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The purpose of this study is to localize the seizure onset zone of patients suffering from drug-resistant epilepsy. During the last two decades, multiple studies proposed the use of Independent Component Analysis (ICA) to analyze ictal electroencephalogram (EEG) recordings. This study aims at evaluating ICA potential with quantitative measurements. In particular, we address the challenging step where the components extracted by ICA of an ictal nature must be selected. We considered a cohort of 10 patients suffering from extratemporal lobe epilepsy who were rendered seizure-free after surgery. Different sets of pre-processing parameters were compared and component features were explored to help distinguish ictal components from others. Quantitative measurements were implemented to determine whether some of the components returned by ICA were located within the resection zone and thus likely to be ictal. Finally, an assistance to the component selection was proposed based on the implemented features. For every seizure, at least one component returned by ICA was localized within the resection zone, with the optimal pre-processing parameters. Three features were found to distinguish components localized within the resection zone: the dispersion of their active brain sources, the ictal rhythm power and the contribution to the EEG variance. Using the implemented component selection assistance based on the features, the probability that the first proposed component yields an accurate estimation reaches 51.43% (without assistance: 24.74%). The accuracy reaches 80% when considering the best result within the first five components. This study confirms the utility of ICA for ictal EEG analysis in extratemporal lobe epilepsy, and suggests relevant features to analyze the components returned by ICA. A component selection assistance is proposed to guide clinicians in their choice for ictal components.

The economic dispatch problem is a fundamental problem in power system operations. An extensive body of literature has focused on providing fast and robust algorithms for solving the various instances of the economic dispatch. In order to capture physical effects such as the power losses of the network or the valve-point loading effect of combined cycle gas turbines, non-convex models of the economic dispatch have been considered. However, these features of the problem render the convergence analysis more challenging, and few methods in the literature provide insights on the global optimality of the derived solution. In this work, we propose an algorithm that efficiently provides a feasible solution, along with a lower bound, to a non-smooth and non-convex instance of the economic dispatch problem. We test our method on extensively studied test cases.

Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth and nonsmooth low-rank optimization where the feasible set is the set Rm×n≤r of all m × n real matrices of rank at most r. In this paper, motivated by the convergence analysis of such algorithms, we study, by computing inner and outer limits, the continuity of the correspondence that maps each X∈Rm×n≤r to the tangent cone to Rm×n≤r at X. We also deduce results about the continuity of the corresponding normal cone correspondence. Finally, we show that our results include as a particular case the a-regularity of the Whitney stratification of Rm×n≤r following from the fact that this set is a real algebraic variety, called the real determinantal variety.

Lévy flights are a specific model of random walks that appear to be ubiquitous across a wide range of fields where the diffusion process is faster than predicted by classical Brownian motion. Lévy flights consist of a succession of random displacements whose step lengths have a heavy-tailed probability distribution. Unlike Brownian motion, whose step-length probability distribution decays exponentially, the distribution of Lévy flights decays algebraically. This leads to heavier tails and hence a larger probability of very long displacements that would be almost impossible with a Brownian motion. This thesis consists in deriving a space-fractional-order diffusion model that explicitly represents the effect of Lévy flights, deriving a numerical algorithm for solving the model equations, and then applying it to study the dispersion of living organisms in the field of life-science problems. In immunology, it has been observed that CD8+ T cells adopt a Lévy flight foraging pattern in response to Toxoplasma gondii infection in the brain. Here, we show that the Lévy search pattern enables T cells to spread over the whole brain tissue and hence they can rapidly destroy infected cells distributed throughout the brain tissue. However, with the Brownian motion assumption, T cells travel through the brain slowly, leading to a slower decline of the infected cells far away from the source of T cells. In nature, the existence of landscape and physiological limitations prevents the occurrence of arbitrary large displacements by the individuals following a Lévy flight. Instead, truncated Lévy flights are introduced, which lead to truncated space-fractional order diffusion models with a truncation parameter. As an application of such equations, we propose a simple epidemic model with the assumption that infected individuals follow a truncated Lévy flight, and then we investigate the effect of different values of the truncation parameter on the epidemic speed. Finally, we explore the, (AGRO - Sciences agronomiques et ingénierie biologique) -- UCL, 2022

