Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

Abstract

A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.

Efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules

“…This model was chosen so that a comparison could be made with the posterior covariance estimates using the adaptively sparse geometric sampling method of Tompkins et al (2011b). Because the method of Tompkins et al (2011b) used direct posterior sampling to estimate the covariances, it is considered to be a "best" estimation of the nonlinear uncertainty associated with this problem and can serve as a benchmark for our tests (see Tompkins et al [2012] for a direct comparison to Bayesian inference). The data domain in this problem is defined by 34 complex-valued radial electric field observations at two frequencies (0.25 and 1.25 Hz) and offsets ranging from 500 to 7000 m (Figure 1).…”

“…Recently, a new method has emerged that solves the same posterior sampling problem, but uses sparse-grid interpolation with orthogonal polynomials as opposed to random sampling (i.e., Tompkins et al, 2011aTompkins et al, , 2011b. This work significantly improves the efficiency by which nonlinear uncertainties can be computed for medium-sized problems (10,000s of parameters), but it is still not known whether this geometric sampling approach can be applied to large parameterizations (see Tompkins et al, 2012). A few studies from electromagnetics and seismics have attempted to overcome difficulties arising from the posterior sampling problem by utilizing the computational efficiency of the deterministic inverse solution while incorporating nonlinearity via probabilistic sampling of certain inputs (Materese, 1995;Alumbaugh, 2000;Alumbaugh and Newman, 2000).…”

“…Examples that represent this type of approach include Metropolis-Hastings, simulated annealing, neighborhood algorithm, and genetic algorithm. While random sampling methods avoid burdensome inversions and can account for nonlinearity, they come at the high cost of inefficiency (e.g., Haario et al, 2001;Tompkins et al, 2011bTompkins et al, , 2012 and thus are limited to modest-sized problems (10 s of unknowns). Recently, a new method has emerged that solves the same posterior sampling problem, but uses sparse-grid interpolation with orthogonal polynomials as opposed to random sampling (i.e., Tompkins et al, 2011aTompkins et al, , 2011b.…”

Uncertainty analysis and probabilistic segmentation of electrical resistivity images: the 2D inverse problem

“…A similar methodology was used in the Geometric sampling approach (Tompkins et al . (,b; ) and Tompkins and Fernández‐Martínez () to explore the uncertainty space in a CSEM inverse problem. Geometric sampling uses Smolyak grids to perform a deterministic sampling in a low dimensional polytope that is found by solving the linear constrained problems stated in .…”

“…This method combines different techniques for model reduction and deterministic sampling using Smolyak grids, providing similar posterior estimates as Monte Carlo methods (Tompkins et al . ). Ciucivara and Willen () introduced a methodology for uncertainty estimation of subsurface resistivity solutions that uses principal component analysis to reduce the dimension and a perturbation method in the reduced space to sample the equivalent models.…”