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Mon, 19 Feb 2018 09:31:07 +0100Mon, 19 Feb 2018 09:31:07 +0100An Iterative Plug-in Algorithm for Optimal Bandwidth Selection in Kernel Intensity Estimation for Spatial Data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5161
A popular model for the locations of fibres or grains in composite materials
is the inhomogeneous Poisson process in dimension 3. Its local intensity function
may be estimated non-parametrically by local smoothing, e.g. by kernel
estimates. They crucially depend on the choice of bandwidths as tuning parameters
controlling the smoothness of the resulting function estimate. In this
thesis, we propose a fast algorithm for learning suitable global and local bandwidths
from the data. It is well-known, that intensity estimation is closely
related to probability density estimation. As a by-product of our study, we
show that the difference is asymptotically negligible regarding the choice of
good bandwidths, and, hence, we focus on density estimation.
There are quite a number of data-driven bandwidth selection methods for
kernel density estimates. cross-validation is a popular one and frequently proposed
to estimate the optimal bandwidth. However, if the sample size is very
large, it becomes computational expensive. In material science, in particular,
it is very common to have several thousand up to several million points.
Another type of bandwidth selection is a solve-the-equation plug-in approach
which involves replacing the unknown quantities in the asymptotically optimal
bandwidth formula by their estimates.
In this thesis, we develop such an iterative fast plug-in algorithm for estimating
the optimal global and local bandwidth for density and intensity estimation with a focus on 2- and 3-dimensional data. It is based on a detailed
asymptotics of the estimators of the intensity function and of its second
derivatives and integrals of second derivatives which appear in the formulae
for asymptotically optimal bandwidths. These asymptotics are utilised to determine
the exact number of iteration steps and some tuning parameters. For
both global and local case, fewer than 10 iterations suffice. Simulation studies
show that the estimated intensity by local bandwidth can better indicate
the variation of local intensity than that by global bandwidth. Finally, the
algorithm is applied to two real data sets from test bodies of fibre-reinforced
high-performance concrete, clearly showing some inhomogeneity of the fibre
intensity.Pak Hang Lodoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5161Mon, 19 Feb 2018 09:31:07 +0100Application of the Heath-Platen Estimator in Pricing Barrier and Bond Options
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5158
In this thesis, we focus on the application of the Heath-Platen (HP) estimator in option
pricing. In particular, we extend the approach of the HP estimator for pricing path dependent
options under the Heston model. The theoretical background of the estimator
was first introduced by Heath and Platen [32]. The HP estimator was originally interpreted
as a control variate technique and an application for European vanilla options was
presented in [32]. For European vanilla options, the HP estimator provided a considerable
amount of variance reduction. Thus, applying the technique for path dependent options
under the Heston model is the main contribution of this thesis.
The first part of the thesis deals with the implementation of the HP estimator for pricing
one-sided knockout barrier options. The main difficulty for the implementation of the HP
estimator is located in the determination of the first hitting time of the barrier. To test the
efficiency of the HP estimator we conduct numerical tests with regard to various aspects.
We provide a comparison among the crude Monte Carlo estimation, the crude control
variate technique and the HP estimator for all types of barrier options. Furthermore, we
present the numerical results for at the money, in the money and out of the money barrier
options. As numerical results imply, the HP estimator performs superior among others
for pricing one-sided knockout barrier options under the Heston model.
Another contribution of this thesis is the application of the HP estimator in pricing bond
options under the Cox-Ingersoll-Ross (CIR) model and the Fong-Vasicek (FV) model. As
suggested in the original paper of Heath and Platen [32], the HP estimator has a wide
range of applicability for derivative pricing. Therefore, transferring the structure of the
HP estimator for pricing bond options is a promising contribution. As the approximating
Vasicek process does not seem to be as good as the deterministic volatility process in the
Heston setting, the performance of the HP estimator in the CIR model is only relatively
good. However, for the FV model the variance reduction provided by the HP estimator is
again considerable.
Finally, the numerical result concerning the weak convergence rate of the HP estimator
for pricing European vanilla options in the Heston model is presented. As supported by
numerical analysis, the HP estimator has weak convergence of order almost 1.Sema Coskundoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5158Thu, 15 Feb 2018 14:13:35 +0100Multifacility Location Problems with Tree Structure and Finite Dominating Sets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5156
Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an ε-approximation for the euclidean norm case polynomial in the input data and 1/ε.Andrea Maier; Thomas Ullmert; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5156Tue, 13 Feb 2018 14:47:15 +0100A local time stepping method for district heating networks
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5140
In this article a new numerical solver for simulations of district heating networks is presented. The numerical method applies the local time stepping introduced in [11] to networks of linear advection equations. In combination with the high order approach of [4] an accurate and very efficient scheme is developed. In several numerical test cases the advantages for simulations of district heating networks are shown.Raul Borsche; Matthias Eimer; Norbert Siedowpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5140Wed, 31 Jan 2018 08:05:43 +0100On Changepoint Detection in a Series of Stimulus-Response Data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5139
In this paper, we demonstrate the power of functional data models for a statistical analysis of stimulus-response experiments which is a quite natural way to look at this kind of data and which makes use of the full information available. In particular, we focus on the detection of a change in the mean of the response in a series of stimulus-response curves where we also take into account dependence in time.Euna Gesare Nyarige; Jürgen Franke; Alexander Fischerarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5139Wed, 31 Jan 2018 08:02:09 +0100Local stationarity for spatial data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5128
Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.Danilo Pezodoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5128Wed, 17 Jan 2018 09:28:34 +0100Two instances of duality in commutative algebra
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5111
In this thesis we address two instances of duality in commutative algebra.
