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Institut
- Mathematik (65) (entfernen)
One of the main tasks in mathematics is to answer the question whether an equation possesses a solution or not. In the 1940- Thom and Glaeser studied a new type of equations that are given by the composition of functions. They raised the following question: For which functions Ψ does the equation F(Ψ)=f always have a solution. Of course this question only makes sense if the right hand side f satisfies some a priori conditions like being contained in the closure of the space of all compositions with Ψ and is easy to answer if F and f are continuous functions. Considering further restrictions to these functions, especially to F, extremely complicates the search for an adequate solution. For smooth functions one can already find deep results by Bierstone and Milman which answer the question in the case of a real-analytic function Ψ. This work contains further results for a different class of functions, namely those Ψ that are smooth and injective. In the case of a function Ψ of a single real variable, the question can be fully answered and we give three conditions that are both sufficient and necessary in order for the composition equation to always have a solution. Furthermore one can unify these three conditions to show that they are equivalent to the fact that Ψ has a locally Hölder-continuous inverse. For injective functions Ψ of several real variables we give necessary conditions for the composition equation to be solvable. For instance Ψ should satisfy some form of local distance estimate for the partial derivatives. Under the additional assumption of the Whitney-regularity of the image of Ψ, we can give sufficient conditions for flat functions f on the critical set of Ψ to possess a solution F(Ψ)=f.
The optimal control of fluid flows described by the Navier-Stokes equations requires massive computational resources, which has led researchers to develop reduced-order models, such as those derived from proper orthogonal decomposition (POD), to reduce the computational complexity of the solution process. The object of the thesis is the acceleration of such reduced-order models through the combination of POD reduced-order methods with finite element methods at various discretization levels. Special stabilization methods required for high-order solution of flow problems with dominant convection on coarse meshes lead to numerical data that is incompatible with standard POD methods for reduced-order modeling. We successfully adapt the POD method for such problems by introducing the streamline diffusion POD method (SDPOD). Using the novel SDPOD method, we experiment with multilevel recursive optimization at Reynolds numbers of Re=400 and Re=10,000.
This paper mainly studies two topics: linear complementarity problems for modeling electricity market equilibria and optimization under uncertainty. We consider both perfectly competitive and Nash–Cournot models of electricity markets and study their robustifications using strict robustness and the -approach. For three out of the four combinations of economic competition and robustification, we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition, this result corresponds to the two classic welfare theorems, which also apply in both considered robust cases that again yield convex robustified problems. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of -robustifications of Nash–Cournot models. Thus, an analog of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.
This thesis introduces a calibration problem for financial market models based on a Monte Carlo approximation of the option payoff and a discretization of the underlying stochastic differential equation. It is desirable to benefit from fast deterministic optimization methods to solve this problem. To be able to achieve this goal, possible non-differentiabilities are smoothed out with an appropriately chosen twice continuously differentiable polynomial. On the basis of this so derived calibration problem, this work is essentially concerned about two issues. First, the question occurs, if a computed solution of the approximating problem, derived by applying Monte Carlo, discretizing the SDE and preserving differentiability is an approximation of a solution of the true problem. Unfortunately, this does not hold in general but is linked to certain assumptions. It will turn out, that a uniform convergence of the approximated objective function and its gradient to the true objective and gradient can be shown under typical assumptions, for instance the Lipschitz continuity of the SDE coefficients. This uniform convergence then allows to show convergence of the solutions in the sense of a first order critical point. Furthermore, an order of this convergence in relation to the number of simulations, the step size for the SDE discretization and the parameter controlling the smooth approximation of non-differentiabilites will be shown. Additionally the uniqueness of a solution of the stochastic differential equation will be analyzed in detail. Secondly, the Monte Carlo method provides only a very slow convergence. The numerical results in this thesis will show, that the Monte Carlo based calibration indeed is feasible if one is concerned about the calculated solution, but the required calculation time is too long for practical applications. Thus, techniques to speed up the calibration are strongly desired. As already mentioned above, the gradient of the objective is a starting point to improve efficiency. Due to its simplicity, finite differences is a frequently chosen method to calculate the required derivatives. However, finite differences is well known to be very slow and furthermore, it will turn out, that there may also occur severe instabilities during optimization which may lead to the break down of the algorithm before convergence has been reached. In this manner a sensitivity equation is certainly an improvement but suffers unfortunately from the same computational effort as the finite difference method. Thus, an adjoint based gradient calculation will be the method of choice as it combines the exactness of the derivative with a reduced computational effort. Furthermore, several other techniques will be introduced throughout this thesis, that enhance the efficiency of the calibration algorithm. A multi-layer method will be very effective in the case, that the chosen initial value is not already close to the solution. Variance reduction techniques are helpful to increase accuracy of the Monte Carlo estimator and thus allow for fewer simulations. Storing instead of regenerating the random numbers required for the Brownian increments in the SDE will be efficient, as deterministic optimization methods anyway require to employ the identical random sequence in each function evaluation. Finally, Monte Carlo is very well suited for a parallelization, which will be done on several central processing units (CPUs).
