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Publication# Functional Inverse Problems on Spheres: Theory, Algorithms and Applications

Abstract

Many scientific inquiries in natural sciences involve approximating a spherical field -namely a scalar quantity defined over a continuum of directions- from generalised samples of the latter (e.g. directional samples, local averages, etc). Such an approximation task is often carried out by means of a convex optimisation problem, assessing an optimal trade-off between a data-fidelity and regularisation term. To solve this problem numerically, scientists typically discretise the spherical domain by means of quasi-uniform spherical point sets. Finite-difference methods for approximating (pseudo-)differential operators on such discrete domains are however unavailable in general, making it difficult to work with generalised Tikhonov (gTikhonov) or Total Variation (gTV) regularisers, favouring physically admissible spherical fields with smooth and sharp variations respectively. To overcome such limitations, canonical spline-based discretisation schemes have been proposed. In the case of gTikhonov regularisation, the optimality of such schemes has been proven for spherical scattered data interpolation problems with quadratic cost functionals. This result is however too restrictive for most practical purposes, since it is restricted to directional samples and Gaussian noise models. Moreover, a similar optimality result for gTV regularisation is still lacking. In this thesis, we propose a unified theoretical and practical spherical approximation framework for functional inverse problems on the hypersphere. More specifically, we consider recovering spherical fields directly in the continuous domain using penalised convex optimisation problems with gTikhonov or gTV regularisation terms. Our framework is compatible with various measurement types as well as non-differentiable convex cost functionals. Via novel representer theorems, we characterise the solutions of the reconstruction problem for both regularisation strategies. For gTikhonov regularisation, we show that the solution is unique and can be expressed as a linear combination of the sampling linear functionals -modelling the acquisition process- primitived twice with respect to the gTikhonov pseudo-differential operator. For gTV regularisation, we show that the solutions are convex combinations of spherical splines with less innovations than available measurements. We use both results to design canonical spline-based discretisation schemes, exact for gTikhonov regularisation and with vanishing approximation error for gTV regularisation. We propose efficient and provably convergent proximal algorithms to solve the discrete optimisation problems resulting from both discretisation schemes. We illustrate the superiority of our continuous-domain spherical approximation framework over traditional methods on a variety of real and simulated datasets in the fields of meteorology, forestry, radio astronomy and planetary sciences. The sampling functionals, cost functions and regularisation strategies considered in each case are diverse, showing the versatility of both our theoretical framework and algorithmic solutions. In the last part of this thesis finally, we design an efficient and locally convergent algorithm for recovering the spatial innovations of periodic Dirac streams with finite rates of innovation, and propose a recurrent neural-network for boosting spherical approximation methods in the context of real-time acoustic imaging.

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Convex optimization

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets

Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternative

We have developed a new derivative-free algorithm based on Radial Basis Functions (RBFs). Derivative-free optimization is an active field of research and several algorithms have been proposed recently. Problems of this nature in the industrial setting are quite frequent. The reason is that in a number of applications the optimization process contains simulation packages which are treated as black boxes. The development of our own algorithm was originally motivated by an application in biomedical imaging: the medical image registration problem. The particular characteristics of this problem have incited us to develop a new optimization algorithm based on trust-region methods. However it has been designed to be generic and to be applied to a wide range of problems. The main originality of our approach is the use of RBFs to build the models. In particular we have adapted the existing theory based on quadratic models to our own models and developed new procedures especially designed for models based on RBFs. We have tested our algorithm called BOOSTERS against state-of-the-art methods (UOBYQA, NEWUOA, DFO). On the medical image registration problem, BOOSTERS appears to be the method of choice. The tests on problems from the CUTEr collection show that BOOSTERS is comparable to, but not better than other methods on small problems (size 2-20). It is performing very well for medium size problems (20-80). Moreover, it is able to solve problems of dimension 200, which is considered very large in derivative-free optimization. We have also developed a new class of algorithms combining the robustness of derivative-free algorithms with the faster rate of convergence characterizing Newtonlike-methods. In fact, they define a new class of algorithms lying between derivative-free optimization and quasi-Newton methods. These algorithms are built on the skeleton of our derivative-free algorithm but they can incorporate the gradient when it is available. They can be interpreted as a way of doping derivative-free algorithms with derivatives. If the derivatives are available at each iteration, then our method can be seen as an alternative to quasi-Newton methods. At the opposite, if the derivatives are never evaluated, then the algorithm is totally similar to BOOSTERS. It is a very interesting alternative to existing methods for problems whose objective function is expensive to evaluate and when the derivatives are not available. In this situation, the gradient can be approximated by finite differences and its costs corresponds to n additional function evaluations assuming that Rn is the domain of definition of the objective function. We have compared our method with CFSQP and BTRA, two gradient-based algorithms, and the results show that our doped method performs best. We have also a theoretical analysis of the medical image registration problem based on maximization of mutual information. Most of the current research in this field is concentrated on registration based on nonlinear image transformation. However, little attention has been paid to the theoretical properties of the optimization problem. In our analysis, we focus on the continuity and the differentiability of the objective function. We show in particular that performing a registration without extension of the reference image may lead to discontinuities in the objective function. But we demonstrate that, under some mild assumptions, the function is differentiable almost everywhere. Our analysis is important from an optimization point of view and conditions the choice of a solver. The usual practice is to use generic optimization packages without worrying about the differentiability of the objective function. But the use of gradient-based methods when the objective function is not differentiable may result in poor performance or even in absence of convergence. One of our objectives with this analysis is also that practitioners become aware of these problems and to propose them new algorithms having a potential interest for their applications.

