Guest Editors' introduction to the special section on statistical and computational issues in inverse problems

In the words of D D Jackson, the data of real-world inverse problems tend to be inaccurate, insufficient and inconsistent (1972 Geophys. J. R. Astron. Soc. 28 97–110). In view of these features, the characterization of solution uncertainty is an essential aspect of the study of inverse problems. The development of computational technology, in particular of multiscale and adaptive methods and robust optimization algorithms, has combined with advances in statistical methods in recent years to create unprecedented opportunities to understand and explore the role of uncertainty in inversion. Following this introductory article, the special section contains 16 papers describing recent statistical and computational advances in a variety of inverse problem settings. Statistical and coqmputational issues in inverse problems This special issue of Inverse Problems presents statistical and computational ideas and tools for the understanding and quantification of uncertainty in the solution of applied inverse problems. At an appropriate level of abstraction, many scientific questions are inverse problems. Solving those inverse problems is an intrinsically interdisciplinary task: in order to make estimates or draw inferences, experiments must be designed; instruments designed, fabricated and deployed; mathematical descriptions of the measurement process formulated; hypotheses 0266-5611/08/034001+05$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1 Inverse Problems 24 (2008) 034001 L Tenorio et al posited; data collected; theory and algorithms developed; and data processed. Often these steps are performed by disjoint groups of researchers who are not aware of the complexity, limitations and opportunities in the steps performed by others. Assumptions made in one step may contradict or fail to account for assumptions and constraints in other steps, limiting the accuracy and utility of the overall process. How can we assess the reliability of images and inferences, taking into account realistic stochastic and systematic error, approximations to the forward problem, approximations in the representation of the unknown, algorithmic and numerical instabilities and uncertainties, finite computational resources, and the fact that only finitely many observations are available? How can prior knowledge be incorporated into inferences to reduce the true uncertainty— without understating the uncertainty by imposing artificial constraints? How can appropriate physics be incorporated in a computational feasible way, without imposing yet more artificial constraints? How should we assess the relative merits of different methods and algorithms? What measures of the performance are scientifically interesting? Is there an estimator or approach that performs ‘best’, in some scientifically interesting sense, for a given forward mapping, and reasonable assumptions or constraints on the unknown function and the model and data errors? If so, how can we compute it, given the frequently vast ranges of scales and very large extents of faithful numerical representations? How well does the estimator perform if the assumptions are wrong? Constraints on the unknown function and assumptions about the data errors are sometimes matters of science and sometimes matters of mathematical or computational convenience or expressions of a subjective preference. The influence of untested assumptions on the inferences and the uncertainty of those inferences should be studied and quantified. The over-arching problem of understanding the accuracy and limitations of mathematical and computational models of complex systems has acquired the name ‘uncertainty quantification’ (UQ) in recent years. UQ is becoming recognized as an important facet of inverse problems, with applications ranging from assessing the reliability of the power grid to validating the strong motion potential of earthquake-prone basins. Some of the statistical tools rely on asymptotics, distributional assumptions, or assumptions about the function to be recovered that are hard (if not impossible) to test, and some may overlook hard questions, such as the bias of estimators. Others are at a level of abstraction rather removed from application. Such is the state of the art. Nonetheless, the core ideas and framework are important and hold promise for real problems. On the computational side, multiscale and/or data-adaptive methods promise to enable simulation and inversion algorithms of unprecedented physical fidelity and efficiency. There is still much left to do to bridge the gap between what practitioners need to compute, to properly quantify uncertainty in the solution of inverse problems, and what can be computed today. This special section illustrates some of the most promising current approaches to closing this gap. Overview of the special section We proceed to provide a brief summary of the papers in the special section: Biros and Dogan [1] describe a multilevel method for an elliptic inverse source problem in which the first-order conditions for an optimum source estimate amount to a Fredholm operator equation. They construct a multilevel preconditioner by solving a constant coefficient approximating problem and use the known spectral decomposition of this approximating problem to construct a hierarchy of subspaces. The target problem is then solved by a conjugate gradient iteration.