Geometric sampling: An approach to uncertainty in high dimensional spaces

Abstract

Uncertainty is always present in inverse problems. The main reasons for that are noise in data and measurement error, solution non-uniqueness, data coverage and bandwidth limitations, physical assumptions and numerical approximations. In the context of nonlinear inversion, the uncertainty problem is that of quantifying the variability in the model space supported by prior information and the observed data. In this paper we outline a general nonlinear inverse uncertainty estimation method that allows for the comprehensive search of model posterior space while maintaining computational efficiencies similar to deterministic inversions. Integral to this method is the combination of model reduction techniques, a constrained mapping approach and a sparse sampling scheme. This approach allows for uncertainty quantification in inverse problems in high dimensional spaces and very costly forward evaluations. We show some results in non linear geophysical inversion (electromagnetic data). © 2010 Springer-Verlag Berlin Heidelberg.