Numerical methods for multiscale inverse problems

We consider the inverse problem of determining the highly oscillatory coefficient $a^\epsilon$ in partial differential equations of the form $-\nabla\cdot (a^\epsilon\nabla u^\epsilon)+bu^\epsilon = f$ from given measurements of the solutions. Here, $\epsilon$ indicates the smallest characteristic wavelength in the problem ($0<\epsilon\ll1$). In addition to the general difficulty of finding an inverse, the oscillatory nature of the forward problem creates an additional challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed and one common approach is to replace the original problem with an effective parameter estimation problem. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, $b = 0$, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, $b > 0$, and exploration seismology, $b < 0$

Indirect Image Registration with Large Diffeomorphic Deformations

The paper adapts the large deformation diffeomorphic metric mapping framework for image registration to the indirect setting where a template is registered against a target that is given through indirect noisy observations. The registration uses diffeomorphisms that transform the template through a (group) action. These diffeomorphisms are generated by solving a flow equation that is defined by a velocity field with certain regularity. The theoretical analysis includes a proof that indirect image registration has solutions (existence) that are stable and that converge as the data error tends so zero, so it becomes a well-defined regularization method. The paper concludes with examples of indirect image registration in 2D tomography with very sparse and/or highly noisy data.Comment: 43 pages, 4 figures, 1 table; revise

Adaptive Minimax Testing for Inverse Problems

This thesis deals with non-parametric hypothesis testing for ill-posed inverse problems, where optimality is measured in a non-asymptotic minimax sense. Loosely speaking, we observe only an approximation of a transformed version of the quantity of interest. Statistical inference, which usually requires an inversion of the transformation, is thus an inverse problem. Particularly challenging are ill-posed inverse problems, where the inverse transformation is not stable. The thesis is divided into two parts, which investigate different ill-posed inverse models: the inverse Gaussian sequence space model with partially unknown operator and a circular convolution model. In both models we derive minimax separation radii of testing, which characterise how much an object has to differ from the null hypothesis to be detectable by a statistical test. We propose two types of testing procedures, an indirect and a direct one. The indirect test is based on a projection-type estimation of the distance to the null and we prove its minimax optimality under mild assumptions. The direct test is instead based on estimating the energy in the image space and thus avoids an inversion of the operator. We highlight the situations in which also the direct test performs optimally. As usual in non-parametric statistics, the performance of our tests depends on the optimal choice of a dimension parameter, which relies on prior knowledge of the underlying structure of the model. We derive adaptive testing strategies by applying a classical Bonferroni aggregation to both the direct and the indirect testing procedures and analyse their performance. Compared with the non-adaptive tests their radii face a deterioration by a log-factor, which we show to be an unavoidable cost to pay for adaptation. Since our minimax optimal testing procedures are based on estimators of a quadratic functional, we further explore the connection between the two problems – quadratic functional estimation and minimax testing – in the circular convolution model. We show how results from one framework can be exploited in the other. Lastly, we consider minimax testing under privacy constraints, where the observations are deliberately transformed before being released to the statistician in order to protect the privacy of an individual

Near-Optimal Recovery of Linear and N-Convex Functions on Unions of Convex Sets

In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, "$N$-convex functionals" (the simplest example being the maximum of several fractional-linear functions) of unknown "signal" known to belong to the union of finitely many convex compact sets from indirect noisy observations of the signal. Our main assumption is that the observation scheme in question is good in the sense of A. Goldenshluger, A. Juditsky, A. Nemirovski, Electr. J. Stat. 9(2) (2015), arXiv:1311.6765, the simplest example being the Gaussian scheme where the observation is the sum of linear image of the signal and the standard Gaussian noise. The proposed estimates, same as upper bounds on their worst-case risks, stem from solutions to explicit convex optimization problems, making the estimates "computation-friendly.