Kaiserslautern - Fachbereich Mathematik
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Linear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time periodicity of solutions is required to single out certain solutions. Here, we would like to make a point of allowing time to be defined on a metric graph or network where on the branching points coupling conditions are imposed such that time can have ramifications and even loops. This not only generalizes the classical setting and allows for more freedom in the modeling of coupled and interacting systems of evolution equations, but it also provides a unified framework for initial value and time-periodic problems. For these time-graph Cauchy problems questions of well-posedness and regularity of solutions for parabolic problems are studied along with the question of which time-graph Cauchy problems cannot be reduced to an iteratively solvable sequence of Cauchy problems on intervals. Based on two different approaches—an application of the Kalton–Weis theorem on the sum of closed operators and an explicit computation of a Green’s function—we present the main well-posedness and regularity results. We further study some qualitative properties of solutions. While we mainly focus on parabolic problems, we also explain how other Cauchy problems can be studied along the same lines. This is exemplified by discussing coupled systems with constraints that are non-local in time akin to periodicity.
Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system SINGULAR, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of SINGULAR, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka’s celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores.
In this paper we present the comparison of experiments and numerical simulations for bubble cutting by a wire. The air bubble is surrounded by water. In the experimental setup an air bubble is injected on the bottom of a water column. When the bubble rises and contacts the wire, it is separated into two daughter bubbles. The flow is modeled by the incompressible Navier–Stokes equations. A meshfree method is used to simulate the bubble cutting. We have observed that the experimental and numerical results are in very good agreement. Moreover, we have further presented simulation results for liquid with higher viscosity. In this case the numerical results are close to previously published results.
Papadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the
Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time
solvability of a certain auxiliary problem determines the class of multiobjective optimization
problems that admit a polynomial-time computable (1+ε, . . . , 1+ε)-approximate Pareto set
(also called an ε-Pareto set). Similarly, in this article, we characterize the class ofmultiobjective
optimization problems having a polynomial-time computable approximate ε-Pareto set
that is exact in one objective by the efficient solvability of an appropriate auxiliary problem.
This class includes important problems such as multiobjective shortest path and spanning
tree, and the approximation guarantee we provide is, in general, best possible. Furthermore,
for biobjective optimization problems from this class, we provide an algorithm that computes
a one-exact ε-Pareto set of cardinality at most twice the cardinality of a smallest such set and
show that this factor of 2 is best possible. For three or more objective functions, however,
we prove that no constant-factor approximation on the cardinality of the set can be obtained
efficiently.
An important ingredient of any moving-mesh method for fluid-structure interaction (FSI) problems is the mesh moving technique (MMT) used to adapt the computational mesh in the moving fluid domain. An ideal MMT is computationally inexpensive, can handle large mesh motions without inverting mesh elements and can sustain an FSI simulation for extensive periods of time without irreversibly distorting the mesh. Here we compare several commonly used MMTs which are based on the solution of elliptic partial differential equations, including harmonic extension, bi-harmonic extension and techniques based on the equations of linear elasticity. Moreover, we propose a novel MMT which utilizes ideas from continuation methods to efficiently solve the equations of nonlinear elasticity and proves to be robust even when the mesh undergoes extreme motions. In addition to that, we study how each MMT behaves when combined with the mesh-Jacobian-based stiffening. Finally, we evaluate the performance of different MMTs on a popular two-dimensional FSI benchmark reproduced by using an isogeometric partitioned solver with strong coupling.
This article is dedicated to the weight set decomposition of a multiobjective (mixed-)integer linear problem with three objectives. We propose an algorithm that returns a decomposition of the parameter set of the weighted sum scalarization by solving biobjective subproblems via Dichotomic Search which corresponds to a line exploration in the weight set. Additionally, we present theoretical results regarding the boundary of the weight set components that direct the line exploration. The resulting algorithm runs in output polynomial time, i.e. its running time is polynomial in the encoding length of both the input and output. Also, the proposed approach can be used for each weight set component individually and is able to give intermediate results, which can be seen as an “approximation” of the weight set component. We compare the running time of our method with the one of an existing algorithm and conduct a computational study that shows the competitiveness of our algorithm. Further, we give a state-of-the-art survey of algorithms in the literature.
