OPUS 4 | Search

Higher Order Asymptotics for the MSE of Robust M-Estimators of Location on Shrinking Total Variation Neighborhoods (2008)

Matthias Brandl

In the setup of shrinking neighbourhoods in sample size n about an ideal (“smooth”) central model, Rieder (1994) determines the optimal asymptotic linear estimator w.r.t. the asymptotic MSE evaluated uniformly on these neighbourhoods. We answer the question to which degree the asymptotic optimality carries over to finite sample size in the context of total variation neighbourhoods. In contrast to usual higher order asymptotics, instead of giving approximations to distribution functions (or densities), we expand the risk directly by application of Edgeworth and Taylor expansions. In the context of determining the exact finite sample risk for sample size n>2 M. Kohl showed in Kohl (2005) that the speed of convergence towards the asymptotic risk is faster by an order in case of total variation compared to convex contamination neighbourhoods. M. Kohl conjectured that this is caused by the higher symmetry of total variation neighbourhoods. Looking at the MSE in the asymptotic optimal setup we get the same results as M. Kohl for finite sample risk. Furthermore we can show by direct expansion of the MSE that for a higher speed of convergence symmetry of the ideal distribution F is essential. We confirm our result by a cross-check in the ideal model and illustrate our theoretical investigations for F = N(0;1). We also deal with the question of an actual realization of a least favourable deviation from the ideal model in a finite context. We settle on the symmetric case and monotone odd influence curves of Hampel-type form, and show that for a certain kind of manipulation mechanism we gain our theoretical results up to the desired order. It shows up that we only get access to the results in the finite context if we require the finite sample to attain the minimum and maximum of the given influence curve with a certain probability already. Depending on this probability we derive a lower bound on the sample length n. We substantiate the sufficiently exact algorithm by determining the amount K of observations to be manipulated as well as the bound c on the observations for having maximum influence on the MSE according to the value of the influence curve. We give a restrictive condition on the distribution of K that lets the probability of the (K:n)-quantile exceeding a now concrete bound c be exponentially negligible. The bound c is explicitly calculated for F = N(0;1) and suitable four-point distributions for K are given that satisfy all the previous claimed conditions. Thus we gain an algorithm generating observations from a least favourable distribution out of a finite total variation neighbourhood w.r.t. an asymptotically optimal risk.

Derived categories of coherent sheaves on rational homogeneous manifolds (2005)

Christian Böhning

Abstract. One way to reformulate the celebrated theorem of Beilinson is that $(\mathcal{O}(-n),\dots , \mathcal{O})$ and $(\Omega^n(n), \dots , \Omega^1 (1), \mathcal{O})$ are strong complete exceptional sequences in $D^b(Coh\,\mathbb{P}^n)$, the bounded derived category of coherent sheaves on $\mathbb{P}^n$. In a series of papers M. M. Kapranov generalized this result to flag manifolds of type $A_n$ and quadrics. In another direction, Y. Kawamata has recently proven existence of complete exceptional sequences on toric varieties. Starting point of the present work is a conjecture of F. Catanese which says that on every rational homogeneous manifold $X=G/P$, where $G$ is a connected complex semisimple Lie group and $P\subset G$ a parabolic subgroup, there should exist a complete strong exceptional poset and a bijection of the elements of the poset with the Schubert varieties in $X$ such that the partial order on the poset is the order induced by the Bruhat-Chevalley order. An answer to this question would also be of interest with regard to a conjecture of B. Dubrovin which has its source in considerations concerning a hypothetical mirror partner of a projective variety $Y$: There is a complete exceptional sequence in $D^b(Coh\, Y)$ if and only if the quantum cohomology of $Y$ is generically semisimple (the complete form of the conjecture also makes a prediction about the Gram matrix of such a collection). A proof of this conjecture would also support M. Kontsevich's homological mirror conjecture, one of the most important open problems in applications of complex geometry to physics today. The goal of this work will be to provide further evidence for F. Catanese's conjecture, to clarify some aspects of it and to supply new techniques. In section 2 it is shown among other things that the length of every complete exceptional sequence on $X$ must be the number of Schubert varieties in $X$ and that one can find a complete exceptional sequence on the product of two varieties once one knows such sequences on the single factors, both of which follow from known methods developed by Rudakov, Gorodentsev, Bondal et al. Thus one reduces the problem to the case $X=G/P$ with $G$ simple. Furthermore it is shown that the conjecture holds true for the sequences given by Kapranov for Grassmannians and quadrics. One computes the matrix of the bilinear form on the Grothendieck $K$-group $K_{\circ}(X)$ given by the Euler characteristic with respect to the basis formed by the classes of structure sheaves of Schubert varieties in $X$; this matrix is conjugate to the Gram matrix of a complete exceptional sequence. Section 3 contains a proof of theorem 3.2.7 which gives complete exceptional sequences on quadric bundles over base manifolds on which such sequences are known. This enlarges substantially the class of varieties (in particular rational homogeneous manifolds) on which those sequences are known to exist. In the remainder of section 3 we consider varieties of isotropic flags in a symplectic resp. orthogonal vector space. By a theorem due to Orlov (thm. 3.1.5) one reduces the problem of finding complete exceptional sequences on them to the case of isotropic Grassmannians. For these, theorem 3.3.3 gives generators of the derived category which are homogeneous vector bundles; in special cases those can be used to construct complete exceptional collections. In subsection 3.4 it is shown how one can extend the preceding method to the orthogonal case with the help of theorem 3.2.7. In particular we prove theorem 3.4.1 which gives a generating set for the derived category of coherent sheaves on the Grassmannian of isotropic 3-planes in a 7-dimensional orthogonal vector space. Section 4 is dedicated to providing the geometric motivation of Catanese's conjecture and it contains an alternative approach to the construction of complete exceptional sequences on rational homogeneous manifolds which is based on a theorem of M. Brion (thm. 4.1.1) and cellular resolutions of monomial ideals a la Bayer/Sturmfels. We give a new proof of the theorem of Beilinson on $\mathbb{P}^n$ in order to show that this approach might work in general. We also prove theorem 4.2.5 which gives a concrete description of certain functors that have to be investigated in this approach.

