Parameter Estimation via Bayesian Inversion: Theory, Methods, and Applications

摘要

Uncertainty quantification is becoming an increasingly important area of investigation in the field of computational simulations. An understanding in the confidence of a simulation result requires information concerning the uncertainties associated with individual sub-models. The development of mathematical models for physical systems resides in the interpretation of experimental results. Inherent to physically interesting mathematical models is the occurrence of unobservable model parameters. The resolution of information concerning model parameters is typically performed through the use of least-squares regression analysis; however, least-squares analysis does not provide adequate information concerning the confidence which may be placed in the parameter estimates. Bayesian inversion provides quantifiable information concerning the confidence which may be placed in the parameter estimates allowing for overall simulation uncertainty quantification. Here, the application of Bayesian statistics to the general discrete inverse problem is presented. Following the presentation of the Bayesian formulation of the general discrete inverse problem, the procedure is applied to two scientifically interesting inverse problems: the reversible-reaction diffusion inverse problem and the Arrhenius inverse problem. The Arrhenius inverse problem is solved using a novel approach developed here. The novel approach is compared to other probabilistic and deterministic approaches to assess the validity of the method.