Using Fréchet sensitivity analysis to interrogate distributed parameters in random systems

As simulation continues to replace experimentation in the design cycle, the need to quantify uncertainty in model outputs due to uncertainties in the model parameters becomes critical. While intelligent sampling methods such as sparse grid collocation has expanded the class of random systems that can be simulated to aid uncertainty quantification, the statistical characterization of the model parameters are rarely known. In previous works, we have proposed a number of methods for identification of the most significant parametric variations as well as an optimization-based framework for estimation of distributed parameters. This work combines these two approaches, we identify a stochastic parameter in a random system then use the Fréchet derivative to determine the most significant (deterministic, spatial) parametric variations. In other words, showing that the Fréchet derivative is Hilbert-Schmidt allows us to compute those parametric variations that have the most local impact on the solution. These variations are used to interrogate the stochastic parameter. We illustrate our methods with numerical example identifying the distributed stochastic parameter in an elliptic boundary value problem.