THE INDEX THEOREM FOR QUASI-TORI (2013)

Tsz On Mario Chan

The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the numbers of negative and positive eigenvalues of the associated hermitian form. In this thesis, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert–Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using L2-methods coupled with the Kazama–Dolbeault isomorphism and Bochner–Kodaira formulas.

Two Irreducible Components of the Moduli Space M can 1,3 (2012)

Yifan Chen

This thesis is devoted to study two families of surfaces of general type: extended Burniat surfaces with K^2=3 and Keum-Naie-Mendes Lopes-Pardini surfaces. We focus on the corresponding subsets in the Gieseker moduli space. Extended Burniat surfaces with K^2=3 were constructed by Bauer and Catanese in the course of studying the tertiary Burniat surfaces and they showed that their closure is an irreducible component of the moduli space. We prove here the union of the loci described by them is indeed a full irreducible component. We also study the local deformations of two families of degenerations of the extended Burniat surfaces. Keum-Naie-Mendes Lopes-Pardini surfaces are the surfaces constructed by Mendes Lopes and Paridini, which realize the Keum-Naie surfaces with K^2=3 as degenerations. We reconstruct a subfamily of such surfaces and investigate their deformations. We show that the closure of the corresponding subset of the Keum-Naie-Mendes Lopes-Pardini surfaces is an irreducible component of the moduli space.

Stability of flat galaxies (2008)

Roman Firt

In this thesis we investigate the existence and properties of stationary solutions of the flat Vlasov-Poisson system. This system of partial differential equations can be used as a model of extremely flat astronomical objects and is a combination between the two-dimensional motion of particles and the three-dimensional interaction through their gravitational potential.

Optimal Control Problems Governed by Nonlinear Partial Differential Equations and Inclusions (2010)

