Filtern
Dokumenttyp
- Dissertation (7) (entfernen)
Sprache
- Englisch (7) (entfernen)
Volltext vorhanden
- ja (7)
Schlagworte
- Optimization (7) (entfernen)
Fachbereich / Organisatorische Einheit
Since their introduction, robots have primarily influenced the industrial world, providing new opportunities and challenges for humans and machinery. With the introduction of lightweight robots and mobile robot platforms, the field of robot applications has been expanded, diversified, and brought closer to society. The increased degree of digitalization and the personalization of goods and products require an enhanced and flexible robot deployment by operating several multi-robot systems along production processes, industrial applications, assembly and packaging lines, transport systems, etc.
Efficient and safe robot operation relies on successful task planning followed by the computation and execution of task-performing motion trajectories. This thesis addresses these issues by developing, implementing, and validating optimization-based methods for task and trajectory planning in robotics, considering certain optimality and performance criteria. The focus is mainly on the time optimality of the presented approaches with respect to both execution and computation time without compromising safe robot use.
Driven by a systematic approach, the basis for the algorithm development is established first by modeling the kinematics and dynamics of the considered robots and identifying required dynamic parameters. In a further step, time-optimal task and trajectory planning algorithms for a single robotic arm are developed. Initially, a hierarchical approach is introduced consisting of two decoupled optimization-based control policies, a binary problem for task planning, and a continuous model predictive trajectory planning problem. The two layers of the hierarchical structure are then merged into a monolithic layer, resulting in a hybrid structure in the form of a mixed-integer optimization problem for inherent task and trajectory planning.
Motivated by a multi-robot deployment, the hierarchical control structure for time-optimal task and trajectory planning is extended for the case of a two-arm robotic system with highly overlapping operational spaces, leading to challenging robot motions with high inter-robot collision potential. To this end, a novel predictive approach for collision avoidance is proposed based on a continuous approximation of the robot geometry, resulting in a nonlinear optimization problem capable of online applications with real-time requirements. Towards a mobile and flexible robot platform, a model predictive path-following controller for an omnidirectional mobile robot is introduced. Here, a time-minimal approach is also applied, which consists of the robot following a given parameterized path as accurately as possible and at maximum speed.
The performance of the proposed algorithms and methods is experimentally analyzed and validated under real conditions on robot demonstrators. Implementation details, including the resulting hardware and software architecture, are presented, followed by a detailed description of the results. Concrete and industry-oriented demonstrators for integrating robotic arms in existing manual processes and the indoor navigation of a mobile robot complete the work.
Due to its performance, the field of deep learning has gained a lot of attention, with neural networks succeeding in areas like \( \textit{Computer Vision} \) (CV), \( \textit{Neural Language Processing} \) (NLP), and \( \textit{Reinforcement Learning} \) (RL). However, high accuracy comes at a computational cost as larger networks require longer training time and no longer fit onto a single GPU. To reduce training costs, researchers are looking into the dynamics of different optimizers, in order to find ways to make training more efficient. Resource requirements can be limited by reducing model size during training or designing more efficient models that improve accuracy without increasing network size.
This thesis combines eigenvalue computation and high-dimensional loss surface visualization to study different optimizers and deep neural network models. Eigenvectors of different eigenvalues are computed, and the loss landscape and optimizer trajectory are projected onto the plane spanned by those eigenvectors. A new parallelization method for the stochastic Lanczos method is introduced, resulting in faster computation and thus enabling high-resolution videos of the trajectory and second-order information during neural network training. Additionally, the thesis presents the loss landscape between two minima along with the eigenvalue density spectrum at intermediate points for the first time.
Secondly, this thesis presents a regularization method for \( \textit{Generative Adversarial Networks} \) (GANs) that uses second-order information. The gradient during training is modified by subtracting the eigenvector direction of the biggest eigenvalue, preventing the network from falling into the steepest minima and avoiding mode collapse. The thesis also shows the full eigenvalue density spectra of GANs during training.
Thirdly, this thesis introduces ProxSGD, a proximal algorithm for neural network training that guarantees convergence to a stationary point and unifies multiple popular optimizers. Proximal gradients are used to find a closed-form solution to the problem of training neural networks with smooth and non-smooth regularizations, resulting in better sparsity and more efficient optimization. Experiments show that ProxSGD can find sparser networks while reaching the same accuracy as popular optimizers.
Lastly, this thesis unifies sparsity and \( \textit{neural architecture search} \) (NAS) through the framework of group sparsity. Group sparsity is achieved through \( \ell_{2,1} \)-regularization during training, allowing for filter and operation pruning to reduce model size with minimal sacrifice in accuracy. By grouping multiple operations together, group sparsity can be used for NAS as well. This approach is shown to be more robust while still achieving competitive accuracies compared to state-of-the-art methods.