In this thesis we study structure-preserving model reduction methods for the efficient and reliable approximation of dynamical systems. A major focus is the approximation of a nonlinear flow problem on networks, which can, e.g., be used to describe gas network systems. Our proposed approximation framework guarantees so-called port-Hamiltonian structure and is general enough to be realizable by projection-based model order reduction combined with complexity reduction. We divide the discussion of the flow problem into two parts, one concerned with the linear damped wave equation and the other one with the general nonlinear flow problem on networks.
The study around the linear damped wave equation relies on a Galerkin framework, which allows for convenient network generalizations. Notable contributions of this part are the profound analysis of the algebraic setting after space-discretization in relation to the infinite dimensional setting and its implications for model reduction. In particular, this includes the discussion of differential-algebraic structures associated to the network-character of our problem and the derivation of compatibility conditions related to fundamental physical properties. Amongst the different model reduction techniques, we consider the moment matching method to be a particularly well-suited choice in our framework.
The Galerkin framework is then appropriately extended to our general nonlinear flow problem. Crucial supplementary concepts are required for the analysis, such as the partial Legendre transform and a more careful discussion of the underlying energy-based modeling. The preservation of the port-Hamiltonian structure after the model-order- and complexity-reduction-step represents a major focus of this work. Similar as in the analysis of the model order reduction, compatibility conditions play a crucial role in the analysis of our complexity reduction, which relies on a quadrature-type ansatz. Furthermore, energy-stable time-discretization schemes are derived for our port-Hamiltonian approximations, as structure-preserving methods from literature are not applicable due to our rather unconventional parametrization of the solution.
Apart from the port-Hamiltonian approximation of the flow problem, another topic of this thesis is the derivation of a new extension of moment matching methods from linear systems to quadratic-bilinear systems. Most system-theoretic reduction methods for nonlinear systems rely on multivariate frequency representations. Our approach instead uses univariate frequency representations tailored towards user-defined families of inputs. Then moment matching corresponds to a one-dimensional interpolation problem rather than to a multi-dimensional interpolation as for the multivariate approaches, i.e., it involves fewer interpolation frequencies to be chosen. The notion of signal-generator-driven systems, variational expansions of the resulting autonomous systems as well as the derivation of convenient tensor-structured approximation conditions are the main ingredients of this part. Notably, our approach allows for the incorporation of general input relations in the state equations, not only affine-linear ones as in existing system-theoretic methods.