Julia Fischer

The focus of this thesis lies on examining the solvability of optimal control problems constrained by nonlinear partial differential equations (PDE) and inclusions (PDI). There exist statements on the existence of solutions for optimal control problems with linear and semi-linear PDEs with monotone parts. The theory for non-monotone PDEs resp. the related optimal control problems is, to the author’s knowledge, incomplete regarding important issues. This concerns particularly the case of PDEs containing mappings, which only satisfy boundedness conditions on restricted sets. At first an optimal control problem is considered, which is characterized by a Laplace equation with Dirichlet boundary conditions and a nonlinear non-monotone Nemytskii operator. Under the decisive assumption of the existence of so called sub- and supersolutions for this differential equation and by introducing a truncation operator we can define an auxiliary problem which is characterized by a pseudomonotone operator. Thereby the solution theory for pseudomonotone operators of Brézis (1968) is applicable. Moreover, starting with the definitions of sub- und supersolution it can be shown, that every solution of the auxiliary problem is a solution of the original problem. The choice of a new optimal control problem which substitutes the original optimal control problem is again governed by the properties of the auxiliary operator. The equivalence of the auxiliary problem to the original problem and the existence of at least one solution can be shown. The technique of applying the Theorem of Lax-Milgram on a linearized problem can be adapted to the semi-linear non-monotone case. This procedure is already known from the theory of semi-linear monotone problems. For optimal control problems with quasi-linear differential equations, different methods are required. As in the semi-linear case, the property of pseudomonotonicity plays a key role in proving the existence of a solution of the quasi-linear PDE. In the proof of the existence of a solution for the optimal control problem other properties of the auxiliary operator are exploited. In the elliptic case operators which satisfy the S+ -property are important. In order to utilize this property, a transformation of the operator to some coercive auxiliary operator is necessary. For this reason a term is added, which penalizes the deviation from the admissible set of states. This term is characterized by a factor, which is derived explicitly in this work. The proof of the existence of a solution of the optimal control problem with parabolic equations is based on the definition of an auxiliary operator, coercivity and the S+ -property of operators. The set of solutions of the considered PDE is compact, but the number of solutions and the situation to each other is unknown. This leads to difficulties in deriving necessary optimality conditions. For this reason a direct approach to solve the optimal control problem with semi-linear PDEs is introduced. It is assumed, that the state constraints coincide with the sub- and the supersolution of the PDE with the upper and lower boundary of the control variable. Using an auxiliary operator, this assumption allows the formulation of an equivalent optimal control problem without pointwise state constraints. Through semi-discretization we can define a family of optimal control problems on a finite dimensional state-space. Existence of a subsequence of solutions of these optimal control problems which converges to a solution of the original problem is shown. Another important class of optimal control problems include differential inclusions which are described by multivalued operators. Quasi-linear elliptic inclusions are examined under global as well as local boundedness conditions. Under the assumption of global boundedness the properties of pseudomonotonicity and coercivity for a multivalued auxiliary operator are proven. The existence of at least one solution for the original inclusion follows from the application of a result from Hu and Papageorgiou (1997) on the auxiliary problem. The existence of at least one solution of the optimal control problem is proven by exploiting the coercivity of the multivalued auxiliary operator and the S+ -property of the non-multivalued part of this mapping. In the case of multivalued mappings of Clarke’s gradient type, the existence of at least one solution of the optimal control problem can be shown under local boundedness conditions. Elliptic as well as parabolic quasi-linear inclusions are considered. The proof is again based on coercivity and the S+ -property of the related auxiliary operators and the embedding properties of the spaces.

Shape Calculus Applied to Elliptic Optimal Control Problems (2012)

Michael Frey

This thesis is devoted to the analysis of a very simple, pointwisely state-constrained optimal control problem of an elliptic partial differential equation. The transfer of an idea from the field of optimal control of ordinary differential equations, which proved fruitful with respect to both theoretical treatment and design of algorithms, is the starting point. On this, the state inequality constraint, which is regarded as an equation inside the active set, is differentiated in order to obtain a control law. A geometrical splitting of the constraints is necessary to carry over this approach to the chosen model problem. The associated assertions are rigorously ensured. The subsequent derivation of a control law in the sense of the abovementioned idea yields an equivalent reformulation of the model problem. The active set appears as an independent and equal optimization variable in this new formulation. Thereby a new class of optimization problem is established, which forms a hybrid of optimal control and shape-/topology optimization: set optimal control. This class is integrated into the very abstract framework of optimization on vector bundles; for that purpose some important notions from the field of calculus on manifolds are introduced and related with shape calculus. First order necessary conditions of the set optimal control problem are derived by means of two different approaches: on the one hand a reduced approach via the elimination of the state variable, which uses a formulation as bilevel optimization problem, is pursued, and on the other hand a formal Lagrange principle is presented. A comparison of the newly obtained optimality conditions with those known form literature yields relations between the Lagrange multipliers; in particular, it becomes apparent that the new approach involves higher regularity. The comparison is embedded to the theory of partial differential-algebraic equations, and it is shown that the new approach yields a reduction of the differential index. Upon investigation of the gradient and the second covariant derivative of the objective functional different Newton- and trial algorithms are presented and discussed in detail. By means of a comparison with the well-established primal-dual active set method different benefits of the new approach become apparent. In particular, the new algorithms can be formulated in function space without any regularization. Some numerical tests illustrate that an efficient and competitive solution of state-constrained optimal control problems is achieved. The whole work gives numerous references to different mathematical disciplines and encourages further investigations. All in all, it should be regarded as a first step towards a more comprehensive perspective on state-constrained optimal control of partial differential equations.

Lotsize optimization leading to a p-median problem with cardinalities (2007)

Constantin Gaul Sascha Kurz Jörg Rambau

We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem.