From the child playing with its toys to the surgeon holding its scalpel, object manipulation in everyday life is omnipresent. This skill relies on collecting information on the state of the manipulated object using our senses, with touch being of first importance. Using the information provided by touch, our central nervous system sends commands to our muscles to perform movements that take into account how slippery an object is, its shape, the characteristics of the desired movement, and more. In this thesis, we studied how finger skin deformations influence object manipulation, as touch and skin deformations are closely linked. First, we present a novel device that allows for the measurement of fingerpad skin deformations during active manipulation and present the link between the grip force and skin deformations during a simple oscillation task. Then, we show how localized fingerpad strains allow to quickly adapt the grip force to the friction when lifting objects. Finally, we show how the grip force is adjusted so that the upper bound of the amount of partial slip and finger skin deformation remain similar across friction conditions. This work contributes to the understanding of the role of touch during object manipulation and provides useful information for the design of friction sensors and somatosensory prostheses., (FSA - Sciences de l'ingénieur) -- UCL, 2022

Large gas power plants are—and will be in the years to come—a major element of power system operations. On one hand, for the flexibility they offer, which is most valuable for the massive integration of renewable energy sources in the energy mix and the associated uncertainty linked to the production of such sources. On the other hand, they can serve as a replacement for nuclear power plants, following the policy of some European countries like Germany and Belgium. These large gas power plants often obey the valve point effect. This physical effect is due to the increase of throttling losses when operating a unit off a valve point, that is, just after opening one of the several fuel admission valves. We show how the consideration of this physical effect makes the dispatch a nonconvex and nonsmooth optimization problem and propose algorithms that aim at efficiently finding a solution as close as possible to the global optimum, along with some guarantees. In the first part of the thesis, we present a three-step algorithm. The first step solves a relaxation of the problem to obtain an infeasible solution that we expect to be close to the feasible set, along with a lower bound to the global optimum. Then, the second step projects this infeasible solution onto the feasible set. Finally, the last step amounts to locally improving the feasible solution that is obtained in the second step via a Riemannian subgradient descent scheme. In the second part of the thesis, we further analyze the second step. The problem is posed abstractly as the projection of a given point onto the intersection of a quadratic hypersurface—or quadric—and a box. We show how to compute the exact projection onto a central non-cylindrical quadric and use splitting methods for the projection onto the full set., (FSA - Sciences de l'ingénieur) -- UCL, 2022

Quasi-Newton methods are very popular in Optimization. They have a long, rich history, and perform extremely well for solving real-life problems. However, almost nothing is known about theoretical efficiency guarantees for these methods. The goal of this work is the advancement of the theory of quasi-Newton methods. This includes both obtaining new convergence estimates for the already existing algorithms and developing new methods with provable efficiency guarantees. In this thesis, we present our results in several directions. First, we provide a new theoretical analysis of local superlinear convergence of classical quasi-Newton methods and establish explicit and non-asymptotic bounds on their rate of convergence. Then, we develop and analyze new quasi-Newton methods which have some advanced features. Specifically, we propose a new family of greedy quasi-Newton methods for which, apart from local superlinear convergence, it is also possible to guarantee the convergence of Hessian approximations. Finally, we study one algorithm which is related to classical quasi-Newton methods, namely, the Ellipsoid Method, and develop a new variant of this method, which has a better dependency on the dimensionality of the problem than the standard one., (FSA - Sciences de l'ingénieur) -- UCL, 2022

