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Institut
- Mathematik (47) (entfernen)
The main achievement of this thesis is an analysis of the accuracy of computations with Loader's algorithm for the binomial density. This analysis in later progress of work could be used for a theorem about the numerical accuracy of algorithms that compute rectangle probabilities for scan statistics of a multinomially distributed random variable. An example that shall illustrate the practical use of probabilities for scan statistics is the following, which arises in epidemiology: Let n patients arrive at a clinic in d = 365 days, each of the patients with probability 1/d at each of these d days and all patients independently from each other. The knowledge of the probability, that there exist 3 adjacent days, in which together more than k patients arrive, helps deciding, after observing data, if there is a cluster which we would not suspect to have occurred randomly but for which we suspect there must be a reason. Formally, this epidemiological example can be described by a multinomial model. As multinomially distributed random variables are examples of Markov increments, which is a fact already used implicitly by Corrado (2011) to compute the distribution function of the multinomial maximum, we can use a generalized version of Corrado's Algorithm to compute the probability described in our example. To compute its result, the algorithm for rectangle probabilities for Markov increments always uses transition probabilities of the corresponding Markov Chain. In the multinomial case, the transition probabilities of the corresponding Markov Chain are binomial probabilities. Therefore, we start an analysis of accuracy of Loader's algorithm for the binomial density, which for example the statistical software R uses. With the help of accuracy bounds for the binomial density we would be able to derive accuracy bounds for the computation of rectangle probabilities for scan statistics of multinomially distributed random variables. To figure out how sharp derived accuracy bounds are, in examples these can be compared to rigorous upper bounds and rigorous lower bounds which we obtain by interval-arithmetical computations.
Shape optimization is of interest in many fields of application. In particular, shape optimization problems arise frequently in technological processes which are modelled by partial differential equations (PDEs). In a lot of practical circumstances, the shape under investigation is parametrized by a finite number of parameters, which, on the one hand, allows the application of standard optimization approaches, but, on the other hand, unnecessarily limits the space of reachable shapes. Shape calculus presents a way to circumvent this dilemma. However, so far shape optimization based on shape calculus is mainly performed using gradient descent methods. One reason for this is the lack of symmetry of second order shape derivatives or shape Hessians. A major difference between shape optimization and the standard PDE constrained optimization framework is the lack of a linear space structure on shape spaces. If one cannot use a linear space structure, then the next best structure is a Riemannian manifold structure, in which one works with Riemannian shape Hessians. They possess the often sought property of symmetry, characterize well-posedness of optimization problems and define sufficient optimality conditions. In general, shape Hessians are used to accelerate gradient-based shape optimization methods. This thesis deals with shape optimization problems constrained by PDEs and embeds these problems in the framework of optimization on Riemannian manifolds to provide efficient techniques for PDE constrained shape optimization problems on shape spaces. A Lagrange-Newton and a quasi-Newton technique in shape spaces for PDE constrained shape optimization problems are formulated. These techniques are based on the Hadamard-form of shape derivatives, i.e., on the form of integrals over the surface of the shape under investigation. It is often a very tedious, not to say painful, process to derive such surface expressions. Along the way, volume formulations in the form of integrals over the entire domain appear as an intermediate step. This thesis couples volume integral formulations of shape derivatives with optimization strategies on shape spaces in order to establish efficient shape algorithms reducing analytical effort and programming work. In this context, a novel shape space is proposed.
The present work considers the normal approximation of the binomial distribution and yields estimations of the supremum distance of the distribution functions of the binomial- and the corresponding standardized normal distribution. The type of the estimations correspond to the classical Berry-Esseen theorem, in the special case that all random variables are identically Bernoulli distributed. In this case we state the optimal constant for the Berry-Esseen theorem. In the proof of these estimations several inequalities regarding the density as well as the distribution function of the binomial distribution are presented. Furthermore in the estimations mentioned above the distribution function is replaced by the probability of arbitrary, not only unlimited intervals and in this new situation we also present an upper bound.
