(Joint work with Alberto Malinverno)
Solving an inverse problem means determining the parameters of a model
given a set of measurements. In solving many practical inverse problems,
accounting for the uncertainty of the solution is very important to aid in
decision-making. A standard approach to do this begins by choosing a model
parametrization and then using a Bayesian approach to make inferences on
the model parameters from measurement data. However, this quantified
uncertainty is a function of the model parametrization and for many inverse
problems, there are many model parametrizations that account for the data
equally well. A well known approach to accounting for model uncertainty is
Bayesian Model Averaging where many model parametrizations are
considered. Significant computational costs are associated with this
method because one must compute the posterior distribution for each model
parametrization.
We consider a family of model parametrizations given by decimated wavelet
bases. By decimated wavelet basis we mean a subset of the model’s
coordinates in a wavelet basis. For linear inverse problems, we
demonstrate new fast algorithms for updating the prior and posterior
covariance matrices when wavelet model parameters are added or deleted from
the decimated basis. We also introduce algorithms for updating the
determinant and Cholesky decomposition of the model’s covariance
matrices. These algorithms deliver order of magnitude savings over
computing these covariance matrices from scratch and make Bayesian Model
Averaging a realistic approach for accounting for uncertainty in inverse
problem solutions.
In order to clarify the role of our model updates, we show that our wavelet
model update algorithms update the model’s posterior distribution after
modifying the model’s local spatial resolution, whereas Kalman filters
provide a means of updating a model when assimilating new measurement data.
These results show a major advantage to be gained by parametrizing models
with wavelets and represent a significant step forward in addressing the
challenging computational problem of dealing with large models that account
for uncertainty.