@phdthesis{Schuette, author = {Sch{\"u}tte, Christof}, title = {Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4063}, number = {SC-99-18}, abstract = {The function of many important biomolecules comes from their dynamic properties and their ability to switch between different {\em conformations}. In a conformation, the large scale geometric structure of the molecule is understood to be conserved, whereas on smaller scales the system may well rotate, oscillate or fluctuate. In a recent article [J. Comp. Phys., 151,1 (1999)], the present author and coworkers demonstrated that (a) conformations can be understood as almost invariant sets of some Markov chain being defined via the Hamiltonian system governing the molecular dynamics and that (b) these sets can efficiently be computed via eigenvectors of the corresponding Markov operator. The persent manuscript reviews the mathematical modelling steps behind the novel concept, includes a rigorous analytical justification of this approach and especially of the numerical details of the algorithm, and illustrates its performance when applied to realistic molecular systems.}, language = {en} } @phdthesis{Schultz, author = {Schultz, R{\"u}diger}, title = {Structure and Stability in Two-Stage Stochastic Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1819}, number = {SC-95-15}, abstract = {{\begin{footnotesize} This thesis is concerned with structural properties and the stability behaviour of two-stage stochastic programs. Chapter~1 gives an introduction into stochastic programming and a summary of the main results of the thesis. In Chapter~2 we present easily verifiable sufficient conditions for the strong convexity of the expected-recourse function in a stochastic program with linear complete recourse. Different levels of randomness in the data are considered. We start with models where only the right-hand side of the constraints is random and extend these results to the situation where also the technology matrix contains random entries. The statements on strong convexity imply new stability estimates for sets of optimal solutions when perturbing the underlying probability measure. We work out H{\"o}lder estimates (in terms of the \$\mbox{L}_1\$-Wasserstein distance) for optimal solution sets to linear recourse models with random technology matrix. In Chapter~3 ({\it joint work with Werner R{\"o}misch, Berlin}) we are aiming at the Lipschitz stability of optimal solution sets to linear recourse models with random right-hand side. To this end , we first adapt the distance notion for the underlying probability measures to the structure of the model and derive a Lipschitz estimate for optimal solutions based on that distance. Here, the strong convexity established in Chapter~2 turns out as an essential assumption. For applications, however, a Lipschitz estimate with respect to a more accesssible probability distance is desirable. Structural properties of the expected-recourse function finally permit such an estimate in terms of the Kolmogorov-Smirnov distance of linear transforms of the underlying measures. The general analysis is specified to estimation via empirical measures. We obtain a law of iterated logarithm, a large deviation estimate and an estimate for the asymptotic distribution of optimal solution sets. Chapters~4 and~5 deal with two-stage linear stochastic programs where integrality constraints occur in the second stage. In Chapter~4 we study basic continuity properties of the expected-recourse function for models with random right-hand side and random technology matrix. The joint continuity with respect to the decision variable and the underlying probability measure leads to qualitative statements on the stability of local optimal values and local optimal solutions. In Chapter~5 we demonstrate that a variational distance of probability measures based on a suitable Vapnik-\v{C}ervonenkis class of Borel sets leads to convergence rates of the H{\"o}lder type for the expected recourse as a function of the underlying probability measure. The rates carry over to the convergence of local optimal values. As an application we again consider estimation via empirical measures. Beside qualitative asymptotic results for optimal values and optimal solutions we obtain a law of iterated logarithm for optimal values. \end{footnotesize}}}, language = {en} } @phdthesis{Schmidt, author = {Schmidt, Frank}, title = {A New Approach to Coupled Interior-Exterior Helmholtz-Type Problems: Theory and Algorithms}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7128}, number = {02-45}, abstract = {The work presents a new approach to the numerical solution of time-harmonic and time-dependent scattering problems. We replace Sommerfeld's radiation condition valid for the Helmholtz equation by a more general concept called pole condition. The pole condition is based on the Laplace transform of the exterior solution and allows a characterization of outgoing waves. Both new insight into the analysis of scattering problems as well as new numerical algorithms are obtained.