On the use of convex underestimators in global optimization

Många kvantitativa problem från vitt skilda områden kan beskrivas som optimeringsproblem. Ett mått på lösningens kvalitet bör optimeras samtidigt som vissa villkor på lösningen uppfylls. Kvalitetsmåttet kallas vanligen objektfunktion och kan beskriva kostnader (exempelvis produktion, logistik), potentialenergi (molekylmodellering, proteinveckning), risk (finans, försäkring) eller något annat relevant mått. I min doktorsavhandling diskuteras speciellt icke-linjär programmering, NLP, i ändliga dimensioner. Problem med enkel struktur, till exempel någon form av konvexitet, kan lösas effektivt. Tyvärr kan inte alla kvantitativa samband modelleras på ett konvext vis. Icke-konvexa problem kan angripas med heuristiska metoder, algoritmer som söker lösningar med hjälp av deterministiska eller stokastiska tumregler. Ibland fungerar det här väl, men heuristikerna kan sällan garantera kvaliteten på lösningen eller ens att en lösning påträffas. För vissa tillämpningar är det här oacceptabelt. Istället kan man tillämpa så kallad global optimering. Genom att successivt dela variabeldomänen i mindre delar och beräkna starkare gränser på det optimala värdet hittas en lösning inom feltoleransen. Den här metoden kallas branch-and-bound, ungefär dela-och-begränsa. För att ge undre gränser (vid minimering) approximeras problemet med enklare problem, till exempel konvexa, som kan lösas effektivt. I avhandlingen studeras tillvägagångssätt för att approximera differentierbara funktioner med konvexa underskattningar, speciellt den så kallade alphaBB-metoden. Denna metod adderar störningar av en viss form och garanterar konvexitet genom att sätta villkor på den perturberade Hessematrisen. Min forskning har lyft fram en naturlig utvidgning av de perturbationer som används i alphaBB. Nya metoder för att bestämma underskattningsparametrar har beskrivits och jämförts. I sammanfattningsdelen diskuteras global optimering ur bredare perspektiv på optimering och beräkningsalgoritmer.

Segmentation of overlapping convex objects

This thesis presents a framework for segmentation of clustered overlapping convex objects. The proposed approach is based on a three-step framework in which the tasks of seed point extraction, contour evidence extraction, and contour estimation are addressed. The state-of-art techniques for each step were studied and evaluated using synthetic and real microscopic image data. According to obtained evaluation results, a method combining the best performers in each step was presented. In the proposed method, Fast Radial Symmetry transform, edge-to-marker association algorithm and ellipse fitting are employed for seed point extraction, contour evidence extraction and contour estimation respectively. Using synthetic and real image data, the proposed method was evaluated and compared with two competing methods and the results showed a promising improvement over the competing methods, with high segmentation and size distribution estimation accuracy.

Computing the persistent homology of range images with alpha shapes

Laser scanning is becoming an increasingly popular method for measuring 3D objects in industrial design. Laser scanners produce a cloud of 3D points. For CAD software to be able to use such data, however, this point cloud needs to be turned into a vector format. A popular way to do this is to triangulate the assumed surface of the point cloud using alpha shapes. Alpha shapes start from the convex hull of the point cloud and gradually refine it towards the true surface of the object. Often it is nontrivial to decide when to stop this refinement. One criterion for this is to do so when the homology of the object stops changing. This is known as the persistent homology of the object. The goal of this thesis is to develop a way to compute the homology of a given point cloud when processed with alpha shapes, and to infer from it when the persistent homology has been achieved. Practically, the computation of such a characteristic of the target might be applied to power line tower span analysis.

Occlusal and dentofacial characteristics of the deciduous dentition and treatment effects of the eruption guidance appliance in the early mixed dentition

The aim of this investigation was to analyze the dental occlusion in the deciduous dentition, and the effects of orthodontic treatment carried out in the early mixed dentition with the eruption guidance appliance. The deciduous occlusion and craniofacial morphology of 486 children (244 girls and 242 boys) were investigated at the onset of the mixed dentition period (mean age 5.1 years, range 4.0-7.8 years). Treatment in the treatment group and follow-up in the control group were started when the first deciduous incisor was exfoliated (T1) and ended when all permanent incisors and first molars were fully erupted (T2). The mean age of the children was 5.1 years (SD 0.5) at T1 and 8.4 years (SD 0.5) at T2. Treatment was carried out with the eruption guidance appliance. Occlusal changes that took place in 167 children were compared with those of 104 untreated control children. Pre- and post-treatment cephalometric radiographs were taken, and the craniofacial morphology of 115 consecutively treated children was compared with that of 104 control children. The prevalence of malocclusion in the deciduous dentition was 68% or 93% depending on how the cut-off value between the acceptable and non-acceptable occlusal characteristic was defined. The early dentofacial features of children with distal occlusion, large overjet and deepbite differed from those with normal occlusion. However, the skeletal pattern of these three malocclusions showed considerable similarity each being characterized by a retrusive mandible, small maxillo-mandibular difference, convex profile, retrusive lower incisors, and large interincisal angle. In the treatment group, overjet and overbite decreased significantly from T1 to T2. Following treatment, a tooth-to-tooth contact was found in 99% of the treated children but only in 24% of the controls. A Class I molar relationship was observed in 90% of the children in the treatment group, and in 48% in the control group. Good alignment of the incisors was observed in 98% of the treated children, whereas upper crowding was found in 32% and lower crowding in 47% of the controls. A significant difference between the groups was found in the mandibular length, midfacial length and maxillo-mandibular differential. The occlusal correction, brought about by the eruption guidance appliance, was achieved mainly through changes in the dentoalveolar region of the mandible. In addition, the appliance seemed to enhance the growth of the mandible. Treatment in the early mixed dentition using the eruption guidance appliance is an effective method to normalize occlusion and reduce further need of orthodontic treatment. Only few spontaneous corrective changes can be expected without active intervention.