The Hadamard product of two holomorphic functions which is defined via a convolution integral constitutes a generalization of the Hadamard product of two power series which is obtained by pointwise multiplying their coefficients. Based on the integral representation mentioned above, an associative law for this convolution is shown. The main purpose of this thesis is the examination of the linear and continuous Hadamard convolution operators. These operators map between spaces of holomorphic functions and send - with a fixed function phi - a function f to the convolution of phi and f. The transposed operator is computed and turns out to be a Hadamard convolution operator, too, mapping between spaces of germs of holomorphic functions. The kernel of Hadamard convolution operators is investigated and necessary and sufficient conditions for those operators to be injective or to have dense range are given. In case that the domain of holomorphy of the function phi allows a Mellin transform of phi, certain (generalized) monomials are identified as eigenfunctions of the corresponding operator. By means of this result and some extract of the theory of growth of entire functions, further propositions concerning the injectivity, the denseness of the range or the surjectivity of Hadamard convolution operators are shown. The relationship between Hadamard convolution operators, operators which are defined via the convolution with an analytic functional and differential operators of infinite order is investigated and the results which are obtained in the thesis are put into the research context. The thesis ends with an application of the results to the approximation of holomorphic functions by lacunary polynomials. On the one hand, the question under which conditions lacunary polynomials are dense in the space of all holomorphic functions is investigated and on the other hand, the rate of approximation is considered. In this context, a result corresponding to the Bernstein-Walsh theorem is formulated.
In dieser Dissertation beschäftigen wir uns mit der konstruktiven und generischen Gewinnung universeller Funktionen. Unter einer universellen Funktion verstehen wie dabei eine solche holomorphe Funktion, die in gewissem Sinne ganze Klassen von Funktionen enthält. Die konstruktive Methode beinhaltet die explizite Konstruktion einer universellen Funktion über einen Grenzprozess, etwa als Polynomreihe. Die generische Methode definiert zunächst rein abstrakt die jeweils gewünschte Klasse von universellen Funktionen. Mithilfe des Baireschen Dichtesatzes wird dann gezeigt, dass die Klasse dieser Funktionen nicht nur nichtleer, sondern sogar G_delta und dicht in dem betrachteten Funktionenraum ist. Beide Methoden bedienen sich der Approximationssätze von Runge und von Mergelyan. Die Hauptergebnisse sind die folgenden: (1) Wir haben konstruktiv die Existenz von universellen Laurentreihen auf mehrfach zusammenhängenden Gebieten bewiesen. Zusätzlich haben wir gezeigt, dass die Menge solcher universeller Laurentreihen dicht im Raum der auf dem betrachteten Gebiet holomorphen Funktionen ist. (2) Die Existenz von universellen Faberreihen auf gewissen Gebieten wurde sowohl konstruktiv als auch generisch bewiesen. (3) Zum einen haben wir konstruktiv gezeigt, dass es so genannte ganze T-universelle Funktionen mit vorgegebenen Approximationswegen gibt. Die Approximationswege sind durch eine hinreichend variable funktionale Form vorgegeben. Die Menge solcher Funktionen ist im Raum der ganzen Funktionen eine dichte G_delta-Menge. Zum anderen haben wir generisch die Existenz von auf einem beschränkten Gebiet T-universellen Funktionen bezüglich gewisser vorgegebener Approximationswege bewiesen. Die Approximationswege sind auch hier genügend allgemein.
In dieser Dissertation beschäftigen wir uns mit der konstruktiven und generischen Gewinnung universeller Funktionen. Unter einer universellen Funktion verstehen wie dabei eine solche holomorphe Funktion, die in gewissem Sinne ganze Klassen von Funktionen enthält. Die konstruktive Methode beinhaltet die explizite Konstruktion einer universellen Funktion über einen Grenzprozess, etwa als Polynomreihe. Die generische Methode definiert zunächst rein abstrakt die jeweils gewünschte Klasse von universellen Funktionen. Mithilfe des Baireschen Dichtesatzes wird dann gezeigt, dass die Klasse dieser Funktionen nicht nur nichtleer, sondern sogar G_delta und dicht in dem betrachteten Funktionenraum ist. Beide Methoden bedienen sich der Approximationssätze von Runge und von Mergelyan. Die Hauptergebnisse sind die folgenden: (1) Wir haben konstruktiv die Existenz von universellen Laurentreihen auf mehrfach zusammenhängenden Gebieten bewiesen. Zusätzlich haben wir gezeigt, dass die Menge solcher universeller Laurentreihen dicht im Raum der auf dem betrachteten Gebiet holomorphen Funktionen ist. (2) Die Existenz von universellen Faberreihen auf gewissen Gebieten wurde sowohl konstruktiv als auch generisch bewiesen. (3) Zum einen haben wir konstruktiv gezeigt, dass es so genannte ganze T-universelle Funktionen mit vorgegebenen Approximationswegen gibt. Die Approximationswege sind durch eine hinreichend variable funktionale Form vorgegeben. Die Menge solcher Funktionen ist im Raum der ganzen Funktionen eine dichte G_delta-Menge. Zum anderen haben wir generisch die Existenz von auf einem beschränkten Gebiet T-universellen Funktionen bezüglich gewisser vorgegebener Approximationswege bewiesen. Die Approximationswege sind auch hier genügend allgemein.
