Computational measure theoretic approach to inverse sensitivity analysis: Methods and analysis

We consider the inverse problem of quantifying the uncertainty of inputs to a finite dimensional map, e.g. determined implicitly by solution of a nonlinear system, given specified uncertainty in a linear functional of the output of the map. The uncertainty in the output functional might be suggested by experimental error or imposed as part of a sensitivity analysis. We describe this problem probabilistically, so that the uncertainty in the quantity of interest is represented by a random variable with a known distribution, and we assume that the map from the input space to the quantity of interest is smooth. We derive an efficient method for determining the unique solution to the problem of inverting through a many-to-one map by computing set-valued inverses of the input space which combines a forward sensitivity analysis with the Implicit Function Theorem. We then derive an efficient computational measure theoretic approach to further invert into the entire input space resulting in an approximate probability measure on the input space.We provide detailed error analysis for inverse problems involving nonlinear ordinary differential equations and semilinear elliptic partial differential equations.