arXiv subjects

Georgios E. Zouraris

Publications and source records attributed to Georgios E. Zouraris.

11 recordsLinked to original sources

arXiv2022

Improved Efficiency of Multilevel Monte Carlo for Stochastic PDE through Strong Pairwise Coupling

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when (MLMC) is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the (MLMC) method with strong pairwise coupling that was developed and studied numerically on filtering problems in [{\it Chernov et al., Numer. Math., 147 (2021), 71-125}], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this feature.

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arXiv2020

A Relaxation/Finite Difference discretization of a 2D Semilinear Heat Equation over a rectangular domain

We consider an initial and Dirichlet boundary value problem for a semilinear, two dimensional heat equation over a rectangular domain. The problem is discretized in time by a version of the Relaxation Scheme proposed by C. Besse (C. R. Acad. Sci. Paris Sér. I, vol. 326 (1998)) for the nonlinear Schrödinger equation and in space by a standard second order finite difference method. The proposed method is unconditionally well-posed and its convergence is established by proving an optimal second order error estimate allowing a mild mesh condition to hold.

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arXiv2020

Error estimation of the Relaxation Finite Difference Scheme for the nonlinear Schrödinger Equation

We consider an initial- and boundary- value problem for the nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference method and in time by the Relaxation Scheme proposed by C. Besse [C. R. Acad. Sci. Paris Sér. I {\bf 326} (1998), 1427-1432]. We provide optimal order error estimates, in the discrete $L_t^{\infty}(H_x^1)$ norm, for the approximation error at the time nodes and at the intermediate time nodes. In the context of the nonlinear Schr{ö}dinger equation, it is the first time that the derivation of an error estimate, for a fully discrete method based on the Relaxation Scheme, is completely addressed.

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arXiv2018

Error Estimation of the Besse Relaxation Scheme for a Semilinear Heat Equation

The solution to the initial and Dirichlet boundary value problem for a semilinear, one dimensional heat equation is approximated by a numerical method that combines the Besse relaxation scheme in time (C. R. Acad. Sci. Paris S{é}r. I, vol. 326 (1998)) with a central finite difference method in space. A new, composite stability argument is developed, leading to an optimal, second-order error estimate in the discrete $L_t^{\infty}(H_x^1)-$norm. It is the first time in the literature where an error estimate for fully discrete approximations based on the Besse relaxation scheme is provided.

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arXiv2017

Crank-Nicolson finite element approximations for a linear stochastic heat equation with additive space-time white noise

We formulate an initial- and Dirichlet boundary- value problem for a linear stochastic heat equation, in one space dimension, forced by an additive space-time white noise. First, we approximate the mild solution to the problem by the solution of the regularized second-order linear stochastic parabolic problem with random forcing proposed by Allen, Novosel and Zhang (Stochastics Stochastics Rep., 64, 1998). Then, we construct numerical approximations of the solution to the regularized problem by combining the Crank-Nicolson method in time with a standard Galerkin finite element method in space. We derive strong a priori estimates of the modeling error made in approximating the mild solution to the problem by the solution to the regularized problem, and of the numerical approximation error of the Crank-Nicolson finite element method.

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arXiv2017

An IMEX Finite Element Method for a linearized Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise

We consider a model initial- and Dirichlet boundary- value problem for a linearized Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we introduce a canvas problem the solution to which is a regular approximation of the mild solution to the problem and depends on a finite number of random variables. Then, fully-discrete approximations of the solution to the canvas problem are constructed using, for discretization in space, a Galerkin finite element method based on $H^2$ piecewise polynomials, and, for time-stepping, an implicit/explicit method. Finally, we derive a strong a priori estimate of the error approximating the mild solution to the problem by the canvas problem solution, and of the numerical approximation error of the solution to the canvas problem.

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arXiv2017

A linear implicit finite difference discretization of the Schrodinger-Hirota Equation

A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved, assuming that $τ$, $h$ and $τ^4/h$ are sufficiently small, where $τ$ is the time-step and $h$ is the space mesh-size. The efficiency of the proposed method is verified by results from numerical experiments.

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arXiv2016

Crank-Nicolson finite element approximations for a linear stochastic fourth order equation with additive space-time white noise

We consider a model initial- and Dirichlet boundary- value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. First, we approximate its solution by the solution of an auxiliary fourth-order stochastic parabolic problem with additive, finite dimensional, spectral-type stochastic load. Then, fully-discrete approximations of the solution to the approximate problem are constructed by using, for the discretization in space, a standard Galerkin finite element method based on $H^2$-piecewise polynomials, and, for time-stepping, the Crank-Nicolson method. Analyzing the convergence of the proposed discretization approach, we derive strong error estimates which show that the order of strong convergence of the Crank-Nicolson finite element method is equal to that reported in [Kosioris and Zouraris MMAN 44 (2010)] for the Backward Euler finite element method.

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arXiv2012

Finite Element Approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise

We consider an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate regularized stochastic parabolic problem discretizing the noise using linear splines. Then fully-discrete approximations to the solution of the regularized problem are constructed using, for the discretization in space, a Galerkin finite element method based on H2-piecewise polynomials, and, for time-stepping, the Backward Euler method. Finally, we derive strong a priori estimates for the modeling error and for the numerical approximation error to the solution of the regularized problem.

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arXiv2012

Monte Carlo Euler approximations of HJM term structure financial models

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

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arXiv2009

Fully-Discrete Finite Element Approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise: II. 2D and 3D Case

We consider an initial- and Dirichlet boundary- value problem for a fourth-order linear stochastic parabolic equation, in two or three space dimensions, forced by an additive space-time white noise. Discretizing the space-time white noise a modeling error is introduced and a regularized fourth-order linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the regularized problem are constructed by using, for discretization in space, a standard Galerkin finite element method based on C1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modeling error and for the approximation error to the solution of the regularized problem.

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