Das erste Beispiel einer so genannten universellen holomorphen Funktion stammt von Birkhoff, welcher im Jahre 1929 die Existenz einer ganzen Funktion beweisen konnte, die gewissermaßen jede ganze Funktion durch geeignete Translationen approximieren kann. In der Folgezeit hat sich der Bereich der "universellen Approximation" zu einem eigenständigen Gebiet innerhalb der komplexen Approximationstheorie entwickelt, und es gibt eine Vielzahl an Ergebnissen über universelle Funktionen. Hierbei wurde sich allerdings fast ausschließlich auf das Studium holomorpher und ganzer Funktionen beschränkt, insbesondere die Klasse der meromorphen Funktionen wurde bisher kaum auf das Phänomen der Universalität hin untersucht. Die vorliegende Arbeit beschäftigt sich mit universeller meromorpher Approximation, und geht der Fragestellung nach, ob meromorphe Funktionen mit gewissen Universalitätseigenschaften existieren, und ob die klassischen Ergebnisse aus der universellen holomorphen Approximation auf den meromorphen Fall erweiterbar sind. Hierbei wird zunächst zwischen Translations- und Streckungsuniversalität unterschieden und bewiesen, dass in beiden Fällen jeweils eine im Raum der meromorphen Funktionen residuale Menge an universellen Funktionen existiert. Weiterhin werden die Eigenschaften dieser Funktionen ausführlich studiert. Anschließend werden meromorphe Funktionen auf Ableitungsuniversalität hin untersucht. Hierbei wird einerseits gezeigt, dass im Allgemeinen keine positiven Ergebnisse möglich sind, während andererseits eine spezielle Klasse meromorpher Funktionen betrachtet wird, für welche universelles Verhalten der sukzessiven Ableitungen nachgewiesen werden kann.
In this thesis, we present a new approach for estimating the effects of wind turbines for a local bat population. We build an individual based model (IBM) which simulates the movement behaviour of every single bat of the population with its own preferences, foraging behaviour and other species characteristics. This behaviour is normalized by a Monte-Carlo simulation which gives us the average behaviour of the population. The result is an occurrence map of the considered habitat which tells us how often the bat and therefore the considered bat population frequent every region of this habitat. Hence, it is possible to estimate the crossing rate of the position of an existing or potential wind turbine. We compare this individual based approach with a partial differential equation based method. This second approach produces a lower computational effort but, unfortunately, we lose information about the movement trajectories at the same time. Additionally, the PDE based model only gives us a density profile. Hence, we lose the information how often each bat crosses special points in the habitat in one night. In a next step we predict the average number of fatalities for each wind turbine in the habitat, depending on the type of the wind turbine and the behaviour of the considered bat species. This gives us the extra mortality caused by the wind turbines for the local population. This value is used for a population model and finally we can calculate whether the population still grows or if there already is a decline in population size which leads to the extinction of the population. Using the combination of all these models, we are able to evaluate the conflict of wind turbines and bats and to predict the result of this conflict. Furthermore, it is possible to find better positions for wind turbines such that the local bat population has a better chance to survive. Since bats tend to move in swarm formations under certain circumstances, we introduce swarm simulation using partial integro-differential equations. Thereby, we have a closer look at existence and uniqueness properties of solutions.