Matching problems with additional resource constraints are generalizations of the classical matching problem. The focus of this work is on matching problems with two types of additional resource constraints: The couple constrained matching problem and the level constrained matching problem. The first one is a matching problem which has imposed a set of additional equality constraints. Each constraint demands that for a given pair of edges either both edges are in the matching or none of them is in the matching. The second one is a matching problem which has imposed a single equality constraint. This constraint demands that an exact number of edges in the matching are so-called on-level edges. In a bipartite graph with fixed indices of the nodes, these are the edges with end-nodes that have the same index. As a central result concerning the couple constrained matching problem we prove that this problem is NP-hard, even on bipartite cycle graphs. Concerning the complexity of the level constrained perfect matching problem we show that it is polynomially equivalent to three other combinatorial optimization problems from the literature. For different combinations of fixed and variable parameters of one of these problems, the restricted perfect matching problem, we investigate their effect on the complexity of the problem. Further, the complexity of the assignment problem with an additional equality constraint is investigated. In a central part of this work we bring couple constraints into connection with a level constraint. We introduce the couple and level constrained matching problem with on-level couples, which is a matching problem with a special case of couple constraints together with a level constraint imposed on it. We prove that the decision version of this problem is NP-complete. This shows that the level constraint can be sufficient for making a polynomially solvable problem NP-hard when being imposed on that problem. This work also deals with the polyhedral structure of resource constrained matching problems. For the polytope corresponding to the relaxation of the level constrained perfect matching problem we develop a characterization of its non-integral vertices. We prove that for any given non-integral vertex of the polytope a corresponding inequality which separates this vertex from the convex hull of integral points can be found in polynomial time. Regarding the calculation of solutions of resource constrained matching problems, two new algorithms are presented. We develop a polynomial approximation algorithm for the level constrained matching problem on level graphs, which returns solutions whose size is at most one less than the size of an optimal solution. We then describe the Objective Branching Algorithm, a new algorithm for exactly solving the perfect matching problem with an additional equality constraint. The algorithm makes use of the fact that the weighted perfect matching problem without an additional side constraint is polynomially solvable. In the Appendix, experimental results of an implementation of the Objective Branching Algorithm are listed.
In the first part of this work we generalize a method of building optimal confidence bounds provided in Buehler (1957) by specializing an exhaustive class of confidence regions inspired by Sterne (1954). The resulting confidence regions, also called Buehlerizations, are valid in general models and depend on a designated statistic'' that can be chosen according to some desired monotonicity behaviour of the confidence region. For a fixed designated statistic, the thus obtained family of confidence regions indexed by their confidence level is nested. Buehlerizations have furthermore the optimality property of being the smallest (w.r.t. set inclusion) confidence regions that are increasing in their designated statistic. The theory is eventually applied to normal, binomial, and exponential samples. The second part deals with the statistical comparison of pairs of diagnostic tests and establishes relations 1. between the sets of lower confidence bounds, 2. between the sets of pairs of comparable lower confidence bounds, and 3. between the sets of admissible lower confidence bounds in various models for diverse parameters of interest.
