Heat equation parameter estimation based on the POD-Galerkin approach

Abstract

The proper orthogonal decomposition (POD), a mathematical procedure initially developed to determine low-dimensional approximated descriptions of high-dimensional functions, has already proved its efficiency for model reduction of linear and non-linear systems. In this paper, the POD and the Galerkin projection methods are combined to estimate the thermal diffusivity parameter of a wall, the behavior of which is assumed to be modeled by the famous heat partial differential equation (PDE). According to this approach, empirical orthogonal functions are first extracted from the POD, then used by the Galerkin method to transform the initial PDE into a set of ordinary differential equations (ODEs). These ODEs explicitly contain the parameter to be identified which is estimated in this contribution by using the Levenberg-Marquardt algorithm. Simulation results are provided to illustrate the effectiveness of this multi-step approach.