Die Ménage-Polynome (engl.: ménage hit polynomials) ergeben sich in natürlicher Weise aus den in der Kombinatorik auftretenden Ménage-Zahlen. Eine Verbindung zu einer gewissen Klasse hypergeometrischer Polynome führt auf die Untersuchung spezieller Folgen von Polynomen vom Typ 3-F-1. Unter Verwendung einer Modifikation der komplexen Laplace-Methode zur gleichmäßigen asymptotischen Auswertung von Parameterintegralen sowie einiger Hilfsmittel aus der Potentialtheorie der komplexen Ebene werden starke und schwache Asymptotiken für die in Rede stehenden Polynomfolgen hergeleitet.
Quadratische Optimierungsprobleme (QP) haben ein breites Anwendungsgebiet, wie beispielsweise kombinatorische Probleme einschließlich des maximalen Cliquenroblems. Motzkin und Straus [25] zeigten die Äquivalenz zwischen dem maximalen Cliquenproblem und dem standard quadratischen Problem. Auch mathematische Statistik ist ein weiteres Anwendungsgebiet von (QP), sowie eine Vielzahl von ökonomischen Modellen basieren auf (QP), z.B. das quadratische Rucksackproblem. In [5] Bomze et al. haben das standard quadratische Optimierungsproblem (StQP) in ein Copositive-Problem umformuliert. Im Folgenden wurden Algorithmen zur Lösung dieses copositiviten Problems von Bomze und de Klerk in [6] und Dür und Bundfuss in [9] entwickelt. Während die Implementierung dieser Algorithmen einige vielversprechende numerische Ergebnisse hervorbrachten, konnten die Autoren nur die copositive Neuformulierung des (StQP)s lösen. In [11] präsentierte Burer eine vollständig positive Umformulierung für allgemeine (QP)s, sogar mit binären Nebenbedingungen. Leider konnte er keine Methode zur Lösung für ein solches vollständig positives Problem präsentieren, noch wurde eine copositive Formulierung vorgeschlagen, auf die man die oben erwähnten Algorithmen modifizieren und anwenden könnte, um diese zu lösen. Diese Arbeit wird einen neuen endlichen Algorithmus zur Lösung eines standard quadratischen Optimierungsproblems aufstellen. Desweiteren werden in dieser Thesis copositve Darstellungen für ungleichungsbeschränkte sowie gleichungsbeschränkte quadratische Optimierungsprobleme vorgestellt. Für den ersten Ansatz wurde eine vollständig positive Umformulierung des (QP) entwickelt. Die copositive Umformulierung konnte durch Betrachtung des dualen Problems des vollständig positiven Problems erhalten werden. Ein direkterer Ansatz wurde gemacht, indem das Lagrange-Duale eines äquivalenten quadratischen Optimierungsproblems betrachtet wurde, das durch eine semidefinite quadratische Nebenbedingung beschränkt wurde. In diesem Zusammenhang werden Bedingungen für starke Dualität vorgeschlagen.
