Kaiserslautern - Fachbereich Mathematik
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- Abstract ODE (1)
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We consider a Darcy flow model with saturation-pressure relation extended
with a dynamic term, namely, the time derivative of the saturation.
This model was proposed in works of J.Hulshof and J.R.King (1998), S.M.Hassanizadeh and W.G.Gray (1993),
F.Stauffer (1978).
We restrict ourself to one spatial dimension and strictly positive
initial saturation. For this case we transform the initial-boundary value
problem into combination of elliptic boundary-value problem and initial
value problem for abstract Ordinary Differential Equation. This splitting
is rather helpful both for theoretical aspects and numerical methods.
The goal of a multicriteria program is to explore different possibilities and their respective compromises which adequately represent the nondominated set. An exact description will in most cases fail because the number of efficient solutions is either too large or even infinite. We approximate the nondominated by computing a finite collection of nondominated points. Different ideas have been applied, including nonnegative weighted scalarization, Tchebycheff weighted scalarization, block norms and epsilon-constraints. Block norms are the building blocks for the inner and outer approximation algorithms proposed by Klamroth. We review these algorithms and propose three different variants. However, block norm based algorithms require to solve a sequence of subproblems, the number of subproblems becomes relatively high for six criteria and even intractable for real applications with nine criteria. Thus, we use bilevel linear programming to derive an approximation algorithm. We finally analyze and compare the approximation quality, running time and numerical convergence of the proposed methods.
Zerlegungen und Parkettierungen der Ebene spielen in vielen wissenschaftlichen, praktischen und künstlerischen Bereichen eine wichtige Rolle. In dieser Abhandlung werden solche diskrete Systeme von Punktmengen betrachtet. Zunächst werden Packungen einfacher Figuren durch Polyominos, diskrete Zerlegungen der Ebene sowie Zerlegungen von Polygonen in Polygone behandelt. Weiterführend werden sowohl Mosaike, als auch Parkette und deren Anwendungsbeispiele vorgestellt.