AN OVERVIEW OF INVERSE MATERIAL IDENTIFICATION WITHIN THE FRAMEWORKS OF DETERMINISTIC AND STOCHASTIC PARAMETER ESTIMATION

Abstract

This work investigates the problem of parameter estimation within the frameworks of deterministic and stochastic parameter estimation methods. For the deterministic methods, we look at constrained and unconstrained optimization approaches. For the constrained optimization approaches we study three different formulations: L 2 , error in constitutive equation method (ECE), and the modified error in constitutive equation (MECE) method. We investigate these formulations in the context of both Tikhonov and total variation (TV) regularization. The constrained optimization approaches are compared with an unconstrained nonlinear least-squares (NLLS) approach. In the least-squares framework we investigate three different formulations: standard, MECE, and ECE. With the stochastic methods, we first investigate Bayesian calibration, where we use Monte Carlo Markov chain (MCMC) methods to calculate the posterior parameter estimates. For the Bayesian methods, we investigate the use of a standard likelihood function, a likelihood function that incorporates MECE, and a likelihood function that incorporates ECE. Furthermore, we investigate the maximum a posteriori (MAP) approach. In the MAP approach, parameters’ full posterior distribution are not generated via sampling; however, parameter point estimates are computed by searching for the values that maximize the parameters’ posterior distribution. Finally, to achieve dimension reduction in both the MCMC and NLLS approaches, we approximate the parameter field with radial basis functions (RBF). This transforms the parameter estimation problem into one of determining the governing parameters for the RBF.