Eine ganze Funktion φ heißt T-universell bezüglich einer gegebenen Folge b:={b_{n}\}_{n \in ℕ komplexer Zahlen mit b_{n} \to \infty$, falls eine geeignete Folge φ(z+b_{n_{k}})\}$ additiver Translationen von φ lokal gleichmäßig in ℂ gegen jede vorgegebene ganze Funktion konvergiert. Ferner nennen wir eine ganze Funktion φ, für welche eine geeignete Folge φ{(n_k)}\}$ ihrer Ableitungen lokal gleichmäßig in ℂ gegen jede vorgegebene ganze Funktion konvergiert, ableitungsuniversell. Die Existenz solcher Funktionen wurde von Birkhoff (1929) und MacLane (1952) bzw. Verallgemeinerungen ihrer Ergebnisse gesichert. In dieser Arbeit wird die Konstruktion solcher Funktionen, die zusätzlich auf jeder Geraden beschränkt sind oder Nullstellen an bestimmten vorgegebenen Punkten besitzen, studiert. Im Besonderen stellte sich hierbei heraus, dass die Menge aller bezüglich einer gegebenen Folge b - welche einer gewissen Bedingung genügt - T-universellen Funktionen, die überdies auf jeder Geraden beschränkt sind, zwar dicht, aber nicht residual im Raum aller ganzen Funktionen versehen mit der lokal-gleichmäßigen Topologie ist. Ebenso überraschend ist die Konstruktion von T-universellen Funktionen, welche eine "regelmäßige Nullstellenasymptotik" besitzen.
In this thesis, we aim to study the sampling allocation problem of survey statistics under uncertainty. We know that the stratum specific variances are generally not known precisely and we have no information about the distribution of uncertainty. The cost of interviewing each person in a stratum is also a highly uncertain parameter as sometimes people are unavailable for the interview. We propose robust allocations to deal with the uncertainty in both stratum specific variances and costs. However, in real life situations, we can face such cases when only one of the variances or costs is uncertain. So we propose three different robust formulations representing these different cases. To the best of our knowledge robust allocation in the sampling allocation problem has not been considered so far in any research.
The first robust formulation for linear problems was proposed by Soyster (1973). Bertsimas and Sim (2004) proposed a less conservative robust formulation for linear problems. We study these formulations and extend them for the nonlinear sampling allocation problem. It is very unlikely to happen that all of the stratum specific variances and costs are uncertain. So the robust formulations are in such a way that we can select how many strata are uncertain which we refer to as the level of uncertainty. We prove that an upper bound on the probability of violation of the nonlinear constraints can be calculated before solving the robust optimization problem. We consider various kinds of datasets and compute robust allocations. We perform multiple experiments to check the quality of the robust allocations and compare them with the existing allocation techniques.
The thesis studies the question how universal behavior is inherited by the Hadamard product. The type of universality that is considered here is universality by overconvergence; a definition will be given in chapter five. The situation can be described as follows: Let f be a universal function, and let g be a given function. Is the Hadamard product of f and g universal again? This question will be studied in chapter six. Starting with the Hadamard product for power series, a definition for a more general context must be provided. For plane open sets both containing the origin this has already been done. But in order to answer the above question, it becomes necessary to have a Hadamard product for functions that are not holomorphic at the origin. The elaboration of such a Hadamard product and its properties are the second central part of this thesis; chapter three will be concerned with them. The idea of the definition of such a Hadamard product will follow the case already known: The Hadamard product will be defined by a parameter integral. Crucial for this definition is the choice of appropriate integration curves; these will be introduced in chapter two. By means of the Hadamard product- properties it is possible to prove the Hadamard multiplication theorem and the Borel-Okada theorem. A generalization of these theorems will be presented in chapter four.
Considering the numerical simulation of mathematical models it is necessary to have efficient methods for computing special functions. We will focus our considerations in particular on the classes of Mittag-Leffler and confluent hypergeometric functions. The PhD Thesis can be structured in three parts. In the first part, entire functions are considered. If we look at the partial sums of the Taylor series with respect to the origin we find that they typically only provide a reasonable approximation of the function in a small neighborhood of the origin. The main disadvantages of these partial sums are the cancellation errors which occur when computing in fixed precision arithmetic outside this neighborhood. Therefore, our aim is to quantify and then to reduce this cancellation effect. In the next part we consider the Mittag-Leffler and the confluent hypergeometric functions in detail. Using the method we developed in the first part, we can reduce the cancellation problems by "modifying" the functions for several parts of the complex plane. Finally, in in the last part two other approaches to compute Mittag-Leffler type and confluent hypergeometric functions are discussed. If we want to evaluate such functions on unbounded intervals or sectors in the complex plane, we have to consider methods like asymptotic expansions or continued fractions for large arguments z in modulus.
