Kaiserslautern - Fachbereich Mathematik
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Faculty / Organisational entity
Scaled boundary isogeometric analysis (SB-IGA) describes the computational domain by proper boundary NURBS together with a well-defined scaling center; see [5]. More precisely, we consider star convex domains whose domain boundaries correspond to a sequence of NURBS curves and the interior is determined by a scaling of the boundary segments with respect to a chosen scaling center. However, providing a decomposition into star shaped blocks one can utilize SB-IGA also for more general shapes. Even though several geometries can be described by a single patch, in applications frequently there appear multipatch structures. Whereas a C0 continuous patch coupling can be achieved relatively easily, the situation becomes more complicated if higher regularity is required. Consequently, a suitable coupling method is inevitably needed for analyses that require global C1 continuity.In this contribution we apply the concept of analysis-suitable G1 parametrizations [2] to the framework of SB-IGA for the C1 coupling of planar domains with a special consideration of the scaling center. We obtain globally C1 regular basis functions and this enables us to handle problems such as the Kirchhoff-Love plate and shell, where smooth coupling is an issue. Furthermore, the boundary representation within SB-IGA makes the method suitable for the concept of trimming. In particular, we see the possibility to extend the coupling procedure to study trimmed plates and shells.The approach was implemented using the GeoPDEs package [1] and its performance was tested on several numerical examples. Finally, we discuss the advantages and disadvantages of the proposed method and outline future perspectives.
We compute three-dimensional displacement vector fields to estimate the deformation of microstructural data sets in mechanical tests. For this, we extend the well-known optical flow by Brox et al. to three dimensions, with special focus on the discretization of nonlinear terms. We evaluate our method first by synthetically deforming foams and comparing against this ground truth and second with data sets of samples that underwent real mechanical tests. Our results are compared to those from state-of-the-art algorithms in materials science and medical image registration. By a thorough evaluation, we show that our proposed method is able to resolve the displacement best among all chosen comparison methods.
We show that every convergent power series with monomial extended Jacobian ideal is right equivalent to a Thom–Sebastiani polynomial. This solves a problem posed by Hauser and Schicho. On the combinatorial side, we introduce a notion of Jacobian semigroup ideal involving a transversal matroid. For any such ideal, we construct a defining Thom–Sebastiani polynomial. On the analytic side, we show that power series with a quasihomogeneous extended Jacobian ideal are strongly Euler homogeneous. Due to a Mather–Yau-type theorem, such power series are determined by their Jacobian ideal up to right equivalence.
The German energy mix, which provides an overview of the sources of electricity available in Germany, is changing as a result of the expansion of renewable energy sources. With this shift towards sustainable energy sources such as wind and solar power, the electricity market situation is also in flux. Whereas in the past there were few uncertainties in electricity generation and only demand was subject to stochastic uncertainties, generation is now subject to stochastic fluctuations as well, especially due to weather dependency. To provide a supportive framework for this different situation, the electricity market has introduced, among other things, the intraday market, products with half-hourly and quarter-hourly time slices, and a modified balancing energy market design. As a result, both electricity price forecasting and optimization issues remain topical.
In this thesis, we first address intraday market modeling and intraday index forecasting. To do so, we move to the level of individual bids in the intraday market and use them to model the limit order books of intraday products. Based on statistics of the modeled limit order books, we present a novel estimator for the intraday indices. Especially for less liquid products, the order book statistics contain relevant information that allows for significantly more accurate predictions in comparison to the benchmark estimator.
Unlike the intraday market, the day ahead market allows smaller companies without their own trading department to participate since it is operated as a market with daily auctions. We optimize the flexibility offer of such a small company in the day ahead market and model the prices with a stochastic multi-factor model already used in the industry. To make this model accessible for stochastic optimization, we discretize it in time and space using scenario trees. Here we present existing algorithms for scenario tree generation as well as our own extensions and adaptations. These are based on the nested distance, which measures the distance between two distributions of stochastic processes. Based on the resulting scenario trees, we apply the stochastic optimization methods of stochastic programming, dynamic programming, and reinforcement learning to illustrate in which context the methods are appropriate.
A characterisation of the spaces \({\mathcal {G}}_K\) and \({\mathcal {G}}_K'\) introduced in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) and Potthoff and Timpel (Potential Anal 4(6):637–654, 1995) is given. A first characterisation of these spaces provided in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) uses the concepts of holomorphy on infinite dimensional spaces. We, instead, give a characterisation in terms of U-functionals, i.e., classic holomorphic function on the one dimensional field of complex numbers. We apply our new characterisation to derive new results concerning a stochastic transport equation and the stochastic heat equation with multiplicative noise.
Covering edges in networks
(2019)
In this paper we consider the covering problem on a networkG=(V,E)withedgedemands. The task is to cover a subsetJ⊆Eof the edges with a minimum numberof facilities within a predefined coverage radius. We focus on both the nodal andthe absolute version of this problem. In the latter, facilities may be placed every-where in the network. While there already exist polynomial time algorithms to solvethe problem on trees, we establish a finite dominating set (i.e., a finite subset ofpoints provably containing an optimal solution) for the absolute version in generalgraphs. Complexity and approximability results are given and a greedy strategy isproved to be a (1+ln(|J|))-approximate algorithm. Finally, the different approachesare compared in a computational study.
Hajós' conjecture asserts that a simple Eulerian graph on n vertices can be decomposed into at most [(n-1)/2] cycles. The conjecture is only proved for graph classes in which every element contains vertices of degree 2 or 4. We develop new techniques to construct cycle decompositions. They work on the common neighborhood of two degree-6 vertices. With these techniques, we find structures that cannot occur in a minimal counterexample to Hajós' conjecture and verify the conjecture for Eulerian graphs of pathwidth at most 6. This implies that these graphs satisfy the small cycle double cover conjecture.
Insurance companies and banks regularly have to face stress tests performed by regulatory instances. To model their investment decision problems that includes stress scenarios, we propose the worst-case portfolio approach. Thus, the resulting optimal portfolios are already stress test prone by construction. A central issue of the worst-case portfolio approach is that neither the time nor the order of occurrence of the stress scenarios are known. Even more, there are no probabilistic assumptions regarding the occurrence of the stresses. By defining the relative worst-case loss and introducing the concept of minimum constant portfolio processes, we generalize the traditional concepts of the indifference frontier and the indifference-optimality principle. We prove the existence of a minimum constant portfolio process that is optimal for the multi-stress worst-case problem. As a main result we derive a verification theorem that provides conditions on Lagrange multipliers and nonlinear ordinary differential equations that support the construction of optimal worst-case portfolio strategies. The practical applicability of the verification theorem is demonstrated via numerical solution of various worst-case problems with stresses. There, it is in particular shown that an investor who chooses the worst-case optimal portfolio process may have a preference regarding the order of stresses, but there may also be stress scenarios where he/she is indifferent regarding the order and time of occurrence.
We present new results on standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring over a computable field K. We prove the semicontinuity of the “highest corner” in a family of ideals, parametrized by the spectrum of a Noetherian domain A. This semicontinuity is used to design a new modular algorithm for computing a standard basis of I if K is the quotient field of A. It uses the computation over the residue field of a “good” prime ideal of A to truncate high order terms in the subsequent computation over K. We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps A = ℤ and A = k[t], k any field and t a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for A = ℤ, since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system SINGULAR and we present several examples illustrating its power.