Parallel solution of the discretised and linearised G-heat equation

Abstract

The present study deals with the numerical solution of the G-heat equation. Since the G-heat equation is defined in an unbounded domain, we firstly state that the solution of the G-heat equation defined in a bounded domain converges to the solution of the G-heat equation when the measure of the domain tends to infinity. Moreover, after time discretisation by an implicit time marching scheme, we define a method of linearisation of each stationary problem, which leads to the solution of a large scale algebraic system. A unified approach analysis of the convergence of the sequential and parallel relaxation methods is given. Finally, we present the results of numerical experiments. . He teaches numerical analysis, optimisation and numerical solution of boundary value problems. His fields of interest are in numerical analysis, large scale nonlinear systems of evolution equations, optimal control, parallel computing and more particularly, domain decomposition methods for the solution of nonlinear boundary values problems; he is interested to apply the obtained theoretical results to applications concerning finance, image processing, mechanics, etc. He is also a scientific expert, advisor and referee for several international scientific committees and journals.

Parallel asynchronous algorithms: A survey

“…To illustrate some applications, let us cite for example non-linear diffusion problems involved in plasma physics or in the modeling of solar ovens [25] and [69], the Hamilton -Jacobi -Bellman problem involved in image processing [35], the obstacle problem modelling financial mathematics problems or mechanical problems (see [107] and [120] to [124]), the Navier -Stokes equation modeling hydrodynamic problems and also, on various architectures, various applications in biology such as protein separation by electrophoresis modeled by problems coupled from this equation with one or more convection -diffusion equations to calculate the various protein concentrations as well as a potential equation describing the electrical behavior of the process (see [125] to [128] or fluid interaction -structure problems where the Navier -Stokes equation is coupled to the Navier equation describing the behavior of the structure [28] and [46]. Other applications related to P.D.E.…”

Synthetic presentation of iterative asynchronous parallel algorithms

“…Many studies have also been achieved concerning numerical solution of linear and non linear partial differential equations; let us cite for example non-linear diffusion problems involved in plasma physics or in the modelling of solar ovens [6] and [42], the Hamilton -Jacobi -Bellman problem involved in image processing [47] , the obstacle problem modelling financial mathematics problems or mechanical problems (see [78] and [86] to [90]), the Navier -Stokes equation modeling hydrodynamic problems and also, on various architectures, various applications in biology such as protein separation by electrophoresis modelled by problems coupled from this equation with one or more convection -diffusion equations to calculate the various protein concentrations as well as a potential equation describing the electrical behavior of the process (see [91] to [94] or fluid interaction -structure problems where the Navier -Stokes equation is coupled to the Navier equation describing the behavior of the structure [19]. Other applications related to P.D.E.…”