOct 11 2015In this paper, we consider a semi-linear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework, we establish existence, uniqueness and space-time regularity of a mild solution to such equation. Unlike ... More
An Efficient Derivative-Free Milstein Scheme for Stochastic Partial Differential Equations with Commutative NoiseSep 28 2015We propose a derivative-free Milstein scheme for stochastic partial differential equations with trace class noise which fulfill a certain commutativity condition. The same theoretical order of convergence with respect to the spatial and time discretizations ... More
Enhancing the Order of the Milstein Scheme for Stochastic Partial Differential Equations with Commutative NoiseSep 28 2015Aug 14 2018We consider a higher-order Milstein scheme for stochastic partial differential equations with trace class noise which fulfill a certain commutativity condition. A novel technique to generally improve the order of convergence of Taylor schemes for stochastic ... More
Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearitiesApr 14 2015In this article we propose a new, explicit and easily implementable numerical method for approximating a class of semilinear stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. We establish strong convergence rates for ... More
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train formatMar 11 2015We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities ... More
Full discretisation of semi-linear stochastic wave equations driven by multiplicative noiseFeb 28 2015Nov 25 2015A fully discrete approximation of the semi-linear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space and a stochastic trigonometric method for the temporal approximation. ... More
Split-step Milstein methods for multi-channel stiff stochastic differential systemsNov 26 2014We consider split-step Milstein methods for the solution of stiff stochastic differential equations with an emphasis on systems driven by multi-channel noise. We show their strong order of convergence and investigate mean-square stability properties for ... More
Weak convergence of finite element approximations of linear stochastic evolution equations with additive Lévy noiseNov 04 2014Feb 03 2015We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive L\'evy noise. We first derive a representation formula for the error which we then apply to study ... More
Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculusAug 04 2014Jun 22 2015This paper deals with the weak error estimates of the exponential Euler method for semi-linear stochastic partial differential equations (SPDEs). A weak error representation formula is first derived for the exponential integrator scheme in the context ... More
Computation of the Response Surface in the Tensor Train data formatJun 11 2014We apply the Tensor Train (TT) approximation to construct the Polynomial Chaos Expansion (PCE) of a random field, and solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization. We compare two strategies of the polynomial ... More
A computational study of the effects of remodelled electrophysiology and mechanics on initiation of ventricular fibrillation in human heart failureJun 06 2014The study of pathological cardiac conditions such as arrhythmias, a major cause of mortality in heart failure, is becoming increasingly informed by computational simulation, numerically modelling the governing equations. This can provide insight where ... More
First-order weak balanced schemes for bilinear stochastic differential equationsMar 24 2014Aug 25 2014We use the linear scalar SDE as a test problem to show that it is possible to construct almost sure stable first-order weak balanced schemes based on the addition of stabilizing functions to the drift terms. Then, we design balanced schemes for multidimensional ... More
Analysis and Approximation of Stochastic Nerve Axon EquationsFeb 19 2014We consider spatially extended conductance based neuronal models with noise described by a stochastic reaction diffusion equation with additive noise coupled to a control variable with multiplicative noise but no diffusion. We only assume a local Lipschitz ... More
Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo MethodsDec 20 2013This work is motivated by the need to develop efficient tools for uncertainty quantification in subsurface flows associated with radioactive waste disposal studies. We consider single phase flow problems in random porous media described by correlated ... More
Layer methods for Navier-Stokes equations with additive noiseDec 20 2013We propose and study a number of layer methods for stochastic Navier-Stokes equations (SNSE) with spatial periodic boundary conditions and additive noise. The methods are constructed using conditional probabilistic representations of solutions to SNSE ... More
An exponential integrator scheme for time discretization of nonlinear stochastic wave equationDec 18 2013Nov 30 2014This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation. A unified framework is first set forth, which covers important cases of additive and multiplicative ... More
On the backward Euler approximation of the stochastic Allen-Cahn equationNov 08 2013Nov 26 2013We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$, and study the semidiscretization in time of the equation by an implicit Euler method. We show that ... More
An exponential Wagner-Platen type scheme for SPDEsSep 18 2013The strong numerical approximation of semilinear stochastic partial differential equations (SPDEs) driven by infinite dimensional Wiener processes is investigated. There are a number of results in the literature that show that Euler-type approximation ... More
Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noiseAug 21 2013Aug 24 2014Novel fully discrete schemes are developed to numerically approximate a semilinear stochastic wave equation driven by additive space-time white noise. Spectral Galerkin method is proposed for the spatial discretization, and exponential time integrators ... More
Numerical Methods and Analysis via Random Field Based Malliavin Calculus for Backward Stochastic PDEsJun 28 2013We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve nonlinear and ... More
From rough path estimates to multilevel Monte CarloMay 24 2013Jun 17 2016New classes of stochastic differential equations can now be studied using rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven ... More