In recent years, the study of dynamical systems has developed into a central research area in mathematics. Actually, in combination with keywords such as "chaos" or "butterfly effect", parts of this theory have been incorporated in other scientific fields, e.g. in physics, biology, meteorology and economics. In general, a discrete dynamical system is given by a set X and a self-map f of X. The set X can be interpreted as the state space of the system and the function f describes the temporal development of the system. If the system is in state x ∈ X at time zero, its state at time n ∈ N is denoted by f^n(x), where f^n stands for the n-th iterate of the map f. Typically, one is interested in the long-time behaviour of the dynamical system, i.e. in the behaviour of the sequence (f^n(x)) for an arbitrary initial state x ∈ X as the time n increases. On the one hand, it is possible that there exist certain states x ∈ X such that the system behaves stably, which means that f^n(x) approaches a state of equilibrium for n→∞. On the other hand, it might be the case that the system runs unstably for some initial states x ∈ X so that the sequence (f^n(x)) somehow shows chaotic behaviour. In case of a non-linear entire function f, the complex plane always decomposes into two disjoint parts, the Fatou set F_f of f and the Julia set J_f of f. These two sets are defined in such a way that the sequence of iterates (f^n) behaves quite "wildly" or "chaotically" on J_f whereas, on the other hand, the behaviour of (f^n) on F_f is rather "nice" and well-understood. However, this nice behaviour of the iterates on the Fatou set can "change dramatically" if we compose the iterates from the left with just one other suitable holomorphic function, i.e. if we consider sequences of the form (g∘f^n) on D, where D is an open subset of F_f with f(D)⊂ D and g is holomorphic on D. The general aim of this work is to study the long-time behaviour of such modified sequences. In particular, we will prove the existence of holomorphic functions g on D having the property that the behaviour of the sequence of compositions (g∘f^n) on the set D becomes quite similarly chaotic as the behaviour of the sequence (f^n) on the Julia set of f. With this approach, we immerse ourselves into the theory of universal families and hypercyclic operators, which itself has developed into an own branch of research. In general, for topological spaces X, Y and a family {T_i: i ∈ I} of continuous functions T_i:X→Y, an element x ∈ X is called universal for the family {T_i: i ∈ I} if the set {T_i(x): i ∈ I} is dense in Y. In case that X is a topological vector space and T is a continuous linear operator on X, a vector x ∈ X is called hypercyclic for T if it is universal for the family {T^n: n ∈ N}. Thus, roughly speaking, universality and hypercyclicity can be described via the following two aspects: There exists a single object which allows us, via simple analytical operations, to approximate every element of a whole class of objects. In the above situation, i.e. for a non-linear entire function f and an open subset D of F_f with f(D)⊂ D, we endow the space H(D) of holomorphic functions on D with the topology of locally uniform convergence and we consider the map C_f:H(D)→H(D), C_f(g):=g∘f|_D, which is called the composition operator with symbol f. The transform C_f is a continuous linear operator on the Fréchet space H(D). In order to show that the above-mentioned "nice" behaviour of the sequence of iterates (f^n) on the set D ⊂ F_f can "change dramatically" if we compose the iterates from the left with another suitable holomorphic function, our aim consists in finding functions g ∈ H(D) which are hypercyclic for C_f. Indeed, for each hypercyclic function g for C_f, the set of compositions {g∘f^n|_D: n ∈ N} is dense in H(D) so that the sequence of compositions (g∘f^n|_D) is kind of "maximally divergent" " meaning that each function in H(D) can be approximated locally uniformly on D via subsequences of (g∘f^n|_D). This kind of behaviour stands in sharp contrast to the fact that the sequence of iterates (f^n) itself converges, behaves like a rotation or shows some "wandering behaviour" on each component of F_f. To put it in a nutshell, this work combines the theory of non-linear complex dynamics in the complex plane with the theory of dynamics of continuous linear operators on spaces of holomorphic functions. As far as the author knows, this approach has not been investigated before.
The main topic of this treatise is the solution of two problems from the general theory of linear partial differential equations with constant coefficients. While surjectivity criteria for linear partial differential operators in spaces of smooth functions over an open subset of euclidean space and distributions were proved by B. Malgrange and L. Hörmander in 1955, respectively 1962, concrete evaluation of these criteria is still a highly non-trivial task. In particular, it is well-known that surjectivity in the space of smooth functions over an open subset of euclidean space does not automatically imply surjectivity in the space of distributions. Though, examples for this fact all live in three or higher dimensions. In 1966, F. Trèves conjectured that in the two dimensional setting surjectivity of a linear partial differential operator on the smooth functions indeed implies surjectivity on the space of distributions. An affirmative solution to this problem is presented in this treatise. The second main result solves the so-called problem of (distributional) parameter dependence for solutions of linear partial differential equations with constant coefficients posed by J. Bonet and P. Domanski in 2006. It is shown that, in dimensions three or higher, this problem in general has a negative solution even for hypoelliptic operators. Moreover, it is proved that the two dimensional case is again an exception, because in this setting the problem of parameter dependence always has a positive solution.
