Fréchet Sensitivity Analysis for Partial Differential Equations with Distributed Parameters

This paper reviews Fréchet sensitivity analysis for partial differential equations with variations in distributed parameters. The Fréchet derivative provides a linear map between parametric variations and the linearized response of the solution. We propose a methodology based on representations of the Fréchet derivative operator to find those variations that lead to the largest changes to the solution (the most significant variations). This includes an algorithm for computing these variations that only requires the action of the Fréchet operator on a given direction (the Gateaux derivative) and its adjoint. This algorithm is applicable since it does not require an approximation of the entire Fréchet operator, but only typical sensitivity analysis software for partial differential equations. The proposed methodology can be utilized to find worst case distributed disturbances and is thus applicable to uncertainty quantification and the optimal placement of sensors and actuators.