In a (linear) parametric optimization problem, the objective value of each feasible solution is an affine function of a real-valued parameter and one is interested in computing a solution for each possible value of the parameter. For many important parametric optimization problems including the parametric versions of the shortest path problem, the assignment problem, and the minimum cost flow problem, however, the piecewise linear function mapping the parameter to the optimal objective value of the corresponding non-parametric instance (the optimal value function) can have super-polynomially many breakpoints (points of slope change). This implies that any optimal algorithm for such a problem must output a super-polynomial number of solutions. We provide a method for lifting approximation algorithms for non-parametric optimization problems to their parametric counterparts that is applicable to a general class of parametric optimization problems. The approximation guarantee achieved by this method for a parametric problem is arbitrarily close to the approximation guarantee of the algorithm for the corresponding non-parametric problem. It outputs polynomially many solutions and has polynomial running time if the non-parametric algorithm has polynomial running time. In the case that the non-parametric problem can be solved exactly in polynomial time or that an FPTAS is available, the method yields an FPTAS. In particular, under mild assumptions, we obtain the first parametric FPTAS for each of the specific problems mentioned above and a (3/2 + ε) -approximation algorithm for the parametric metric traveling salesman problem. Moreover, we describe a post-processing procedure that, if the non-parametric problem can be solved exactly in polynomial time, further decreases the number of returned solutions such that the method outputs at most twice as many solutions as needed at minimum for achieving the desired approximation guarantee.
Various regulatory initiatives (such as the pan-European PRIIP-regulation or the German chance-risk classification for state subsidized pension products) have been introduced that require product providers to assess and disclose the risk-return profile of their issued products by means of a key information document. We will in this context outline a concept for a (forward-looking) simulation-based approach and highlight its application and advantages. For reasons of comparison, we further illustrate the performance of approximation methods based on a projection of observed returns into the future such as the Cornish–Fisher expansion or bootstrap methods.
Seit 1993 veranstaltet der Fachbereich Mathematik der TU Kaiserslautern jährlich die mathematischen Modellierungswochen. Die Veranstaltung erwuchs parallel zu der steigenden Relevanz angewandter mathematischer Forschungsgebiete, wie der Techno- und der Wirtschaftsmathematik. Sie soll dazu dienen, Schülerinnen und Schülern die Bedeutung mathematischer Arbeitsweisen in der heutigen Berufswelt, insbesondere in Industrie und Wirtschaft, begreifbar zu machen. Darüber hinaus bietet die Modellierungswoche den teilnehmenden Lehrkräften einen Einblick in die Projektarbeit mit offenen Fragestellungen im Rahmen der mathematischen Modellierung. In diesem Report beschreiben wir die Projekte, die während der Modellierungswoche im Dezember 2022 durchgeführt wurden.
Das MINT-EC-Girls-Camp: Math-Talent-School richtet sich an mathematikbegeisterte Schülerinnen von MINT-EC-Schulen, die Einblicke in die Berufswelt von Mathematikerinnen und Mathematikern bekommen möchten. Die Veranstaltung veranschaulicht den Schülerinnen die steigende Relevanz angewandter mathematischer Forschungsgebiete, wie der Techno- und der Wirtschaftsmathematik. Sie soll dazu dienen, Schüler:innen die Bedeutung mathematischer Arbeitsweisen in der heutigen Berufswelt, insbesondere in Industrie und Wirtschaft, begreifbar zu machen. Die Talent-School wird organisiert von MINT-EC und dem Felix-Klein-Zentrum für Mathematik. Die fachwissenschaftliche Betreuung der Schülerinnen während dieser Talent-School wurde durch Mitarbeitende des Kompetenzzentrums für Mathematische Modellierung in MINT-Projekten in der Schule (KOMMS) der TU Kaiserslautern und des Fraunhofer ITWM umgesetzt. In diesem Report beschreiben wir die Projekte, die während der Talent-School im Oktober 2022 durchgeführt wurden.