Abstract
Inferential solutions to inverse problems provide substantial advantages over deterministic methods, such as: quantitative estimates with posterior (data-dependent) error estimates, predictive densities, model comparison and direct support for optimal decisions. The ability to include arbitrary forward maps, and hence use high-level representations of the unknowns, allows structure-preserving model reduction and also allows 'classification' to be performed within the 'imaging' step. Since inferential methods make (provably) optimal use of data, the ability to reduce data to a minimal set gives cost savings in applications where collecting data is expensive. The price of these advantages is presently the relatively high computational cost of sampling algorithms for computing estimates. Hence the most significant advances are in computational methods for sample-based inference in inverse problems. In this article we review the inferential formulation of inverse problems, some reasons why it is necessary to take on the extra machinery of inferential solutions, the 'basement level' methods for computing inferential solutions, and summarize some recent advances in computational methods for inferential solutions to inverse problems.