Application of Two Bayesian Filters to Estimate Unknown Heat Fluxes in a Natural Convection Problem

Abstract

Sequential Monte Carlo (SMC) or particle filter methods, which have been originally introduced in the beginning of the 1950s, became very popular in the last few years in the statistical and engineering communities. Such methods have been widely used to deal with sequential Bayesian inference problems in the fields like economics, signal processing, and robotics, among others. SMC methods are an approximation of sequences of probability distributions of interest, using a large set of random samples, named particles. These particles are propagated along time with a simple Sampling Importance distribution. Two advantages of this method are: they do not require the restrictive hypotheses of the Kalman filter, and they can be applied to nonlinear models with non-Gaussian errors. This paper uses two SMC filters, namely the SIR (sampling importance resampling filter) and the ASIR (auxiliary sampling importance resampling filter) to estimate a heat flux on the wall of a square cavity encasing a liquid undergoing natural convection. Measurements, which contain errors, taken at the boundaries of the cavity were used in the estimation process. The mathematical model as well as the initial condition are supposed to have some errors, which were taken into account in the probabilistic evolution model used for the filter. Also, the results using different grid sizes and patterns for the direct and inverse problems were used to avoid the so-called inverse crime. In these results, additional errors were considered due to the different location of the grid points used. The final results were remarkably good when using the ASIR filter.