Nonnegative Matrix Factorization (NMF) is a popular data analysis tool for nonnegative data, able to extract meaningful features from a dataset, compress it and filter its noise. To do so, this method factorizes an input matrix Y as the product of two factors, the low-rank nonnegative matrices A and X. Recently, several interesting works have analyzed the possibility of imposing some additional structure to the NMF model, by requiring factors A and X to be based on nonnegative parametrizable functions, such as nonnegative splines or polynomials. When the input data is structured, for example for smooth continuous signals, this leads to factorizations that are less sensitive to noise and better able to find the characteristic elements of the dataset. This thesis is a continuation of that work. We first study NMF using polynomials and splines. We focus on the Hierarchical Alternating Least Squares (HALS) algorithm, which has the advantage of quickly finding a good solution, and converging to a stationary point under mild conditions. We generalize this algorithm and observe that it obtains good results and is competitive with existing approaches. We also seek to generalize the NMF model in a more formal way, so that any parameterizable function could be used in the factors. This leads to the definition of the H-NMF (NMF on Hilbert spaces), allowing to factorize a two-variable function (such as a matrix) as the sum of a small number of products of one-variable functions (the characteristic elements). This generalization is proved to have characteristics and theoretical guarantees similar to those of the standard NMF problem. Our generalization of HALS uses projections onto the functions under consideration (e.g. nonnegative polynomial or splines). As these projections can be costly and difficult to compute, we consider to approximate them. We are able to identify several cases where the algorithm keeps good convergence properties, and use these results to describe heuri, (FSA - Sciences de l'ingénieur) -- UCL, 2022

Multi-agent systems are a powerful tool used to model a wide variety of problems involving interconnected independent entities collaborating to achieve some objective. The overwhelming majority of studies based on this model assume the composition of the systems remains constant, and yet at the same time aim for robustness and scalability. In this thesis, we question this assumption by considering open multi-agent systems, subject to arrivals and departures of agents which cannot be neglected. In the first part, we highlight and study general properties of such systems by analyzing the well-known consensus problem and the Gossip algorithm. On the one hand, we propose an appropriate way to represent open systems whose size changes with time, applied to this type of problems. On the other hand, we introduce tools and methods to characterize the performance of algorithms in such systems, whose convergence is prevented by nature. In the second part, we study the behavior of decentralized optimization algorithms solving the resource allocation problem subject to replacements of agents. Using two distinct types of metrics, we analyze the evolution of the error achieved by such algorithms in expectation. In particular, we characterize the noxious influence of replacements on their efficiency and behavior., (FSA - Sciences de l'ingénieur) -- UCL, 2022

Quality control of meteorological data is an important part of atmospheric analysis and prediction, as missing or erroneous data can have a negative impact on the accuracy of these environmental products. In practice, the presence of missing data in the weather series is quite common and problematic for many uses. We have compared the performance of matrix completion methods with the state of the art to solve this missing data problem. The experimental results were carried out using the daily minimum and maximum temperature measurements of the network of weather stations operated by the Royal Meteorological Institute (RMI) of Belgium.

We study a continuous-time system that solves the optimization problem over the set of orthogonal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but eventually lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical metric of the Stiefel manifold. We show that the vector field of the proposed flow can be interpreted as the sum of a Riemannian gradient on a generalized Stiefel manifold and a normal vector. Moreover, we prove that the proposed flow globally converges to the set of critical points, and any local minimum and isolated critical point is asymptotically stable.

Birds have inspired human innovation and technology for centuries. The first documented human flying attempt is in the late 800, from Abbas ibn Firnas who created what is considered the first aviation experiment. Later in the years, around the 15th century, Leonardo da Vinci dedicated long time of his research to formally understand bird's flight, leading him to the famous ornithopter invention. Biological fliers are still nowadays a source of scientific inspiration for unsolved research questions. Their capability of flying such a long range during migrations, responding to environmental perturbations and maneuvering are still beyond current engineering advances. This Thesis aims at contributing in the field of flight stability of migratory birds, trying to understand the biological and kinematic parameters that govern flapping flight. From a physical point of view, the flapping motion represents a forcing term in the equations of motion of bird flight. Generally speaking, due to this action, there are no fixed points of equilibrium. The problem of studying stability of bird flight therefore is re-formulated via a limit cycle analysis of such equations of motion. We tackled this problem via a numerical framework that relies on a multiple-shooting algorithm that both detects limit cycles, and assesses their stability via Floquet theory. We provide evidence that the generation and repartition of the pitching moment are the main mechanisms that regulate longitudinal stability. Our numerical results suggest that open loop stability cannot be achieved in absence of the tail surface. The open loop stability can however be achieved introducing the tail surface. Nevertheless we showed that the latter configuration is not ideal for long range flights, both in terms of power consumption and bird's agility. We deduced that to satisfy the observed flight configuration with furled tail, while maintaining stable flight, birds developed biocontrol schemes for continuosly adapting, (SC - Sciences) -- UCL, 2022

