Optimal control of mass transfer in peritoneal dialysis

The aim of this work is to set up mathematical models and numerical methods to investigate the mass transfer process occurring during the peritoneal dialysis (PD) therapy. More precisely the final goal is the set up of tools to find for each patient submitted to PD the best therapy profile to purify blood and remove water from the patients. First, we build up specific mathematical models to represent the physical phenomena we are interested in. As discussed in chapter 1, starting from the Kedem-Katchalsky equations, we end up with a systems of nonlinear ordinary differential equations describing the various aspects of the physical problem. Then, we propose a method based on nonlinear programming techniques to solve the inverse problem arising from the need to assess the peritoneal membrane characteristics which are not directly measurable on the patient. Thanks to its flexibility we are able to support the main standard tests nowadays in use to assess the kinetic properties of the peritoneum. We devise a suitable parametrization of the control function (DPD Dynamic Peritoneal Dialysis) that allows to improve the standard PD profiles with a larger set of treatments. Then we propose an optimization algorithm to improve the PD efficiency. Moreover, in the framework of control theory, we devise an algorithm based on the maximum principle of Pontryagin for switched systems to investigate deeply the PD optimal control problem. Afterwards we carry out numerical simulations, investigating the main inputs influencing the peritoneal dialysis efficiency. Specifically we show an extensive comparison between the APD (Automated Peritoneal Dialysis) and DPD (Dynamic Peritoneal Dilaysis) in order to assess the conditions under which DPD allows to improve the PD performance. Then a numerical investigation based on the algorithm devised for switched systems is carried out to assess the adequacy of DPD. Moreover we set up a procedure to reach an efficiency target and minimize the patient's exposure to glucose in order to improve the PD biocompatibility. Finally, we present the validation results of the mathematical model in order to verify its accuracy. A comparison between the APD and DPD is presented. Moreover we carry out a statistical analysis to assess the error distribution related to the most relevant quantities in order to evaluate strengths and weaknesses of this model and to identify the needs for a further improvement.

From Classical to Unsupervised-Deep-Learning Methods for Solving Inverse Problems in Imaging.

Harshit Gupta

In this thesis, we propose new algorithms to solve inverse problems in the context of biomedical images. Due to ill-posedness, solving these problems require some prior knowledge of the statistics of the underlying images. The traditional algorithms, in the field, assume prior knowledge related to smoothness or sparsity of these images. Recently, they have been outperformed by the second generation algorithms which harness the power of neural networks to learn required statistics from training data. Even more recently, last generation deep-learning-based methods have emerged which require neither training nor training data. This thesis devises algorithms which progress through these generations. It extends these generations to novel formulations and applications while bringing more robustness. In parallel, it also progresses in terms of complexity, from proposing algorithms for problems with 1D data and an exact known forward model to the ones with 4D data and an unknown parametric forward model. We introduce five main contributions. The last three of them propose deep-learning-based latest-generation algorithms that require no prior training. 1) We develop algorithms to solve the continuous-domain formulation of inverse problems with both classical Tikhonov and total-variation regularizations. We formalize the problems, characterize the solution set, and devise numerical approaches to find the solutions. 2) We propose an algorithm that improves upon end-to-end neural-network-based second generation algorithms. In our method, a neural network is first trained as a projector on a training set, and is then plugged in as a projector inside the projected gradient descent (PGD). Since the problem is nonconvex, we relax the PGD to ensure convergence to a local minimum under some constraints. This method outperforms all the previous generation algorithms for Computed Tomography (CT). 3) We develop a novel time-dependent deep-image-prior algorithm for modalities that involve a temporal sequence of images. We parameterize them as the output of an untrained neural network fed with a sequence of latent variables. To impose temporal directionality, the latent variables are assumed to lie on a 1D manifold. The network is then tuned to minimize the data fidelity. We obtain state-of-the-art results in dynamic magnetic resonance imaging (MRI) and even recover intra-frame images. 4) We propose a novel reconstruction paradigm for cryo-electron-microscopy (CryoEM) called CryoGAN. Motivated by generative adversarial networks (GANs), we reconstruct a biomolecule's 3D structure such that its CryoEM measurements resemble the acquired data in a distributional sense. The algorithm is pose-or-likelihood-estimation-free, needs no ab initio, and is proven to have a theoretical guarantee of recovery of the true structure. 5) We extend CryoGAN to reconstruct continuously varying conformations of a structure from heterogeneous data. We parameterize the conformations as the output of a neural network fed with latent variables on a low-dimensional manifold. The method is shown to recover continuous protein conformations and their energy landscape.

EPFL2020

Numerical simulation of a sailing boat

Matteo Lombardi

In this work, various aspects concerning the numerical simulation of a sailing boat are investigated. The attention is focused on simulation of the free-surface, the fluid-structure interaction (FSI) between wind and sails, and the dynamics of the whole boat. Some preliminary work on shape optimization is also presented. Free-surface simulations are carried out both on classical academic benchmark problems and for the prediction of the wave pattern around racing yachts. The comparison with experimental results and numerical data obtained with other CFD codes has proven the validity of the proposed set-up. In this framework, the Level Set method has been implemented and validated as a possible solution to the problem of interface diffusion arising in planing conditions with the standard Volume of Fluid approach. The FSI problem governing the interaction between wind and sails is solved via a strongly coupled segregated approach based on the standard Dirichlet-Neumann coupling. The structural solution is based on a MITC4 finite-element shell code. The fluid and structural meshes are non-conforming and the exchange of information at the interface is obtained via radial basis functions (RBF) interpolation. The fluid mesh motion is accomplished either through the mapping generated using the radial basis functions or via a method based on the inverse distance weighting (IDW) interpolations. The attention is paid to the methodological aspects of this complex problem and to the analysis of the numerical results. The fluid-structure interaction simulations of one and two sail configurations, with different trimmings, are presented; both steady and transient simulations are performed. The results obtained are very encouraging and show the potential of the proposed model. The dynamic motion of the Series 60 hull and the Alinghi’s AC 32 monohull complete of bulb, keel and sails have also been investigated. For the latter in particular, the free sink, trim and roll case has proven to be particularly interesting, with the boat rolling considerably on the side due to the aerodynamic loads exerted by the wind on the sails. Here, the whole boat has been considered as a rigid body but the integration of the FSI module for the sails into the full boat dynamic system is already under development. Finally, a preliminary analysis of shape optimization techniques applied to sailing boat elements have been investigated: in particular the attention has been focused on the drag minimization of (pseudo) bulb geometries under the constraint of fixed volume/fixed righting moment. The shape parametrization have been achieved either using directly the surface element nodes or via the Free Form Deformation (FFD) technique while the optimization algorithm has been based either on the solution of the adjoint Navier-Stokes equations or via pseudo finite-difference algorithms. The algorithm and developments mentioned have been implemented in a common open-source framework, the library OpenFOAM®.

EPFL2012

The sliding Frank-Wolfe algorithm and its application to super-resolution microscopy

Quentin Alain Denoyelle, Gabriel Peyré, Emmanuel Emilien Louis Soubies

This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-knownsparse regularisation method (also known as LASSO or basis pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.