Sensitivity analysis of partial differential equations: A case for functional sensitivity

Abstract

Sensitivity analysis allows for analyzing the effects of parameter uncertainty. For functional parameters, the sensitivity of the system is described by the functional derivatives of the output variables with respect to the parameters. Approximation of each of the functional parameters by a finite number of scalars (via the finite element representation) allows one to use elementary sensitivity analysis. The functional sensitivities are easily approximated from elementary sensitivities and, being objective quantities, they allow one to evaluate the numerical quality of sensitivities. The grid density necessary for computing functional sensitivities may differ significantly from the grid required for the numerical solution of the governing equation. Many problems in applied mathematics and engineering do not have closed-form solutions and therefore require numerical treatment. The governing equation and the functional parameters, when discretized, lead to a system of ordinary differential equations. The effects of parameter uncertainty can be studied via sensitivity analysis. It is usually performed on the discretized system, yielding elementary sensitivities of the numerical model instead of functional sensitivities of the continuous system under consideration. Our goal is to demonstrate the ease with which approximations of functional sensitivities may be recovered from elementary sensitivities and the advantages the functional sensitivities offer.