In a paper of 1996 the british mathematician Graham R. Allan posed the question, whether the product of two stable elements is again stable. Here stability describes the solvability of a certain infinite system of equations. Using a method from the theory of homological algebra, it is proved that in the case of topological algebras with multiplicative webs, and thus in all common locally convex topological algebras that occur in standard analysis, the answer of Allan's question is affirmative.
In splitting theory of locally convex spaces we investigate evaluable characterizations of the pairs (E, X) of locally convex spaces such that each exact sequence 0 -> X -> G -> E -> 0 of locally convex spaces splits, i.e. either X -> G has a continuous linear left inverse or G -> E has a continuous linear right inverse. In the thesis at hand we deal with splitting of short exact sequences of so-called PLH spaces, which are defined as projective limits of strongly reduced spectra of strong duals of Fréchet-Hilbert spaces. This class of locally convex spaces contains most of the spaces of interest for application in the theory of partial differential operators as the space of Schwartz distributions , the space of real analytic functions and various spaces of ultradifferentiable functions and ultradistributions. It also contains non-Schwartz spaces as B(2,k,loc)(Ω) and spaces of smooth and square integrable functions that are not covered by the current theory for PLS spaces. We prove a complete characterizations of the above problem in the case of X being a PLH space and E either being a Fréchet-Hilbert space or a strong dual of one by conditions of type (T ). To this end, we establish the full homological toolbox of Yoneda Ext functors in exact categories for the category of PLH spaces including the long exact sequence, which in particular involves a thorough discussion of the proper concept of exactness. Furthermore, we exhibit the connection to the parameter dependence problem via the Hilbert tensor product for hilbertizable locally convex spaces. We show that the Hilbert tensor product of two PLH spaces is again a PLH space which in particular proves the positive answer to Grothendieck- problème des topologies. In addition to that we give a complete characterization of the vanishing of the first derivative of the functor proj for tensorized PLH spectra if one of the PLH spaces E and X meets some nuclearity assumptions. To apply our results to concrete cases we establish sufficient conditions of (DN)-(Ω) type and apply them to the parameter dependence problem for partial differential operators with constant coefficients on B(2,k,loc)(Ω) spaces as well as to the smooth and square integrable parameter dependence problem. Concluding we give a complete solution of all the problems under consideration for PLH spaces of Köthe type.
This work investigates the industrial applicability of graphics and stream processors in the field of fluid simulations. For this purpose, an explicit Runge-Kutta discontinuous Galerkin method in arbitrarily high order is implemented completely for the hardware architecture of GPUs. The same functionality is simultaneously realized for CPUs and compared to GPUs. Explicit time steppings as well as established implicit methods are under consideration for the CPU. This work aims at the simulation of inviscid, transsonic flows over the ONERA M6 wing. The discontinuities which typically arise in hyperbolic equations are treated with an artificial viscosity approach. It is further investigated how this approach fits into the explicit time stepping and works together with the special architecture of the GPU. Since the treatment of artificial viscosity is close to the simulation of the Navier-Stokes equations, it is reviewed how GPU-accelerated methods could be applied for computing viscous flows. This work is based on a nodal discontinuous Galerkin approach for linear hyperbolic problems. Here, it is extended to non-linear problems, which makes the application of numerical quadrature obligatory. Moreover, the representation of complex geometries is realized using isoparametric mappings. Higher order methods are typically very sensitive with respect to boundaries which are not properly resolved. For this purpose, an approach is presented which fits straight-sided DG meshes to curved geometries which are described by NURBS surfaces. The mesh is modeled as an elastic body and deformed according to the solution of closest point problems in order to minimize the gap to the original spline surface. The sensitivity with respect to geometry representations is reviewed in the end of this work in the context of shape optimization. Here, the aerodynamic drag of the ONERA M6 wing is minimized according to the shape gradient which is implicitly smoothed within the mesh deformation approach. In this context a comparison to the classical Laplace-Beltrami operator is made in a Stokes flow situation.