The work consists of two parts.
In the first part an optimization problem of structures of linear elastic material with contact modeled by Robin-type boundary conditions is considered. The structures model textile-like materials and possess certain quasiperiodicity properties. The homogenization method is used to represent the structures by homogeneous elastic bodies and is essential for formulations of the effective stress and Poisson's ratio optimization problems. At the micro-level, the classical one-dimensional Euler-Bernoulli beam model extended with jump conditions at contact interfaces is used. The stress optimization problem is of a PDE-constrained optimization type, and the adjoint approach is exploited. Several numerical results are provided.
In the second part a non-linear model for simulation of textiles is proposed. The yarns are modeled by hyperelastic law and have no bending stiffness. The friction is modeled by the Capstan equation. The model is formulated as a problem with the rate-independent dissipation, and the basic continuity and convexity properties are investigated. The part ends with numerical experiments and a comparison of the results to a real measurement.
This thesis deals with the solution of special problems arising in financial engineering or financial mathematics. The main focus lies on commodity indices. Chapter 1 addresses the important issue for the financial engineering practice of developing well-suited models for certain assets (here: commodity indices). Descriptive analysis of the Dow Jones-UBS commodity index compared to the Standard & Poor 500 stock index provides us with first insights of some features of the corresponding distributions. Statistical tests of normality and mean reversion then helps us in setting up a model for commodity indices. Additionally, chapter 1 encompasses a thorough introduction to commodity investment, history of commodities trading and the most important derivatives, namely futures and European options on futures. Chapter 2 proposes a model for commodity indices and derives fair prices for the most important derivatives in the commodity markets. It is a Heston model supplemented with a stochastic convenience yield. The Heston model belongs to the model class of stochastic volatility models and is currently widely used in stock markets. For the application in the commodity markets the stochastic convenience yield is included in the drift of the instantaneous spot return process. Motivated by the results of chapter 1 it seems reasonable to model the convenience yield by a mean reverting Ornstein-Uhlenbeck process. Since trading desks only apply and consider models with closed form solutions for options I derive such formulas for commodity futures by solving the corresponding partial differential equation. Additionally, semi-closed form formulas for European options on futures are determined. The Cauchy problem with respect to these options is more challenging than the first one. A solution can be provided. Unlike equities, which typically entitle the holder to a continuing stake in a corporation, commodity futures contracts normally specify a certain date for the delivery of the underlying physical commodity. In order to avoid the delivery process and maintain a futures position, nearby contracts must be sold and contracts that have not yet reached the delivery period must be purchased (so called rolling). Optimal trading days for selling and buying futures are determined by applying statistical tests for stochastic dominance. Besides the optimization of the rolling procedure for commodity futures we dedicate ourselves in chapter 3 with the optimization of the weightings of the commodity futures that make up the index. To this end, I apply the Markowitz approach or mean-variance optimization. The mean-variance optimization penalizes up-side and down-side risk equally, whereas most investors do not mind up-side risk. To overcome this, I consider in the next step other risk measures, namely Value-at-Risk and Conditional Value-at-Risk. The Conditional Value-at-Risk is generalized to discontinuous cumulative distribution functions of the loss. For continuous loss distributions, the Conditional Value-at-Risk at a given confidence level is defined as the expected loss exceeding the Value-at-Risk. Loss distributions associated with finite sampling or scenario modeling are, however, discontinuous. Various risk measures involving discontinuous loss distributions shall be introduced and compared. I then apply the theoretical results to the field of portfolio optimization with commodity indices. Furthermore, I uncover graphically the behavior of these risk measures. For this purpose, I consider the risk measures as a function of the confidence level. Based on a special discrete loss distribution, the graphs demonstrate the different properties of these risk measures. The goal of the first section of chapter 4 is to apply the mathematical concept of excursions for the creation of optimal highly automated or algorithmic trading strategies. The idea is to consider the gain of the strategy and the excursion time it takes to realize the gain. In this section I calculate formulas for the Ornstein-Uhlenbeck process. I show that the corresponding formulas can be calculated quite fast since the only function appearing in the formulas is the so called imaginary error function. This function is already implemented in many programs, such as in Maple. My main contribution of this topic is the optimization of the trading strategy for Ornstein-Uhlenbeck processes via the Banach fixed-point theorem. The second section of chapter 4 deals with statistical arbitrage strategies, a long horizon trading opportunity that generates a riskless profit. The results of this section provide an investor with a tool to investigate empirically if some strategies (for example momentum strategies) constitute statistical arbitrage opportunities or not.