}, language = {en} } @phdthesis{Reich, author = {Reich, Sebastian}, title = {Dynamical Systems, Numerical Integration, and Exponentially Small Estimates}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3622}, number = {SC-98-19}, abstract = {Dynamical systems with two well-separated time-scales are investigated using normal form theory. Exponential estimates for the normal form truncation error are derived and applied to the numerical integration of differential equations (backward error analysis) and the reduction of highly oscillatory Hamiltonian systems (constrained dynamics and correcting potentials). The theoretical results are used to formulate new algorithms for the time integration of conservative Hamiltonian systems (projected multiple time stepping, soft constraints, rigid bodies, symplectic variable step-size methods).}, language = {en} } @phdthesis{Martin, author = {Martin, Alexander}, title = {Integer Programs with Block Structure}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3911}, number = {SC-99-03}, abstract = {In this thesis we study and solve integer programs with block structure, i.\,e., problems that after the removal of certain rows (or columns) of the constraint matrix decompose into independent subproblems. The matrices associated with each subproblem are called blocks and the rows (columns) to be removed linking constraints (columns). Integer programs with block structure come up in a natural way in many real-world applications. The methods that are widely used to tackle integer programs with block structure are decomposition methods. The idea is to decouple the linking constraints (variables) from the problem and treat them at a superordinate level, often called master problem. The resulting residual subordinate problem then decomposes into independent subproblems that often can be solved more efficiently. Decomposition methods now work alternately on the master and subordinate problem and iteratively exchange information to solve the original problem to optimality. In Part I we follow a different approach. We treat the integer programming problem as a whole and keep the linking constraints in the formulation. We consider the associated polyhedra and investigate the polyhedral consequences of the involved linking constraints. The variety and complexity of the new inequalities that come into play is illustrated on three different types of real-world problems. The applications arise in the design of electronic circuits, in telecommunication and production planning. We develop a branch-and-cut algorithm for each of these problems, and our computational results show the benefits and limits of the polyhedral approach to solve these real-world models with block structure. Part II of the thesis deals with general mixed integer programming problems, that is integer programs with no apparent structure in the constraint matrix. We will discuss in Chapter 5 the main ingredients of an LP based branch-and-bound algorithm for the solution of general integer programs. Chapter 6 then asks the question whether general integer programs decompose into certain block structures and investigate whether it is possible to recognize such a structure. The remaining two chapters exploit information about the block structure of an integer program. In Chapter 7 we parallelize parts of the dual simplex algorithm, the method that is commonly used for the solution of the underlying linear programs within a branch-and-cut algorithm. In Chapter 8 we try to detect small blocks in the constraint matrix and to derive new cutting planes that strengthen the integer programming formulation. These inequalities may be associated with the intersection of several knapsack problems. We will see that they significantly improve the quality of the general integer programming solver introduced in Chapter 5.}, language = {en} } @phdthesis{Lang, author = {Lang, Jens}, title = {Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4089}, number = {SC-99-20}, abstract = {This monograph has been written to illustrate the interlocking of theory, algorithm, and application in developing solution techniques for complex PDE systems. A deep theoretical understanding is necessary to produce a powerful idea leading to a successful algorithm. Efficient and robust implementation is the key to make the algorithm perform satisfactorily. The extra insight obtained by solving real--life problems brings out the structure of the method more clearly and suggests often ways to improve the numerical algorithm. It is my intention to impart the beauty and complexity found in both the theoretical investigation of the adaptive algorithm proposed here, i.e., the coupling of Rosenbrock methods in time and multilevel finite elements in space, and its realization. I hope that this method will find many more interesting applications.}, language = {en} } @phdthesis{Krumke, author = {Krumke, Sven}, title = {Online Optimization: Competitive Analysis and Beyond}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6925}, number = {02-25}, abstract = {Traditional optimization techniques assume, in general, knowledge of all data of a problem instance. There are many cases in practice, however, where decisions have to be made before complete information about the data is available. In fact, it may be necessary to produce a part of the problem solution as soon as a new piece of information becomes known. This is called an \emph{online situation}, and an algorithm is termed \emph{online}, if it makes a decision (computes a partial solution) whenever a new piece of data requests an action. \emph{Competitive analysis} has become a standard yardstick to measure