Hyperbolic type metrics and distortion of quasiconformal map pings

This Ph.D. thesis consists of four original papers. The papers cover several topics from geometric function theory, more specifically, hyperbolic type metrics, conformal invariants, and the distortion properties of quasiconformal mappings. The first paper deals mostly with the quasihyperbolic metric. The main result gives the optimal bilipschitz constant with respect to the quasihyperbolic metric for the M¨obius self-mappings of the unit ball. A quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is also proved. The second paper studies some distortion estimates for the class of quasiconformal self-mappings fixing the boundary values of the unit ball or convex domains. The distortion is measured by the hyperbolic metric or hyperbolic type metrics. The results provide explicit, asymptotically sharp inequalities when the maximal dilatation of quasiconformal mappings tends to 1. These explicit estimates involve special functions which have a crucial role in this study. In the third paper, we investigate the notion of the quasihyperbolic volume and find the growth estimates for the quasihyperbolic volume of balls in a domain in terms of the radius. It turns out that in the case of domains with Ahlfors regular boundaries, the rate of growth depends not merely on the radius but also on the metric structure of the boundary. The topic of the fourth paper is complete elliptic integrals and inequalities. We derive some functional inequalities and elementary estimates for these special functions. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.

Computational models for and from biology : simple gene assembly and reaction systems

The advancement of science and technology makes it clear that no single perspective is any longer sufficient to describe the true nature of any phenomenon. That is why the interdisciplinary research is gaining more attention overtime. An excellent example of this type of research is natural computing which stands on the borderline between biology and computer science. The contribution of research done in natural computing is twofold: on one hand, it sheds light into how nature works and how it processes information and, on the other hand, it provides some guidelines on how to design bio-inspired technologies. The first direction in this thesis focuses on a nature-inspired process called gene assembly in ciliates. The second one studies reaction systems, as a modeling framework with its rationale built upon the biochemical interactions happening within a cell. The process of gene assembly in ciliates has attracted a lot of attention as a research topic in the past 15 years. Two main modelling frameworks have been initially proposed in the end of 1990s to capture ciliates’ gene assembly process, namely the intermolecular model and the intramolecular model. They were followed by other model proposals such as templatebased assembly and DNA rearrangement pathways recombination models. In this thesis we are interested in a variation of the intramolecular model called simple gene assembly model, which focuses on the simplest possible folds in the assembly process. We propose a new framework called directed overlap-inclusion (DOI) graphs to overcome the limitations that previously introduced models faced in capturing all the combinatorial details of the simple gene assembly process. We investigate a number of combinatorial properties of these graphs, including a necessary property in terms of forbidden induced subgraphs. We also introduce DOI graph-based rewriting rules that capture all the operations of the simple gene assembly model and prove that they are equivalent to the string-based formalization of the model. Reaction systems (RS) is another nature-inspired modeling framework that is studied in this thesis. Reaction systems’ rationale is based upon two main regulation mechanisms, facilitation and inhibition, which control the interactions between biochemical reactions. Reaction systems is a complementary modeling framework to traditional quantitative frameworks, focusing on explicit cause-effect relationships between reactions. The explicit formulation of facilitation and inhibition mechanisms behind reactions, as well as the focus on interactions between reactions (rather than dynamics of concentrations) makes their applicability potentially wide and useful beyond biological case studies. In this thesis, we construct a reaction system model corresponding to the heat shock response mechanism based on a novel concept of dominance graph that captures the competition on resources in the ODE model. We also introduce for RS various concepts inspired by biology, e.g., mass conservation, steady state, periodicity, etc., to do model checking of the reaction systems based models. We prove that the complexity of the decision problems related to these properties varies from P to NP- and coNP-complete to PSPACE-complete. We further focus on the mass conservation relation in an RS and introduce the conservation dependency graph to capture the relation between the species and also propose an algorithm to list the conserved sets of a given reaction system.

Extreme Observables and Optimal Measurements in Quantum Theory

Optimization of quantum measurement processes has a pivotal role in carrying out better, more accurate or less disrupting, measurements and experiments on a quantum system. Especially, convex optimization, i.e., identifying the extreme points of the convex sets and subsets of quantum measuring devices plays an important part in quantum optimization since the typical figures of merit for measuring processes are affine functionals. In this thesis, we discuss results determining the extreme quantum devices and their relevance, e.g., in quantum-compatibility-related questions. Especially, we see that a compatible device pair where one device is extreme can be joined into a single apparatus essentially in a unique way. Moreover, we show that the question whether a pair of quantum observables can be measured jointly can often be formulated in a weaker form when some of the observables involved are extreme. Another major line of research treated in this thesis deals with convex analysis of special restricted quantum device sets, covariance structures or, in particular, generalized imprimitivity systems. Some results on the structure ofcovariant observables and instruments are listed as well as results identifying the extreme points of covariance structures in quantum theory. As a special case study, not published anywhere before, we study the structure of Euclidean-covariant localization observables for spin-0-particles. We also discuss the general form of Weyl-covariant phase-space instruments. Finally, certain optimality measures originating from convex geometry are introduced for quantum devices, namely, boundariness measuring how ‘close’ to the algebraic boundary of the device set a quantum apparatus is and the robustness of incompatibility quantifying the level of incompatibility for a quantum device pair by measuring the highest amount of noise the pair tolerates without becoming compatible. Boundariness is further associated to minimum-error discrimination of quantum devices, and robustness of incompatibility is shown to behave monotonically under certain compatibility-non-decreasing operations. Moreover, the value of robustness of incompatibility is given for a few special device pairs.