Es wird die Existenz einer Potenzreihe vom Konvergenzradius 1 bewiesen, so dass die mit einer zweifach unendlichen Matrix A (deren komplexe Einträge drei Bedingungen erfüllen müssen) gebildeten A -Transformierten außerhalb des (einfach zusammenhängenden) Holomorphiegebietes der Potenzreihe überkonvergieren. Das Hauptergebnis der Arbeit ist ein Satz über die Existenz einer universellen Potenzreihe vom Konvergenzradius 1, so dass deren A "Transformierte stetige Funktionen auf kompakten, holomorphe Funktionen auf offenen Mengen (in beiden Fällen liegen die Mengen im Komplement des einfach zusammenhängenden Holomorphiegebietes der Potenzreihe) approximieren und sich zusätzlich zur fast-überall-Approximation messbarer Funktionen auf messbaren Mengen (im Komplement des Holomorphiegebietes der Potenzreihe gelegen) eignen. Als wichtige Konsequenz dieses Hauptergebnisses ergibt sich für den Fall, dass das Holomorphiegebietes der Potenzreihe der Einheitskreis ist, die Existenz einer universellen trigonometrischen Reihe, so dass deren A "Transformierte auf dem Rand des Einheitskreises stetige Funktionen approximieren und zusätzlich messbare Funktionen fast-überall auf [0,2π] approximieren
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 this thesis, global surrogate models for responses of expensive simulations are investigated. Computational fluid dynamics (CFD) have become an indispensable tool in the aircraft industry. But simulations of realistic aircraft configurations remain challenging and computationally expensive despite the sustained advances in computing power. With the demand for numerous simulations to describe the behavior of an output quantity over a design space, the need for surrogate models arises. They are easy to evaluate and approximate quantities of interest of a computer code. Only a few number of evaluations of the simulation are stored for determining the behavior of the response over a whole range of the input parameter domain. The Kriging method is capable of interpolating highly nonlinear, deterministic functions based on scattered datasets. Using correlation functions, distinct sensitivities of the response with respect to the input parameters can be considered automatically. Kriging can be extended to incorporate not only evaluations of the simulation, but also gradient information, which is called gradient-enhanced Kriging. Adaptive sampling strategies can generate more efficient surrogate models. Contrary to traditional one-stage approaches, the surrogate model is built step-by-step. In every stage of an adaptive process, the current surrogate is assessed in order to determine new sample locations, where the response is evaluated and the new samples are added to the existing set of samples. In this way, the sampling strategy learns about the behavior of the response and a problem-specific design is generated. Critical regions of the input parameter space are identified automatically and sampled more densely for reproducing the response's behavior correctly. The number of required expensive simulations is decreased considerably. All these approaches treat the response itself more or less as an unknown output of a black-box. A new approach is motivated by the assumption that for a predefined problem class, the behavior of the response is not arbitrary, but rather related to other instances of the mutual problem class. In CFD, for example, responses of aerodynamic coefficients share structural similarities for different airfoil geometries. The goal is to identify the similarities in a database of responses via principal component analysis and to use them for a generic surrogate model. Characteristic structures of the problem class can be used for increasing the approximation quality in new test cases. Traditional approaches still require a large number of response evaluations, in order to achieve a globally high approximation quality. Validating the generic surrogate model for industrial relevant test cases shows that they generate efficient surrogates, which are more accurate than common interpolations. Thus practical, i.e. affordable surrogates are possible already for moderate sample sizes. So far, interpolation problems were regarded as separate problems. The new approach uses the structural similarities of a mutual problem class innovatively for surrogate modeling. Concepts from response surface methods, variable-fidelity modeling, design of experiments, image registration and statistical shape analysis are connected in an interdisciplinary way. Generic surrogate modeling is not restricted to aerodynamic simulation. It can be applied, whenever expensive simulations can be assigned to a larger problem class, in which structural similarities are expected.