In the first part, we consider value semigroups of non irreducible singular algebraic curves
and their fractional ideals. These are submonoids of Z^n closed under minima, with a conductor and which fulfill special compatibility properties on their elements. Subsets of Z^n
fulfilling these three conditions are known in the literature as good semigroups and their ideals, and their class strictly contains the class of value semigroup ideals. We examine
good semigroups both independently and in relation with their algebraic counterpart. In the combinatoric setting, we define the concept of good system of generators, and we
show that minimal good systems of generators are unique. In relation with the algebra side, we give an intrinsic definition of canonical semigroup ideals, which yields a duality
on good semigroup ideals. We prove that this semigroup duality is compatible with the Cohen-Macaulay duality under taking values. Finally, using the duality on good semigroup ideals, we show a symmetry of the Poincaré series of good semigroups with special properties.
In the second part, we treat Macaulay’s inverse system, a one-to-one correspondence
which is a particular case of Matlis duality and an effective method to construct Artinian k-algebras with chosen socle type. Recently, Elias and Rossi gave the structure of the inverse system of positive dimensional Gorenstein k-algebras. We extend their result by establishing a one-to-one correspondence between positive dimensional level k-algebras and certain submodules of the divided power ring. We give several examples to illustrate
our result.Laura Tozzodoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5111Tue, 19 Dec 2017 09:28:00 +0100Having a Plan B for Robust Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5097
We continue in this paper the study of k-adaptable robust solutions for combinatorial optimization problems with bounded uncertainty sets. In this concept not a single solution needs to be chosen to hedge against the uncertainty. Instead one is allowed to choose a set of k different solutions from which one can be chosen after the uncertain scenario has been revealed. We first show how the problem can be decomposed into polynomially many subproblems if k is fixed. In the remaining part of the paper we consider the special case where k=2, i.e., one is allowed to choose two different solutions to hedge against the uncertainty. We decompose this problem into so called coordination problems. The study of these coordination problems turns out to be interesting on its own. We prove positive results for the unconstrained combinatorial optimization problem, the matroid maximization problem, the selection problem, and the shortest path problem on series parallel graphs. The shortest path problem on general graphs turns out to be NP-complete. Further, we present for minimization problems how to transform approximation algorithms for the coordination problem to approximation algorithms for the original problem. We study the knapsack problem to show that this relation does not hold for maximization problems in general. We present a PTAS for the corresponding coordination problem and prove that the 2-adaptable knapsack problem is not at all approximable.André Chasseinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5097Tue, 12 Dec 2017 08:18:05 +0100Duty Rostering for Physicians at a Department of Orthopedics and Trauma Surgery
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5081
This paper presents a case study of duty rostering for physicians at a department of orthopedics and trauma surgery. We provide a detailed description of the rostering problem faced and present an integer programming model that has been used in practice for creating duty rosters at the department for more than a year. Using real world data, we compare the model output to a manually generated roster as used previously by the department and analyze the quality of the rosters generated by the model over a longer time span. Moreover, we demonstrate how unforeseen events such as absences of scheduled physicians are handled.Clemens Thielenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5081Mon, 04 Dec 2017 10:58:44 +0100Order-semi-primal lattices
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5073
Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5073Fri, 10 Nov 2017 14:43:42 +0100Representations by order-polynomially complete lattices
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5072
Bernd Kilgus; Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5072Fri, 10 Nov 2017 14:38:25 +0100Strictly order primal algebras
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5071
Otfried Lüders; Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5071Fri, 10 Nov 2017 14:33:13 +0100Pre-fixed points of polynomial functions in lattices
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5070
Marcel Erne; Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5070Fri, 10 Nov 2017 14:25:34 +0100Domain decomposition for kinetic problems with strongly contrasted Knudsen numbers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5069
A nonequilibrium situation governed by kinetic equations with strongly contrasted Knudsen numbers in different subdomains is discussed. We consider a domain decomposition problem for Boltzmann- and Euler equations, establish the correct coupling conditions and prove the validity of the obtained coupled solution . Moreover numerical examples comparing different types of coupling conditions are presented.Axel Klarreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5069Fri, 10 Nov 2017 14:20:20 +0100Projective resolutions associated to projections
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5066
Theo de Jong; Duco van Stratenreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5066Fri, 10 Nov 2017 13:00:48 +0100Conjugated operatorideals and the \(\mathcal{A}-\)Local reflexivity principle
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5065
Frank Oertelreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5065Fri, 10 Nov 2017 12:54:35 +0100Regularized approximation methods with perturbations for ill-posed operator equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5064
We are concerned with a parameter choice strategy for the Tikhonov regularization \((\tilde{A}+\alpha I)\tilde{x}\) = T* \(\tilde{y}\)+ w where \(\tilde{A}\) is a (not necessarily selfadjoint) approximation of T*T and T*\(\tilde y\)+ w is a perturbed form of the (not exactly computed) term T*y. We give conditions for convergence and optimal convergence rates.M. Thamban Nair; Eberhard Schockreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5064Fri, 10 Nov 2017 11:57:22 +0100On the Convergence at lnfinity of Solutions with Finite Dirichlet Integral to the Exterior Dirichlet Problem for the Steady Plane Navier-Stokes System of Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5061
Dan Socolescureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5061Fri, 10 Nov 2017 09:55:59 +0100Polynomial functions of modular lattices
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5060
A polynomial function \(f : L \to L\) of a lattice \(\mathcal{L}\) = \((L; \land, \lor)\) is generated by the identity function id \(id(x)=x\) and the constant functions \(c_a (x) = a\) (for every \(x \in L\)), \(a \in L\) by applying the operations \(\land, \lor\) finitely often. Every polynomial function in one or also in several variables is a monotone function of \(\mathcal{L}\).