Die schwache Lösung des dritten Randwertproblems der statischen Elastizitätstheorie in $L^q$ für das Differentialgleichungssystem $\Delta\underline{u}+\lambda\nabla div\underline{u}=div\underline{\underline{f}}$ im beschränkten Gebiet und Außengebiet (2006)

Alexander Gerlach

In dieser Arbeit wird die Lamégleichung $$\Delta\underline{u}+\lambda \nabla div\underline{u}=div \underline{\underline{f}}$$ mit den Randbedingungen (Wobei $T^{(j)}(x)=(T^{(j)}_1(x),...,T^{(j)}_n(x)),\;j=1,...,n-1$ die Basis des Tangentialraumes von $\partial\Omega$ in $x$ und $\underline{N}$ die äußere Normale ist.) I) $$\left.\sum_{i,k=1}^n \partial_i u_k T_k^{(j)} N_i\right|_{\partial\Omega}= \left.\sum_{i,k=1}^n f_{ik}T_k^{(j)} N_i\right|_{\partial\Omega}$$ und $$\left.<\underline{u},\underline{N}>\right|_{\partial\Omega}=0,$$ II) $$\left.\sum_{i,k=1}^n\left[ \partial_i u_k T_k^{(j)} N_i+ \partial_k u_i T_k^{(j)} N_i\right]\right|_{\partial\Omega}=\left.\sum_{i,k=1}^n f_{ik}T_k^{(j)} N_i\right|_{\partial\Omega}$$ und $$\left.<\underline{u},\underline{N}>\right|_{\partial\Omega}=0$$ im Rahmen der schwachen $L^q$-Theorie für beschränkte Gebiete und Außengebiete untersucht. Weiter wird die Existenz eines $\underline{u}\in \underline{Y}^{1,q}(\Omega)$ mit (Randbedingung I) $$<\nabla\underline{u},\nabla\underline{\Phi}>_\Omega+\lambda_1<div\underline{u},div\underline{\Phi}>_\Omega=\sum_{i,k=1}^n\underset{\Omega}{\int}f_{ik}\partial_i \Phi_k dx\text{ für alle }\underline{\Phi}\in\underline{Y}^{1,q'}(\Omega)$$ beziehungsweise ein $\underline{u}$ in einem passend gewähltem Teilraum $\underline{Z}^q(\Omega)\subset \underline{Y}^{1,q}(\Omega)$ mit (Randbedingung II) $$\frac{1}{2}<\epsilon(\underline{u}),\epsilon(\underline{\Phi})>_\Omega+\left(\lambda_2-1\right)<div\underline{u},div\underline{\Phi}>_\Omega=\sum_{i,k=1}^n\underset{\Omega}{\int}f_{ik}\partial_i \Phi_k dx\text{ für alle }\underline{\Phi}\in\underline{Z}^{q'}(\Omega).$$ gezeigt. Eine schwache Lösung, die regulär bis zum Rande angenommen wird, erfüllt dann die Randbedingungen I beziehungsweise II.

Über eine Erweiterung der Methode von Soshnikov zur Untersuchung des größten Eigenwerts auf unsymmetrische Verteilungen (2013)

Felix Grimme

Seit der Entdeckung des Halbkreisgesetzes durch Wigner werden reell-symmetrische Zufallsmatrix-Ensembles untersucht. Soshnikov hat in einer bahnbrechenden Arbeit gezeigt, dass für Wigner-Ensembles $A_n=(\xi_\ij)_{1\le i\le j\le n}$ mit symmetrisch verteilten Einträgen die Verteilung des größten Eigenwerts in einer geeigneten Skalierung für $n\to\infty$ universelles Verhalten zeigt und schwach gegen die Tracy-Widom-Verteilung, die Verteilung des Gauß'schen orthogonalen Ensembles, konvergiert. Für den Beweis nutzt Soshnikov die Momentenmethode. Hierbei wird die Analyse der Verteilungsfunktion des größten Eigenwerts auf die Analyse von Erwartungswerten von Spuren hoher Matrixpotenzen zurückgeführt (die Exponenten wachsen mit $n^{2/3}$). Die Spuren werden via $\tr A_n^{p_n}=\sum_{(i_0,\ldots,i_{p-1})\in[n]^p}\xi_{i_0,i_1}\xi_{i_1,i_2}\ldots\xi_{i_{p-1},i_0}$ als Summe über geschlossene Pfade kombinatorisch interpretiert. In der Analyse gilt es herauszufinden, welche Klassen von Pfaden (die mit den Momenten der Matrixeinträge in Verbindung stehen) die Spuren in der Asymptotik $n\to\infty$ dominieren. Es stellt sich heraus das dies Pfade sind, die jede ihrer Kanten genau zweimal durchlaufen. Das bedeutet, dass die Spuren asymptotisch nur von den für alle Matrixeinträge gleichen zweiten Momenten abhängen, sie sind also asymptotisch für alle betrachteten Ensembles universell. Diese Methode wird in der vorliegenden Arbeit auf Wigner-Ensembles mit nicht notwendig symmetrischen Verteilungen der Einträge erweitert. Die Kombinatorik ist in diesem Fall komplexer. Resultat der Arbeit ist, dass die Methode von Soshnikov funktioniert, wenn folgende Bedingungen erfüllt sind: die ersten und dritten Momente der Einträge sind~0 für die 97.\ Momente existiert eine in~$n$ gleichmäßige Schranke.

Refine

Author

Year of publication

Document Type

Language

Keywords

Institute

70 search hits