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 $Q(\lambda)$ is the map from input parameters to quantities of interest (QoI).
We see that the updated density is given by the product of the initial density and the ratio of the observed to the predicted (the pushforward of the initial). We usually assume the initial and observed densities are given, but practically, we only need the ability to sample from the initial and the observed can be approximated from data. The predicted density, in the denominator of the ratio, must be approximated using some form of density estimation. The algorithm in MrHyDE follows these steps:

1. Generate a set of samples from the initial density.
2. Evaluate the model and the QoI for each of these samples.
3. Construct an approximation of the predicted density using Gaussian kernel density estimation.
4. Compute the ratio of observed and predicted densities at each sample point.
5. Output the value of the ratio for each sample.
6. (Optional) Perform rejection sampling using the ratio and output a set of samples from the updated density.

Note that in the above procedure, all of the computations take place in the QoI space, which is often much lower-dimensional than the input space.

While MrHyDE provides this 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, 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.

To perform DCI entirely within MrHyDE, the following settings should be specified in the input file:

ADD INPUT FILE