Multilevel Monte Carlo methodsApr 19 2013The author's presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and ... More
On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process ObservationsMar 05 2013This paper studies Galerkin approximations applied to the Zakai equation of stochastic filtering. The basic idea of this approach is to project the infinite-dimensional Zakai equation onto some finite-dimensional subspace generated by smooth basis functions; ... More
Lattice Approximation for Stochastic Reaction Diffusion Equations with One-Sided Lipschitz ConditionJan 27 2013Oct 06 2014We consider strong convergence of the finite differences approximation in space for stochastic reaction diffusion equations with multiplicative noise under a one-sided Lipschitz condition only. We derive convergence with an implicit rate depending on ... More
Weak convergence for a spatial approximation of the nonlinear stochastic heat equationDec 21 2012Mar 14 2016We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result ... More
A trigonometric method for the linear stochastic wave equationMar 16 2012Sep 07 2012A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretisation and a stochastic trigonometric scheme for the temporal approximation. ... More
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemesMar 09 2012Sep 19 2012We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. ... More
Pathwise Holder convergence of the implicit Euler scheme for semi-linear SPDEs with multiplicative noiseJan 21 2012In this article we prove pathwise Holder convergence with optimal rates of the implicit Euler scheme for semi-linear parabolic stochastic differential equations with multiplicative noise, set in a UMD Banach space X. We assume the non-linearities to satisfy ... More
Cubature Methods For Stochastic (Partial) Differential Equations In Weighted SpacesJan 19 2012The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded payoffs. We first introduce a recently described flexible functional analytic framework, so called ... More
The Effect of Finite Element Discretisation on the Stationary Distribution of SPDEsOct 20 2011This article studies the effect of discretisation error on the stationary distribution of stochastic partial differential equations (SPDEs). We restrict the analysis to the effect of space discretisation, performed by finite element schemes. The main ... More
Convergence rates for the full Gaussian rough pathsAug 04 2011May 04 2012Under the key assumption of finite {\rho}-variation, {\rho}\in[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 ... More
A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noiseJun 09 2011Apr 01 2012In this paper a new Runge-Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a higher order ... More
Interacting time-fractional and $Δ^ν$ PDEs systems via Brownian-time and Inverse-stable-Lévy-time Brownian sheetsMay 03 2011Feb 26 2012Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system ... More
A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential EquationsNov 11 2010We construct normed spaces of real-valued functions with controlled growth on possibly infinite-dimensional state spaces such that semigroups of positive, bounded operators $(P_t)_{t\ge 0}$ thereon with $\lim_{t\to 0+}P_t f(x)=f(x)$ are in fact strongly ... More
Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz DomainsNov 08 2010We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. ... More
From Brownian-time Brownian sheet to a fourth order and a Kuramoto-Sivashinsky-variant interacting PDEs systemsJul 29 2010Apr 26 2011We introduce $n$-parameter $\Rd$-valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each "time" parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of $n$ linear, fourth order, ... More
A linearized Kuramoto-Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle processMay 20 2010We introduce a new imaginary-Brownian-time-Brownian-angle process, which we also call the linear-Kuramoto-Sivashinsky process (LKSP). Building on our techniques in two recent articles involving the connection of Brownian-time processes to fourth order ... More
A Milstein scheme for SPDEsJan 15 2010Jan 27 2012This article studies an infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity ... More
On some Non Asymptotic Bounds for the Euler SchemeJan 08 2010We obtain non asymptotic bounds for the Monte Carlo algorithm associated to the Euler discretization of some diffusion processes. The key tool is the Gaussian concentration satisfied by the density of the discretization scheme. This Gaussian concentration ... More
Weak order for the discretization of the stochastic heat equation driven by impulsive noiseNov 24 2009Mar 10 2010Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive cylindrical process ... More
The analysis of stochastic stability of stochastic models that describe tumor-immune systemsSep 08 2009In this paper we investigate some stochastic models for tumor-immune systems. To describe these models, we used a Wiener process, as the noise has a stabilization effect. Their dynamics are studied in terms of stochastic stability in the equilibrium points, ... More
Mathematical analysis of stochastic models for tumor-immune systemsJun 15 2009In this paper we investigate some stochastic models for tumor-immune systems. To describe these models we used a Wiener process, as the noise has a stabilization effect. Their dynamics are studied in terms of stochastic stability in the equilibrium points, ... More
Weak order for the discretization of the stochastic heat equationOct 29 2007In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$ dX_t+AX_t dt = Q^{1/2} d W_t, \quad X_0=x ... More
On implicit and explicit discretization schemes for parabolic SPDEs in any dimensionNov 03 2006We study the speed of convergence of the explicit and implicit space-time discretization schemes of the solution $u(t,x)$ to a parabolic partial differential equation in any dimension perturbed by a space-correlated Gaussian noise. The coefficients only ... More
Equation-free, multiscale computation for unsteady random diffusionApr 13 2005We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form ... More
Solution of the Monge-Ampere Equation on Wiener Space for log-concave measuresMar 29 2004In this work we prove that the unique 1-convex solution of the Monge problem contructed from the solution of the Monge-Kantorovitch problem between the Wiener measure and a target measure which has a log-concave density w.r.to the Wiener measure is also ... More