We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson’s theorem. It is formulated as minimizing a trace cost function over the symplectic Stiefel manifold. We first investigate various theoretical aspects of this optimization problem such as characterizing the sets of critical points, saddle points, and global minimizers as well as proving that non-global local minimizers do not exist. Based on our recent results on constructing Riemannian structures on the symplectic Stiefel manifold and the associated optimization algorithms, we then propose solving the symplectic eigenvalue problem in the framework of Riemannian optimization. Moreover, a connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices is discussed. Numerical examples are presented

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de~Groote, who showed that for the multiplication of $2\times2$ matrices with $7$ active multiplications, all algorithms are essentially equivalent to Strassen's algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices, (e.g., $2\times3$ by $3\times2$ or $3\times3$ by $3\times3$ matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.

Low-rank matrix completion is the problem of recovering the missing entries of a data matrix by using the assumption that the true matrix admits a good low-rank approximation. Much attention has been given recently to exploiting correlations between the column/row entities to improve the matrix completion quality. In this paper, we propose preconditioned gradient descent algorithms for solving the low-rank matrix completion problem with graph Laplacian-based regularizers. Experiments on synthetic data show that our approach achieves significant speedup compared to an existing method based on alternating minimization. Experimental results on real world data also show that our methods provide low-rank solutions of similar quality in comparable or less time than the state-of-the-art method.

This paper concerns the minimax center of a collection of linear subspaces. For k-dimensional subspaces of an n-dimensional vector space, this can be cast as finding the center of a minimum enclosing ball on a Grassmann manifold. For subspaces of differing dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the problem in this context leads to a candidate minimax center on each of the constituent manifolds, but does not provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. We propose an optimization problem parametrized by the rank of the minimax center. The solution is computed with a subgradient algorithm applied to the dual problem. By scaling the objective and penalizing the information lost by the rank-k minimax center, we jointly recover an optimal dimension, ⁎ , and a subspace at the center of the minimum enclosing ball, ⁎ , that best represents the data.

We propose a differential geometric construction for families of low-rank covariance matrices, via interpolation on low-rank matrix manifolds. In contrast with standard parametric covariance classes, these families offer significant flexibility for problem-specific tailoring via the choice of “anchor” matrices for the interpolation. Moreover, their low-rank facilitates computational tractability in high dimensions and with limited data. We employ these covariance families for both interpolation and identification, where the latter problem comprises selecting the most representative member of the covariance family given a data set. In this setting, standard procedures such as maximum likelihood estimation are nontrivial because the covariance family is rank-deficient; we resolve this issue by casting the identification problem as distance minimization. We demonstrate the power of these differential geometric families for interpolation and identification in a practical application: wind field covariance approximation for unmanned aerial vehicle navigation.

Matrix and tensor completion arise in many different real-world applications related to the inference or acquisition of data. In this thesis, we investigate matrix and tensor completion in the aspects of models, algorithms and regularization. With respect to models and algorithms, we tackle these problems via low-rank matrix optimization using a least-squares model. The rank-constrained models are known not only for their computational efficiency but also the capability of extracting the most important information in the data that has an intrinsically low rank. By exploiting manifold structures in these rank-constrained models, we develop Riemannian algorithms and analyse the convergence properties of these algorithms. In particular, we study algorithms designed via Riemannian preconditioning and provide novel results that explain their efficiency on the quotient manifold of fixed-rank matrices. We also investigate the usage of a certain graph-based regularizer in the matrix and tensor completion problems. The penalty function in the graph-based regularization is an extension of the Frobenius norm in the problem variable, in the sense that the entries in the variable are not penalized with uniform weights but with respect to the intensities of some pairwise relations encoded in a graph Laplacian matrix. The graph Laplacian-based penalty function promotes matrix or tensor solutions such that the pairwise similarities among its entries conform to the underlying graph structure. Besides traditional graph models, we propose a graph learning algorithm for learning a desired graph structure from data., (FSA - Sciences de l'ingénieur) -- UCL, 2021

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on $\mathrm{Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on $\mathrm{Sp}(2p,2n)$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on $\mathrm{Sp}(2p,2n)$ akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves, and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods.

