Sensitivity and uncertainty quantification of random distributed parameter systems

Abstract

As simulation continues to replace experimentation in the design cycle, the need to quantify uncertainty in model parameters and its effect on simulation results becomes critical. While intelligent sampling methods, such as sparse grid collocation, have expanded the class of random systems that can be simulated with uncertainty quantification, the statistical characterization of the model parameters are rarely known.  In previous works, we have proposed an
optimization-based framework for estimating distributed parameters as well
as a number of methods for identifying  the most significant parametric variations.  In this work, we consider the combination of these methods. We propose the use of
Fr\'echet sensitivity analysis to determine the most significant parametric variations (MSPVs). These MSPVs inform the generation of low order
Karhunen-Lo\`eve expansions of the unknown parameter.  These expansions can be used to build smooth, finite noise approximations of the parameter identification problem based on the theory of infinite dimensional constrained optimization in the space of functions with bounded mixed derivatives.
Alternatively, we can interrogate the identified parametric distributions to determine the tendency of the MSPVs.  We illustrate our methods with a numerical example of identifying the distributed stochastic parameter in an elliptic boundary value problem.