Fast Bayesian inversion for high dimensional inverse problems

Abstract

We investigate the use of learning approaches to handle Bayesian inverse problems in a computationally efficient way when the signals to be inverted present a moderately high number of dimensions and are in large number. We propose a tractable inverse regression approach which has the advantage to produce full probability distributions as approximations of the target posterior distributions. In addition to provide confidence indices on the predictions, these distributions allow a better exploration of inverse problems when multiple equivalent solutions exist. We then show how these distributions can be used for further refined predictions using importance sampling, while also providing a way to carry out uncertainty level estimation if necessary. The relevance of the proposed approach is illustrated both on simulated and real data in the context of a physical model inversion in planetary remote sensing. The approach shows interesting capabilities both in terms of computational efficiency and multimodal inference.

Appendix

1.1 Computation times

The simulations ran on a laptop with 4 cores (at 2.5 Ghz). Table 6 shows computation times. For each experiment, the time is divided in two parts, the time for the learning step (GLLiM inference) and the time for the prediction step, which consists either of mixture merging, mode-finding and importance sampling (Examples 1 to 6) or of noise level estimation via the EM algorithm. Most experiments run in few minutes. The complexity of the forward model and the way it is implemented can take an important part in the resulting running time. This appears in the comparison of examples 2 and 3 which mainly differ in the choice of F. The Hapke model (Example 3) benefits from a more efficient implementation which explains running time twice smaller for similar settings. For equivalent forward model implementations, the time depends mainly on the size and dimensionality of the learning set and on the number of inversions to be performed. Learning sets have equivalent complexity in our experiments, except for the high dimensional example (Example 4, $D=336$). Higher computation times are observed in case of massive inversions. In particular, Example 6 with 156,100 inversions takes few hours. We believe it is the first time that such spatial and spectral parametric maps are obtained due to the intractability of other methods in this setting.

Fig. 10

Inversion of Nontronite laboratory observations. $\bar{{\mathbf {x}}}_{IMIS-G}$ is in red, $\bar{{\mathbf {x}}}_{IMIS-centroid,1}$ and $\bar{{\mathbf {x}}}_{IMIS-centroid,2}$ are in blue. Relative reconstruction errors are shown with a logarithmic scale in the top right plot

1.2 Centroids predictions for the Nontronite dataset

Figure 10 provides a complementary analysis to the results in Fig. 5, for Example 5. The centroids predictions show that, for parameters ${\bar{\theta }}, b,c$, the marginal posteriors are not concentrated around their means, yielding a large range of high probability predictions including the centroids, as shown by the reconstruction errors.

1.3 Massive inversion of spatial and spectral Mars data

Figure 11 shows the maps for parameters b and c after inversion of the real Mars data described in Sect. 7.2.5.

Fig. 11

Mars South pole dataset. Parameters b (top) and c (bottom) averaged over spectral dimension, predicted using $\bar{{\mathbf {x}}}_{IMIS-G}$ (left) or $\hat{{\mathbf {x}}}_{best}$ (right)