Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Mathematical Methods in Tomography

This is the seventh Oberwolfach conference on the mathematics of tomography, the first one taking place in 1980. Tomography is the most popular of a series of medical and scientific imaging techniques that have been developed since the mid seventies of the last century

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

Estimating planetary heat flow from a shallow subsurface heat flow measurement

This study investigates the feasibility of estimating planetary heat flow from a shallow subsurface heat flow measurement with a Function Specification Inversion (FSI) model. Heat flow is a product of the thermal conductivity and gradient at depth; these are measured and therefore contain errors. The model estimates other parameters, as well as the former, while not explicitly accounting for temperature dependent thermal properties. The heat flow is decomposed into steady state basal (planetary) and unsteady state (related to the surface temperature variation) heat flow components. Surface heat flow is typically several orders of magnitude higher than the planetary heat flow; therefore unsteady components in a shallow subsurface heat flow measurement may mask the planetary heat flow. The extent of masking positively correlates with the skin depth and amplitude of the surface heat flow, and negatively correlates with the magnitude of the planetary heat flow. The planetary heat flow is estimated by inverting the temperature measurement and optimising the basal heat flow. The basal heat flow is most effectively optimized from instantaneous measurements, taken when the surface temperature is relatively constant. Long-period measurements, while more accurately optimized, introduce more unsteady temperature gradients, thereby increasing the ill-determinacy and instability of the problem. The model tolerates errors up to 25% in simultaneous optimization of several unknown parameters, with related errors in the optimized basal heat flow. On Mars, the heat flow is optimized to within 10% for measurements over at least twice the skin depth and 0.5 of a Martian year, or at least five times the skin depth and 0.25 of a Martian year. On Mercury, temperature amplitudes control optimized heat flow accuracy; sensor penetration depths well below three skin depths are required. On Vesta, very low heat flows render FSI ineffective with a noise amplitude of 1 mK