The symplectic Stiefel manifold, denoted by Sp(2p,2n) , is the set of linear symplectic maps between the standard symplectic spaces R2p and R2n . When p=n , it reduces to the well-known set of 2n×2n symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space R2n×2p . The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expression of the Riemannian gradient with respect to the Euclidean metric, which is then used in optimization algorithms. Numerical experiments on the nearest symplectic matrix problem and the symplectic eigenvalue problem illustrate the effectiveness of Euclidean-based algorithms.

The increasing integration of intermittent renewable production requires more flexible assets in power systems. This thesis studies potential paths in order to increase the remuneration of these flexible assets. Specifically, this thesis develops trading strategies for the Continuous Intraday Market, and analyzes different options for implementing a scarcity pricing mechanism in the European electricity market design. The contributions of the dissertation are organized in three chapters. Chapter 2 presents a method for trading the production of a storage unit in the Continuous Intraday Market. We model this problem in the Markov Decision Process framework. We present an approach based on Policy Function Approximation for tackling the problem. We provide relevant parameters for defining our policy, and demonstrate the effectiveness of our approach by comparing it to a method employed in the industry on real historical data. In chapter 3, we characterize an optimal policy for trading a fixed quantity in a simplified version of the Continuous Intraday Market. We use this analytical solution as a basis for developing a Value Function Approximation algorithm and an alternative Stochastic Dual Dynamic Programming algorithm that can trade under a more realistic set of assumptions. Chapter 4 proposes a methodology for analysing different market design options for implementing scarcity pricing in the European markets. The methodology relies on analytical insights derived under an assumption of price-taking behavior. These insights are validated by a simulation model which represents the European balancing market as a Markov Decision Process. Our results highlight the benefit of introducing a market in the European market design for trading balancing capacity in real time., (FSA - Sciences de l'ingénieur) -- UCL, 2021

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.

The access to constantly increasing computational capacities has revolutionized the way engineering is seen. We are now able to produce a large quantity of data, thanks to cheap sensors. However, processing such data remains costly both in computational time and in energy. One of the reasons is that the structure of the data is often omitted or unknown. The search space becomes so large that finding the solution to simple problems often turns out to be finding a needle in a haystack. A classical problem in data processing is called the “fitting problem”. It consists in fitting a d-dimensional curve to a set of data points associated to d parameters. The curve must pass sufficiently close to the data points while being regular enough. When the underlying structure of the data points (i.e., the manifold) is known, one can impose to the curve to preserve this structure (i.e., to remain on the manifold), such that the search space is drastically reduced. The goal of this thesis is to develop methods to (approximately) solve this fitting problem; the bet is to require “less” (less computational capabilities, power, storage, time) by leveraging “more” knowledge on the search space. The objective is the following: provide a toolbox that produces a differentiable fitting curve to data points on manifolds, based on very few and simple geometric tools, at low computational cost and storage capacity, all this while maintaining an acceptable quality of the solutions. The algorithms are applied to different illustrative problems. In 3D shape reconstruction, the data points are organs contours acquired via MRI, and the parameter is the acquisition depth; in wind fields estimation, the data points belong to the manifold of positive semi-definite matrices of given rank, and the parameters are the prevalent wind amplitudes and angles. We also show the performances of our algorithms in applications for parametric model order reduction., (FSA - Sciences de l'ingénieur) -- UCL, 2020

We address the problem of extracting common information from multiple datasets. More specifically, we look for a common subspace minimizing the maximal dissimilarity with all datasets and we propose an algorithm derived from the first order necessary conditions of optimality. On synthetic datasets the proposed method gives as good results as a Riemannian based approach, but also provides an evaluation on how far the iterate is from a critical point.

Low-rank matrix completion is the problem of recovering the missing entries of a data matrix by using the assumption that a good low-rank approximation to the true matrix is possible. Much attention has been paid recently to exploiting correlations between the column/row entities through side information to improve the matrix completion quality. In this paper, we propose an efficient algorithm for solving the low-rank matrix completion with graph-based regularizers. Experiments on synthetic data show that our approach achieves significant speedup compared to the alternating minimization scheme.

