Scalable Solutions for Nonlinear Inverse Uncertainty Using Model Reduction, Constraint Mapping, and Sparse Sampling

We present 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 a parameter reduction technique, like Principal Component Analysis, a parameter bounds mapping routine, a sparse sampling scheme, and a forward solver. Parameter reduction, based on linearized model covariances, is used to reduce the model space by orders of magnitude. Parameter constraints are then mapped to this reduced space, using a linear programming scheme, defining a bounded posterior polytope. Sparse deterministic grids are employed to sample this feasible model region, while forward evaluations determine which model samples are equi-probable. The resulting ensemble represents the equivalent model space, consistent with Principal Components, that is used to infer uncertainty measures. The number of forward evaluations is determined adaptively and minimized by finding the sparsest sampling required for convergence of uncertainty measures. We demonstrate, with a surface electromagnetic example, that this method has the potential to reduce the nonlinear inverse uncertainty problem to a deterministic sampling problem in only a few dimensions, requiring limited forward solves, and resulting in an optimally sparse representation of the posterior model space.