Cette thèse est divisée en trois parties. Dans la première partie, nous commençons par décrire des résultats très connus en théorie du contrôle géométrique tels que le théorème de Chow-Rashevsky, la condition de rang de Kalman, l'application Entrée-Sortie et le test linéaire. De plus, nous définissons et nous étudions brièvement la contrôlabilité locale au voisinage d'un contrôle de référence au premier et au second ordre. Dans la deuxième partie, nous donnons une preuve élémentaire du lemme de Franks linéaire pour les flots géodésiques qui utilise des techniques basiques de théorie du contrôle géométrique. Dans la dernière partie, étant donnée une variété Riemanienne compacte, nous prouvons un lemme de Franks uniforme au second ordre pour les flots géodésiques et on applique le résultat à la théorie de la persistance. Dans cette partie, nous introduisons avec plus de détails les notions de contrôlabilité locale au premier et au second ordre. En effet, nous donnons un résultat de contrôlabilité au second ordre dont la preuve est longue et technique. / This thesis is devided into three parts. In the first part we begin by describing some well known results in geometric control theory such as the Chow Rashevsky Theorem, the Kalman rank condition, the End-Point Mapping and the linear test. Moreover, we define and study briefly local controllability around a reference control at first and second order. In the second part we provide an elementary proof of the Franks lemma for geodesic flows using basic tools of geometric control theory. In the last part, given a compact Riemannian manifold, we prove a uniform Franks' lemma at second order for geodesic flows and apply the result in persistence theory. In this part we introduce with more details notions of local controllability at first and second order. In fact, we provide a second order controllability result whose proof is long and technical.

Från om med 1 januari 2024 ska alla kommuner ha en vattentjänstplan. En vattentjänstplan ska innehålla varje kommuns långsiktiga plan för att tillgodose allmänna vattentjänster i framtiden samt åtgärder som behöver vidtas vid skyfall för att skydda VA-anläggningar. Lagändringen infördes 1 januari 2023 vilket har gett kommuner en snäv tidsplan att ta fram denna plan. Arbetet har undersökt vad vattentjänstplanen i Västerås kommun behöver innehålla för att uppfylla kravet om åtgärder vid skyfall, identifiera punkter i spill- och dagvattennätet som potentiellt är sårbara för skyfall och ge förslag på skyfallsåtgärder. För att uppfylla syftet har en risk- och sårbarhetsanalys utförts för att identifiera punkter i spill- och dagvattennätet som är sårbara för skyfall. Analysen inkluderade en workshop med nyckelpersoner på Mälarenergi Vatten AB och en riskmatris som användes som bedömningsunderlag. Från riskmatrisen identifierades punkter som var potentiellt sårbara för skyfall och en lågpunktskartering utfördes i SCALGO Live på utvalda punkter. De regnhändelser som utfördes i karteringen var 10-, 20- och 100-årsregn. Lågpunktskarteringen jämfördes även med en skyfallskartering med markavrinning och ledningsnät, vilket är en kartering av hög detaljeringsgrad, för att undersöka ifall lågpunktskartering kan vara lämpligt underlag till en vattentjänstplan. Resultatet av risk- och sårbarhetsanalysen var att sju punkter, som gavs som förslag under workshopen, hade höga riskvärden och var potentiellt sårbara för skyfall. Tre av sju punkter valdes till vidare analys: Branthovda, Skiljebo och Önsta-Gryta, alla belägna i Västerås tätort. Samtliga av dessa tre punkter var i dagvattennätet. Lågpunktskarteringen i SCALGO Live som utfördes över dessa tre punkter visade stora översvämningar vid ett 100-årsregn. Skyfallsåtgärder som föreslogs för platserna var magasinerings ytor och skyfallsled. Vid jämförelse av lågpunktskartering och skyfallskartering med markavrinning och ledningsnät visade skyfallskarteringen en mindre översvämning för Branthovda och Skiljebo. I Önsta-Gryta var skillnaden mellan karteringarna minimal. Detta var då skyfallskarteringens resultat visar på att dagvattenledningarna i området var överbelastade redan vid ett 10-årsregn, vilket liknade villkoret i lågpunktskarteringen att dagvattenledningarna antas vara fulla. Med detta kan endast en lågpunktskartering visa ett områdes potential till att var sårbara för skyfall, men säger inget om hur spill- eller dagvattennätet påverkas. Dock kan en lågpunktskartering hjälpa till att identifiera områden i tätorter som skulle kunna vara sårbara för översvämningar. / As of January 1st, 2024, all municipalities must have a water service plan. A water service plan must contain each municipality's long-term plan to provide public water services in the future and solutions that need to be taken in the event of a cloudburst to protect water and sewage facilities. The change in law was introduced on January 1st, 2023, which has given municipalities a tight timetable to develop this plan. The study has investigated what the water service plan in Västerås municipality needs to contain in order to fulfill the requirement for solutions in the event of cloudbursts, identify points in the waste and stormwater network that are potentially vulnerable to cloudbursts, and provide suggestions for torrential rain measures. In order to fulfill the purpose, a risk and vulnerability analysis has been carried out to identify points in the waste and stormwater network that are potentially vulnerable to cloudbursts. The analysis included a workshop, with key individuals at Mälarenergi Vatten AB, and a risk matrix that was used as an assessment basis. From the risk matrix, points that were potentially vulnerable to cloudbursts were identified and a low-point mapping was performed in SCALGO Live at the selected points. The rain events performed in the mapping were 10-, 20- and 100-year rainfalls. The low-point mapping was compared with a cloudburst mapping with land runoff and conduit network, which is a mapping with a high degree of detail, to investigate whether low-point mapping can be a suitable basis for a water service plan. The result of the risk and vulnerability analysis was that seven points, which were given as suggestions during the workshop, had high-risk values and were potentially vulnerable to cloudbursts. Three out of the seven points were selected for further analysis: Branthovda, Skiljebo, and Önsta-Gryta, all of them located in Västerås city. All of these sensitive points were in the stormwater network. The low-point mapping in SCALGO Live performed over these three points showed major flooding during a 100-year rainfall event. The proposed cloudburst solutions for the sites were storage areas and cloudburst roads. When comparing low-point mapping and cloudburst mapping with ground runoff and conduit networks, the cloudburst mapping showed a minor flood for Branthovda and Skiljebo. In Önsta-Gryta, the difference between the mappings was minimal. This was due to the results of the cloudburst mapping showing that the stormwater pipes in the area were overloaded even with a 10-year rain, which was similar to the condition in the low-point mapping that the stormwater pipes are assumed to be filled. With this, only a low point mapping can show an area's potential for being vulnerable to cloudbursts but does not say anything about how the waste or stormwater network is affected. However, low point mapping can help identify areas in built-up areas that could be vulnerable to flooding.

No description available.

This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.

MSc (Microbiology) / Department of Microbiology / See the attached abstract below

This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.

No description available.