If every monotone function of \(\mathcal{L}\)is a polynomial function then \(\mathcal{L}\) is called orderpolynomially complete. In this paper we give a new characterization of finite order-polynomially lattices. We consider doubly irreducible monotone functions and point out their relation to tolerances, especially to central relations. We introduce chain-compatible lattices
and show that they have a non-trivial congruence if they contain a finite interval and an infinite chain. The consequences are two new results. A modular lattice \(\mathcal{L}\) with a finite interval is order-polynomially complete if and only if \(\mathcal{L}\) is finite projective geometry. If \(\mathcal{L}\) is simple modular lattice of infinite length then every nontrivial interval is of infinite length and has the same cardinality as any other nontrivial interval of \(\mathcal{L}\). In the last sections we show the descriptive power of polynomial functions of
lattices and present several applications in geometry.Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5060Fri, 10 Nov 2017 09:47:24 +0100On derived varieties
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5059
Derived varieties play an essential role in the theory of hyperidentities. In [11] we have shown that derivation diagrams are a useful tool in the analysis of derived algebras and varieties. In this paper this tool is developed further in order to use it for algebraic constructions of derived algebras. Especially the operator \(S\) of subalgebras, \(H\) of homomorphic irnages and \(P\) of direct products are studied. Derived groupoids from the groupoid \(N or (x,y)\) = \(x'\wedge y'\) and from abelian groups are considered. The latter class serves as an example for fluid algebras and varieties. A fluid variety \(V\) has no derived variety as a subvariety and is introduced as a counterpart for solid varieties. Finally we use a property of the commutator of derived algebras in order to show that solvability and nilpotency are preserved under derivation.Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5059Fri, 10 Nov 2017 09:22:23 +0100The Tangent Space at a Special Sympletic Instanton Bundle on \(P_{2n+1}\)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5058
Giorgio Ottaviani; Günther Trautmannreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5058Thu, 09 Nov 2017 15:32:15 +0100Error estimates for Tikhonov regularization with unbounded regularizing operators
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5056
It is shown that Tikhonov regularization for ill- posed operator equation
\(Kx = y\) using a possibly unbounded regularizing operator \(L\) yields an orderoptimal algorithm with respect to certain stability set when the regularization parameter is chosen according to the Morozov's discrepancy principle. A more realistic error estimate is derived when the operators \(K\) and \(L\) are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also the estimates available under the Hilbert scale approach.M. Thamban Nairreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5056Thu, 09 Nov 2017 12:01:16 +0100The moduli scheme M(0,2,4) over \(\mathbb{P}_3\)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5054
Rosa Maria Miro-Roig; Günther Trautmannreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5054Thu, 09 Nov 2017 11:52:42 +0100On the Variance of the Number of Pivot Steps Required by the Simplex Algorithm
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5053
The article provides an asymptotic probabilistic analysis of the variance of the number of pivot steps required by phase II of the "shadow vertex algorithm" - a parametric variant of the simplex algorithm, which has been proposed by Borgwardt [1] . The analysis is done for data which satisfy a rotationally
invariant distribution law in the \(n\)-dimensional unit ball.Karl-Heinz Küferreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5053Thu, 09 Nov 2017 11:28:09 +0100On the Variance of Additive Random Variables on Stochastic Polyhedra
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5052
Let \(a_i i:= 1,\dots,m.\) be an i.i.d. sequence taking values in \(\mathbb{R}^n\). Whose convex hull is interpreted as a stochastic polyhedron \(P\). For a special class of random variables which decompose additively relative to their boundary simplices, eg. the volume of \(P\), integral representations of their first two moments are given which lead to asymptotic estimations of variances for special "additive variables" known from stochastic approximation theory in case of rotationally symmetric distributions.Karl-Heinz Küferreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5052Thu, 09 Nov 2017 11:14:30 +0100