the quality of online algorithms. One compares the solution produced by an online algorithm to that of an optimal (clairvoyant) offline algorithm. An online algorithm is called \$c\$-competitive if on every input the solution it produces has cost'' at most \$c\$~times that of the optimal offline algorithm. This situation can be imagined as a game between an online player and a malicious adversary. Although competitive analysis is a worst-case analysis and henceforth pessimistic, it often allows important insights into the problem structure. One can obtain an idea about what kind of strategies are promising for real-world systems and why. On the other hand there are also cases where the offline adversary is simply too powerful and allows only trivial competitiveness results. This phenomenon is called hitting the triviality barrier''. We investigate several online problems by means of competitive analysis. We also introduce new concepts to overcome the weaknesses of the standard approach and to go beyond the triviality barrier.}, language = {en} } @phdthesis{Helmberg, author = {Helmberg, Christoph}, title = {Semidefinite Programming for Combinatorial Optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6022}, number = {00-34}, abstract = {This book offers a self-contained introduction to the field of semidefinite programming, its applications in combinatorial optimization, and its computational methods. We equip the reader with the basic results from linear algebra on positive semidefinite matrices and the cone spanned by them. Starting from linear programming, we introduce semidefinite programs and discuss the associated duality theory. We then turn to semidefinite relaxations of combinatorial optimization and illustrate their interrelation. In the second half we deal with computational methods for solving semidefinite programs. First, the interior point approach, its iteration complexity, and implementational issues are discussed. Next, we explain in great detail the spectral bundle method, which is particularly suited for large scale semidefinite programming. One of the most successful techniques in integer linear programming is the cutting plane approach which improves an initial relaxation by adding violated inequalities. We explore possibilities to combine the two solution methods with the cutting plane approach in order to strengthen semidefinite relaxations of combinatorial optimization problems.}, language = {en} } @phdthesis{Gatermann, author = {Gatermann, Karin}, title = {Computer Algebra Methods for Equivariant Dynamical Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4140}, number = {SC-99-26}, abstract = {An introductory chapter on Groebner bases is given which also includes new results on the detection of Groebner bases for sparse polynomial systems. Algorithms for the computation of invariants and equivariants for finite groups, compact Lie groups and algebraic groups are presented and efficient implementation and time comparision are discussed. This chapter also inlcudes improvements of the computation of Noether normalisation and Stanley decomposition. These results are applied in symmetric bifurcation theory and equivariant dynamics. As preparation of the investigation of the orbit space reduction three methods are compared for solving symmetric polynomial systems exactly. The method of orbit space reduction is improved by using the Cohen-Macaulayness of the invariant ring and nested Noether normalization. Finally this is applied for a case of mode interaction in the Taylor-Couette problem.}, language = {en} } @phdthesis{Bornemann, author = {Bornemann, Folkmar A.}, title = {Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3170}, number = {SC-97-48}, abstract = {We present a particular method for the explicit elimination of rapidly oscillating micro-scales in certain singularly perturbed conservative mechanical systems. Non-linear effects call for a non-trivial averaging procedure that we call {\em homogenization in time.} This method is based on energy principles and weak convergence techniques. Since non-linear functionals are in general {\em not} weakly sequentially continuous, we have to study {\em simultaneously} the weak limits of all those non-linear quantities of the rapidly oscillating components which are of importance for the underlying problem. Using the physically motivated concepts of {\em virial theorems}, {\em adiabatic invariants}, and {\em resonances}, we will be able to establish sufficiently many relations between all these weak limits, allowing to calculate them explicitly. Our approach will be {\em paradigmatical} rather than aiming at the largest possible generality. This way, we can show most clearly how concepts and notions from the physical background of the underlying mathematical problem enter and help to determine relations between weak limit quantities. In detail we will discuss natural mechanical systems with a strong constraining potential on Riemannian manifolds, the questions of realization of holonomic constraints, and singular limits of mixed quantum-classical coupling models. This latter class of problems also leads to a new proof for the adiabatic theorem of quantum mechanics. The strength of our methodology will be illustrated by applications to problems from plasma physics, molecular dynamics and quantum chemistry.}, language = {en} }