Data‐Consistent Inversion

Data-consistent inversion (DCI) is a measure-theoretic inversion technique which seeks to solve specific class of stochastic inverse problems. Namely, given an observed (or target) probability measure/density on quantities of interest and a computational model, DCI seeks a probability measure/density on model inputs such that the corresponding push-forward measure/density matches the observed/target. In other words, it seeks a pullback probability measure.

Of course, as with many inverse problems, the solution is not guaranteed to exist or to be unique. Both existence and uniqueness can be obtained if one regularizes the problem by incorporating prior knowledge or an initial guess about the model inputs. Existence of a solution can be guaranteed through what we call the predictability assumption which guarantees that the push-forward of the initial guess through the computational model can predict all of the data. This is mathematically described as requiring that the observed measure is absolutely continuous with respect to the push-forward of the initial.
Incorporating this initial information also regularizes the problem in the sense that the solution is unique given a choice of initial. The DCI solution to this inverse problem is called the updated measure/density:

where $\pi^{\text{init}}{\Lambda}$ is the initial density, $\pi^{\text{obs}}{\cal D}$ is the observed density, $\pi^{\text{pred}}{\cal D}$ is the predicted density (the pushforward of the initial), and $\pi^{\text{up}}{\Lambda}$ is the updated density.

MrHyDE provides the ability to post-process a forward UQ study using samples from the initial density and can either reweight these samples or perform rejection sampling to provide information about the updated density. However, DCI is fairly straightforward to implement since it only requires some form of density estimation (on the QoI, not on the input parameters) and simple rejection sampling. Most of the published work on DCI used data from high-fidelity computational models, such as those provided by MrHyDE, but the actual DCI scripts were in Matlab or Python.