In the thesis the author presents a mathematical model which describes the behaviour of the acoustical pressure (sound), produced by a bass loudspeaker. The underlying physical propagation of sound is described by the non--linear isentropic Euler system in a Lagrangian description. This system is expanded via asymptotical analysis up to third order in the displacement of the membrane of the loudspeaker. The differential equations which describe the behaviour of the key note and the first order harmonic are compared to classical results. The boundary conditions, which are derived up to third order, are based on the principle that the small control volume sticks to the boundary and is allowed to move only along it. Using classical results of the theory of elliptic partial differential equations, the author shows that under appropriate conditions on the input data the appropriate mathematical problems admit, by the Fredholm alternative, unique solutions. Moreover, certain regularity results are shown. Further, a novel Wave Based Method is applied to solve appropriate mathematical problems. However, the known theory of the Wave Based Method, which can be found in the literature, so far, allowed to apply WBM only in the cases of convex domains. The author finds the criterion which allows to apply the WBM in the cases of non--convex domains. In the case of 2D problems we represent this criterion as a small proposition. With the aid of this proposition one is able to subdivide arbitrary 2D domains such that the number of subdomains is minimal, WBM may be applied in each subdomain and the geometry is not altered, e.g. via polygonal approximation. Further, the same principles are used in the case of 3D problem. However, the formulation of a similar proposition in cases of 3D problems has still to be done. Next, we show a simple procedure to solve an inhomogeneous Helmholtz equation using WBM. This procedure, however, is rather computationally expensive and can probably be improved. Several examples are also presented. We present the possibility to apply the Wave Based Technique to solve steady--state acoustic problems in the case of an unbounded 3D domain. The main principle of the classical WBM is extended to the case of an external domain. Two numerical examples are also presented. In order to apply the WBM to our problems we subdivide the computational domain into three subdomains. Therefore, on the interfaces certain coupling conditions are defined. The description of the optimization procedure, based on the principles of the shape gradient method and level set method, and the results of the optimization finalize the thesis.
For the last decade, optimization of beam orientations in intensity-modulated radiation therapy (IMRT) has been shown to be successful in improving the treatment plan. Unfortunately, the quality of a set of beam orientations depends heavily on its corresponding beam intensity profiles. Usually, a stochastic selector is used for optimizing beam orientation, and then a single objective inverse treatment planning algorithm is used for the optimization of beam intensity profiles. The overall time needed to solve the inverse planning for every random selection of beam orientations becomes excessive. Recently, considerable improvement has been made in optimizing beam intensity profiles by using multiple objective inverse treatment planning. Such an approach results in a variety of beam intensity profiles for every selection of beam orientations, making the dependence between beam orientations and its intensity profiles less important. This thesis takes advantage of this property to accelerate the optimization process through an approximation of the intensity profiles that are used for multiple selections of beam orientations, saving a considerable amount of calculation time. A dynamic algorithm (DA) and evolutionary algorithm (EA), for beam orientations in IMRT planning will be presented. The DA mimics, automatically, the methods of beam's eye view and observer's view which are recognized in conventional conformal radiation therapy. The EA is based on a dose-volume histogram evaluation function introduced as an attempt to minimize the deviation between the mathematical and clinical optima. To illustrate the efficiency of the algorithms they have been applied to different clinical examples. In comparison to the standard equally spaced beams plans, improvements are reported for both algorithms in all the clinical examples even when, for some cases, fewer beams are used. A smaller number of beams is always desirable without compromising the quality of the treatment plan. It results in a shorter treatment delivery time, which reduces potential errors in terms of patient movements and decreases discomfort.
Traffic flow on road networks has been a continuous source of challenging mathematical problems. Mathematical modelling can provide an understanding of dynamics of traffic flow and hence helpful in organizing the flow through the network. In this dissertation macroscopic models for the traffic flow in road networks are presented. The primary interest is the extension of the existing macroscopic road network models based on partial differential equations (PDE model). In order to overcome the difficulty of high computational costs of PDE model an ODE model has been introduced. In addition, steady state traffic flow model named as RSA model on road networks has been dicsussed. To obtain the optimal flow through the network cost functionals and corresponding optimal control problems are defined. The solution of these optimization problems provides an information of shortest path through the network subject to road conditions. The resulting constrained optimization problem is solved approximately by solving unconstrained problem invovling exact penalty functions and the penalty parameter. A good estimate of the threshold of the penalty parameter is defined. A well defined algorithm for solving a nonlinear, nonconvex equality and bound constrained optimization problem is introduced. The numerical results on the convergence history of the algorithm support the theoretical results. In addition to this, bottleneck situations in the traffic flow have been treated using a domain decomposition method (DDM). In particular this method could be used to solve the scalar conservation laws with the discontinuous flux functions corresponding to other physical problems too. This method is effective even when the flux function presents more than one discontinuity within the same spatial domain. It is found in the numerical results that the DDM is superior to other schemes and demonstrates good shock resolution.