Claire Marianne Charlotte Capelo

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility of using such data-driven algorithms to solve classic physical and mathematical problems. In particular, we try to model the solution of an inverse continuum mechanics problem in the context of linear elasticity using deep neural networks. To better address the inverse function, we start first by studying the simplest related task,consisting of a building block of the actual composite problem. By empirically proving the learnability of simpler functions, we aim to draw conclusions with respect to the initial problem.The basic inverse problem that motivates this paper is that of a 2D plate with inclusion under specific loading and boundary conditions. From measurements at static equilibrium,we wish to recover the position of the hole. Although some analytical solutions have been formulated for 3D-infinite solids - most notably Eshelby’s inclusion problems - finite problems with particular geometries, material inhomogeneities, loading and boundary conditions require the use of numerical methods which are most often efficient solutions to the forward problem, the mapping from the parameter space to the measurement/signal space, i.e. in our case computing displacements and stresses knowing the size and position of the inclusion. Using numerical data generated from the well-defined forward problem,we train a neural network to approximate the inverse function relating displacements and stresses to the position of the inclusion. The preliminary results on the 2D-finite problem are promising, but the black-box nature of neural networks is a huge issue when it comes to understanding the solution.For this reason, we study a 3D-infinite continuous isotropic medium with unique concentrated load, for which the Green’s function gives an analytical mathematical expression relating relative position of the point force and the displacements in the solid. After de-riving the expression of the inverse, namely recovering the relative position of the force from the Green’s matrix computed at a given point in the medium, we are able to study the sensitivity of the inverse function. From both the expression of the Green’s function and its inverse, we highlight what issues might arise when training neural networks to solve the inverse problem. As the Green’s function is not bijective, bijection must been forced when training for regression. Moreover, due to displacements growing to infinity as we approach the singularity at zero, the training domain must be constrained to be some distance away from the singularity. As we train a neural network to fit the inverse of the Green’s function, we show that the input parameters should include the least possible redundant information to ensure the most efficient training.We then extend our analysis to two point forces. As more loads are added, bijection is harder to enforce as permutations of forces must be taken into account and more collisions may arise, i.e. multiple specific combinations of forces might yield the same measurements.One obvious solution is to increase the number of nodes where displacements are measured to limit the possibility of collision. Through new experiments, we show again that the best training is achieved for the least possible amount of nodes, as long as the training data generated is indeed bijective. As the medium is elastic, we propose a neural network architecture that matches the composite nature of the inverse problem. We also present another formulation of the problem which is invariant to permutations of the forces,namely multilabel classification, and yields good performance in the two-load case.Finally, we study the composite inverse function for 2, 3, 4 and 5 forces. By comparing training and accuracy for different neural network architectures, we expose the model easiest to train. Moreover, the evolution of the final accuracy as more loads are added indicates that deep-neural networks (DNNs) are not well suited to fit a non-linear mapping from and to a high-dimensional space. The results are more convincing for multilabel classification.

2020Thomas Jean Debarre, Julien René Fageot, Harshit Gupta, Michaël Unser

We study continuous-domain linear inverse problems with generalized total-variation (gTV) regularization, expressed in terms of a regularization operator L. It has recently been proved that such inverse problems have sparse spline solutions, with fewer jumps than the number of measurements. Moreover, the type of spline solely depends on L (L-splines) and is independent of the measurements. The continuous-domain inverse problem can be recast in an exact way as a finite-dimensional problem by restricting the search space to splines with knots on a uniform finite grid. However, expressing the L-spline coefficients in the dictionary basis of the Green's function of L is ill-suited for practical problems due to its infinite support. Instead, we propose to formulate the problem in the B-spline dictionary basis, which leads to better-conditioned problems. As we make the grid finer, we show that a solution of the continuous-domain problem can be approached arbitrarily closely with functions of this search space. This result motivates our proposed multiresolution algorithm, which computes sparse solutions of our inverse problem. We demonstrate that this algorithm is computationally feasible for 1D signals when L is an ordinary differential operator.