We consider the manifold of rank-p positive-semidefinite matrices of size n, seen as a quotient of the set of full-rank n-by-p matrices by the orthogonal group in dimension p. The resulting distance coincides with the Wasserstein distance between centered degenerate Gaussian distributions. We obtain expressions for the Riemannian curvature tensor and the sectional curvature of the manifold. We also provide tangent vectors spanning planes associated with the extreme values of the sectional curvature.

—We consider model order reduction of linear timevarying systems on a finite time interval using balanced truncation. A standard way to perform MOR is to first numerically integrate the associated pair of differential Lyapunov equations for the two gramians, then compute projection matrices using the square root method, and finally formulate the reduced systems at each time instant of a chosen grid. This approach is well-knownfordeliveringgoodapproximation,butrathercostly in computation and storage requirement. Furthermore, if one needs to compute the reduced system for any new time instant thatisnotincludedinthechosengrid,thementionedprocedure must be performed again without explicitly making use of the already computed data. For dealing with such a situation, we propose to store the projection matrices corresponding to a simplified sparse time grid and to use them to recover the projection subspaces at any other time instant via curve interpolation on the Grassmann manifold. By doing this, we can avoid the repetition of solving the differential Lyapunov equations which is the most expensive step in the procedure and therefore, as shown in a numerical example, accelerate the online reduction process.

We present several interpolation schemes on the manifold of fixed-rank positive-semidefinite (PSD) matrices. We explain how these techniques can be used for model order reduction of parameterized linear dynamical systems, and obtain preliminary results on an application.

With the constantly growing capability to measure and store data, dealing with different datasets representing similar phenomena is becoming increasingly common. An example in bioinformatics is gene expression data, where the same disease can be studied in different hospitals at different times by measuring the gene expression levels of patients. It is then desirable to merge the different measures obtained, in order to improve the robustness of the subsequent analysis. In this thesis, we investigate different ways of extracting common information from multiple datasets with common features. We propose two different types of approach: removing the differences across datasets, and, keeping the common information. To remove the differences across datasets, we use a matrix factorization approach: we develop a new spatiotemporal version of independent component analysis that we apply to all concatenated datasets, and use the resulting components to model the differences to be removed. The approach is validated on gene expression datasets, and compared to other existing methods in the wider context of a classification task. In the second part of the thesis, we propose a minimax formulation to model the problem of finding a common subspace across a set of subspaces associated with the datasets. We develop multiple algorithms to solve this minimax problem,with some convergence guarantees, and compare their performances on synthetic and real data., (FSA - Sciences de l'ingénieur) -- UCL, 2019

This thesis contributes to the emerging topic of data fitting algorithms that rely on the structure of the data. It tackles two problems: averaging a set of positive-definite (PD) matrices, and fitting a curve to a set of positive-semidefinite (PSD) matrices. Averaging a set of PD matrices is a subtask in several medical imaging and image processing algorithms. It is also used in classifiers for electroencephalogram signals in brain-computer interfaces. This last application motivates the development of fast and incremental averaging tools on PD matrices, to allow online learning and adaptation of the classifier. We propose and study accelerated versions of two algorithms for averaging PD matrices. The first is an incremental gradient descent algorithm on the manifold. The second computes the mean of the data points in a decentralized way, by relying on the expression for the mean of two data points. Finally, we apply our incremental algorithm to the EEG classification problem. For the second task, we restrict ourselves to the set of fixed-rank PSD matrices, to take advantage of the differentiable structure of this set. This restriction is applicable, for example, when working with low-rank estimations of large PSD matrices, or with large covariance matrices characterizing a dynamical process of lower dimension. We compute expressions for the main tools required for applying curve fitting algorithms to the manifold of fixed-rank PSD matrices. We finally apply these algorithms to two problems: parametric model order reduction and wind field estimation., (FSA - Sciences de l'ingénieur) -- UCL, 2019

Nonsmooth Riemannian optimization is a still scarcely explored subfield of optimization theory that concerns the general problem of minimizing (or maximizing), over a domain endowed with a manifold structure, a real-valued function that is not everywhere differentiable. The purpose of this paper is to illustrate, by means of nine concrete examples, that nonsmooth Riemannian optimization finds numerous applications in engineering and the sciences.