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.

I denna uppsats jämförs designförslag för visuell korrigerande feedback via projektor i hemmatränings miljö för att se vilken design och vilka design kombinationer som skulle vara användbart och tydligast att förstå för en användare. I och med covid 19 har det lett till att det är mer aktuellt att träna hemifrån istället för på gym för att minska smittrisken. Men i och med att det ofta inte finns samma stödjande verktyg hemma som på gym (ex. speglar och instruktörer) så kan det leda till att övningar utförs fel och kan leda till skador. Tidigt inne i utforskandet av digitala verktyg för hemmaträning kunde man se en kunskapslucka. Forskning och förslag finns runt området men få verktyg för just hemmaträning. Genom att använda tidigare forskning och tidigare kurslitteratur som bas till designförslagen så skapades det tre olika designförslag. De förslagen användes i kvalitativa intervjuer för att bedöma och kritisera de förslagen som presenterades. Efter intervjuerna evaluerades designförslagen för att landa i ett användbart och tydligt designförslag av visuell korrigerande feedback via projektor. / In this paper, design for visual corrective feedback through projection in at-home-exercising environments is compared to see what design and what design combinations would be usable and most understandable. With the Covid 19 pandemic it has become more actual to start exercising from home instead of going to gyms and increase the infection risk of the virus. However, with most homes not having all the helpful tools that the gyms might have (eg. mirrors and instructors), leads to practitioners not exercising the correct way which in turn might lead to the practitioners receiving injuries rather than the desired results. Early on in the research on digital tools for at home exercising, a knowledge gap came to view. Research and drafts do exist but solutions for at home exercising were rare to nonexistent. With the help of previous research and course literature as a base for design, three proposals were made. Those proposals were used in qualitative interviews to judge and critique the presented proposals. After the interviews, the presented design proposals were reevaluated to give a usable and understandable design proposal of visual corrective feedback through projection.

Variational Auto-Encoders (VAEs) are one of the most famous deep generative models. After showing that standard VAEs may not preserve the topology, that is the shape of the data, between the input and the latent space, we tried to modify them so that the topology is preserved. This would help in particular for performing interpolations in the latent space. Our main contribution is two folds. Firstly, we propose successfully the InvMap-VAE which is a simple way to turn any dimensionality reduction technique, given its embedding, into a generative model within a VAE framework providing an inverse mapping, with all the advantages that this implies. Secondly, we propose the Witness Simplicial VAE as an extension of the Simplicial Auto-Encoder to the variational setup using a Witness Complex for computing a simplicial regularization. The Witness Simplicial VAE is independent of any dimensionality reduction technique and seems to better preserve the persistent Betti numbers of a data set than a standard VAE, although it would still need some further improvements. Finally, the two first chapters of this master thesis can also be used as an introduction to Topological Data Analysis, General Topology and Computational Topology (or Algorithmic Topology), for any machine learning student, engineer or researcher interested in these areas with no background in topology. / Variations autokodare (VAE) är en av de mest kända djupa generativa modellerna. Efter att ha visat att standard VAE inte nödvändigtvis bevarar topologiska egenskaper, det vill säga formen på datan, mellan inmatningsdatan och det latenta rummet, försökte vi modifiera den så att topologin är bevarad. Det här skulle i synnerhet underlätta när man genomför interpolering i det latenta rummet. Denna avhandling består av två centrala bidrag. I första hand så utvecklar vi InvMap-VAE, som är en enkel metod att omvandla vilken metod inom dimensionalitetsreducering, givet dess inbäddning, till en generativ modell inom VAE ramverket, vilket ger en invers avbildning och dess tillhörande fördelar. För det andra så presenterar vi Witness Simplicial VAE som en förlängning av en Simplicial Auto-Encoder till dess variationella variant genom att använda ett vittneskomplex för att beräkna en simpliciel regularisering. Witness Simplicial VAE är oberoende av dimensionalitets reducerings teknik och verkar bättre bevara Betti-nummer av ett dataset än en vanlig VAE, även om det finns utrymme för förbättring. Slutligen så kan de första två kapitlena av detta examensarbete också användas som en introduktion till Topologisk Data Analys, Allmän Topologi och Beräkningstopologi (eller Algoritmisk Topologi) till vilken maskininlärnings student, ingenjör eller forskare som är intresserad av dessa ämnesområden men saknar bakgrund i topologi.