H14F-05:
Quantifying Uncertainty in Physics-Based Models with Measure Theory
H14F-05:
Quantifying Uncertainty in Physics-Based Models with Measure Theory
Monday, 15 December 2014: 5:00 PM
Abstract:
The ultimate goal in scientific inference is to predict some unobserved behavior of a system often described by a specific set of quantities of interest (QoI) computed from the solution to a mathematical model. For example, given a contaminant transport model in a regional aquifer with specified porosity, initial concentrations, etc., we may analyze remediation strategies in order to achieve threshold tolerances in certain wells. Solution to this prediction problem is complicated by several sources of uncertainty. A primary source of uncertainty is in the specification of the input data and parameters that characterize the physical properties of a given state of the system. It is often impossible to experimentally observe the input data and parameters that characterize the physical properties of the modeled system, and any observable QoI are generally of the state of the system itself and may differ substantially from the QoI to be predicted. Thus, we must first solve an inverse problem using observable QoI to quantify uncertainties in the inputs to the model.The inverse problem is complicated by several issues. The solution of the model induces a "QoI map" from the space of model inputs to the observable QoI computed from the solution of the model. This QoI map is generally a "many-to-few" map that reduces the dimension implying that the deterministic inverse problem has set-valued solutions. Additionally, available data on the QoI are subject to natural variation and experimental/observational error modeled as probability measures implying that solutions of the inverse problem and forward prediction problem are given in terms of probability measures.
We describe a novel measure-theoretic framework for the formulation and solution of the stochastic inverse problem for a deterministic physics-based model requiring minimal assumptions. A computational algorithm and error analysis are described accounting for both deterministic and stochastic sources of error.