Inverse Problems and the Quantification of Uncertainty

Abstract: Inverse problems are often formulated as functional equations or optimization problems. Thus the estimation of errors in the solutions amounts to the study of error propagation for these two classes of mathematical problems, at least from the deterministic perspective. A solution algorithm produces a solution estimate in which data error translates into both intrinsic (to the problem) and algorithmic (or numerical) estimation error. All real world data exhibits both data uncertainty (measurement error) and model incompatibility. Therefore any solution estimate which matches the data with some tolerance must be accepted as a "solution". The sets of solutions defined in this way are often very large, and the problem becomes one of determining what inference can be drawn about common features of the models which these solutions represent. Several factors have a large influence on the quality of the answer to this question, for example the choice of data misfit measure, the amount of misfit tolerated, and the type of additional data supplied to specify an acceptable model. These influences are fairly well understood for linear inverse problems, in which model and predicted data are linearly related. I will review the linear theory and present some examples of its application. Deterministic and statistical approaches to error estimation for linear inverse problems are parallel in several respects, and I will note some of the similarities and differences. Nonlinear inverse problems can exhibit a much wilder range of error propagation effects, many of which are at present only poorly understood. Some of the additional subtlety supplied by nonlinearity is nicely illustrated by examples from seismology.

Abstract: Inverse problems can be viewed as special cases of statistical estimation problems. From that perspective, one often can study inverse problems using standard statistical measures of uncertainty, such as bias, variance, mean squared error and other measures of risk, confidence sets, and so on. It is useful to distinguish between the intrinsic uncertainty of an inverse problem and the uncertainty of applying any particular technique for "solving" the inverse problem. The intrinsic uncertainty depends crucially on the prior constraints on the unknown (including prior probability distributions in the case of Bayesian analyses), on the forward operator, on the statistics of the observational errors, and on the nature of the properties of the unknown one wishes to estimate. I will try to convey some geometrical intuition for uncertainty, and the relation between the intrinsic uncertainty of linear inverse problems and the uncertainty of some common techniques applied to them.

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