Parameter estimation for the heat equation on perforated domains

Abstract

In this effort we investigate the behavior of a model derived from homogenization theory as the model solution in parameter estimation procedures for simulated data for heat flow in a porous medium. We consider data simulated from a model on a perforated domain with isotropic flow and data simulated from a model on a homogeneous domain with anisotropic flow. We consider both ordinary and generalized least squares parameter estimation procedures.

Thermal based methods for damage detection and characterization in porous materials

“…Based on the encouraging results in [5] which verified that a homogenization approach provides good approximations to solutions u rand for problems on porous domains, we will use a model motivated by homogenization theory as the model solution in our inverse problems. We will also simulate data from this model as in [5] to understand the effect of the error (as compared to data simulated with u rand D ) associated with the approximation derived from homogenization theory on the inverse problem. Methods used in [2,4,5,15,[17][18][19][20][21][22]25] can be used to establish the good approximation of u rand D (t, x; q) by the homogenization solution u D (t, x; q) where u D (t, x; q) is given by…”

“…We will also simulate data from this model as in [5] to understand the effect of the error (as compared to data simulated with u rand D ) associated with the approximation derived from homogenization theory on the inverse problem. Methods used in [2,4,5,15,[17][18][19][20][21][22]25] can be used to establish the good approximation of u rand D (t, x; q) by the homogenization solution u D (t, x; q) where u D (t, x; q) is given by…”

“…In this effort, we include this porosity in simulated thermal data and subsequent analysis of our models. We have conducted a succession of proof of concept investigations towards the goal of verifying the use of homogenization to model porosity in the detection and characterization of damage [3][4][5]. In [3], we used ideas developed in [15] to develop a method of simulating the flash heat experiment in porous materials.…”

“…We considered different pore size distributions in [3] and observed little difference in distribution of pore size at the levels of porosity that we consider (low because manufacturing techniques at this point are reasonably good). Our first attempt at considering the effect of approximations derived from homogenization theory in inverse problems was in [5] where we considered the effect of the model discrepancy the estimation of parameters and uncertainty in parameter estimates for parameters in the partial differential equation that governs the flow of heat in the flash heat experiment using procedures detailed in [7,8,10,24]. In this article, we extend these results to consider verification of the use of homogenization as a model solution in inverse problems where we estimate both the thermal diffusivity as well as the geometry of the boundary itself.…”

“…Fiber optics were used in [30] with flash heating to detect delaminations in composite materials. Thus we build on our previous work [3][4][5] which relies heavily on results in [7,8,10,15,17,19,21,22,24,29]. In section 2 we discuss two models that we use to simulate data; we then consider damage detection in section 3 in the context of statistical model comparison problems, and finally we use ordinary least squares parameter estimation to consider damage characterization in section 4.…”