@misc{GoetschelWeiser2013, author = {G{\"o}tschel, Sebastian and Weiser, Martin}, title = {Lossy Compression for PDE-constrained Optimization: Adaptive Error Control}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-18575}, year = {2013}, abstract = {For the solution of optimal control problems governed by nonlinear parabolic PDEs, methods working on the reduced objective functional are often employed to avoid a full spatio-temporal discretization of the problem. The evaluation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the ad-joint equation. The state enters into the adjoint equation, requiring the storage of a full 4D data set. If Newton-CG methods are used, two additional trajectories have to be stored. To get numerical results which are accurate enough, in many case very fine discretizations in time and space are necessary, which leads to a significant amount of data to be stored and transmitted to mass storage. Lossy compression methods were developed to overcome the storage problem by reducing the accuracy of the stored trajectories. The inexact data induces errors in the reduced gradient and reduced Hessian. In this paper, we analyze the influence of such a lossy trajectory compression method on Newton-CG methods for optimal control of parabolic PDEs and design an adaptive strategy for choosing appropriate quantization tolerances.}, language = {en} } @misc{Lehmann2013, author = {Lehmann, Felix}, title = {Inexaktheit in Newton-Lagrange-Verfahren f{\"u}r Optimierungsprobleme mit Elliptischen PDGL-Nebenbedingungen}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-41972}, year = {2013}, abstract = {Bei der numerischen L{\"o}sung von Optimalsteuerungsproblemen mit elliptischen partiellen Differentialgleichungen als Nebenbedingung treten unvermeidlich Diskretisierungs- und Iterationsfehler auf. Man ist aus Aufwandsgr{\"u}nden daran interessiert die dabei entstehenden Fehler nicht sehr klein w{\"a}hlen zu m{\"u}ssen. In der Folge werden die linearisierten Nebenbedingungen in einem Composite-Step-Verfahren nicht exakt erf{\"u}llt. In dieser Arbeit wird der Einfluss dieser Ungenauigkeit auf das Konvergenzverhalten von Newton-Lagrange-Verfahren untersucht. Dabei sollen mehrere einschl{\"a}gige lokale Konvergenzresultate diskutiert werden. Anschließend wird ein konkretes Composite-Step-Verfahren formuliert, in dem die Genauigkeit der inneren Iterationsverfahren adaptiv gesteuert werden kann. Am Ende der Arbeit wird an zwei Musterproblemen die hohe {\"U}bereinstimmung der analytischen Voraussagen und der tats{\"a}chlichen Performanz der dargestellten Methoden demonstriert.}, language = {de} } @misc{GoetschelvonTycowiczPolthieretal.2013, author = {G{\"o}tschel, Sebastian and von Tycowicz, Christoph and Polthier, Konrad and Weiser, Martin}, title = {Reducing Memory Requirements in Scientific Computing and Optimal Control}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42695}, year = {2013}, abstract = {In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data.}, language = {en} } @misc{Lubkoll2011, type = {Master Thesis}, author = {Lubkoll, Lars}, title = {Optimal Control in Implant Shape Design}, pages = {67}, year = {2011}, language = {en} } @misc{Schiela2008, author = {Schiela, Anton}, title = {State constrained optimal control problems with states of low regularity}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-509}, number = {08-24}, year = {2008}, abstract = {We consider first order optimality conditions for state constrained optimal control problems. In particular we study the case where the state equation has not enough regularity to admit existence of a Slater point in function space. We overcome this difficulty by a special transformation. Under a density condition we show existence of Lagrange multipliers, which have a representation via measures and additional regularity properties.}, language = {en} } @misc{Weiser2003, author = {Weiser, Martin}, title = {Interior Point Methods in Function Space}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7578}, number = {03-35}, year = {2003}, abstract = {A primal-dual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a short step pathfollowing method is established.}, language = {en} } @misc{SchielaWeiser2005, author = {Schiela, Anton and Weiser, Martin}, title = {Superlinear Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8490}, number = {05-15}, year = {2005}, abstract = {A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global linear convergence we show, that this method converges locally almost quadratically, if the optimal solution satisfies a function space analogue to a non-degeneracy condition. In numerical experiments we observe, that a prototype implementation of our method behaves in compliance with our theoretical results.}, language = {en} } @misc{WeiserSchiela2004, author = {Weiser, Martin and Schiela, Anton}, title = {Function space interior point methods for PDE constrained optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8027}, number = {04-27}, year = {2004}, abstract = {A primal-dual interior point method for optimal control problems with PDE constraints is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed. Numerical results from an inexact continuation method applied to a model problem are shown.}, language = {en} } @phdthesis{Goetschel2015, author = {G{\"o}tschel, Sebastian}, title = {Adaptive Lossy Trajectory Compression for Optimal Control of Parabolic PDEs}, year = {2015}, abstract = {Optimal control problems governed by nonlinear, time-dependent PDEs on three-dimensional spatial domains are an important tool in many fields, ranging from engineering applications to medicine. For the solution of such optimization problems, methods working on the reduced objective functional are often employed to avoid a full spatio-temporal discretization of the problem. The evaluation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, requiring the storage of a full 4D data set. If Newton-CG methods are used, two additional trajectories have to be stored. To get numerical results that are accurate enough, in many cases very fine discretizations in time and space are necessary, leading to a significant amount of data to be stored and transmitted to mass storage. This thesis deals with the development and analysis of methods for lossy compression of such finite element solutions. The algorithms are based on a change of basis to reduce correlations in the data, combined with quantization. This is achieved by transforming the finite element coefficient vector from the nodal to the hierarchical basis, followed by rounding the coefficients to a prescribed precision. Due to the inexact reconstruction, and thus inexact data for the adjoint equation, the error induced in the reduced gradient, and reduced Hessian, has to be controlled, to not impede convergence of the optimization. Accuracy requirements of different optimization methods are analyzed, and computable error estimates for the influence of lossy trajectory storage are derived. These tools are used to adaptively control the accuracy of the compressed data. The efficiency of the algorithms is demonstrated on several numerical examples, ranging from a simple linear, scalar equation to a semi-linear system of reaction-diffusion equations. In all examples considerable reductions in storage space and bandwidth requirements are achieved, without significantly influencing the convergence behavior of the optimization methods. Finally, to go beyond pointwise error control, the hierarchical basis transform can be replaced by more sophisticated wavelet transforms. Numerical experiments indicate that choosing suitable norms for error control allows higher compression factors.}, language = {en} } @misc{ZhangWangHartmannetal.2014, author = {Zhang, Wei and Wang, Han and Hartmann, Carsten and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Applications of the cross-entropy method to importance sampling and optimal control of diffusions}, issn = {1438-0064}, doi = {10.1137/14096493X}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-49720}, year = {2014}, abstract = {We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method.}, language = {en} } @misc{SarichBanischHartmannetal.2013, author = {Sarich, Marco and Banisch, Ralf and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Markov State Models for Rare Events in Molecular Dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42420}, year = {2013}, abstract = {Rare but important transition events between long lived states are a key feature of many molecular systems. In many cases the computation of rare event statistics by direct molecular dynamics (MD) simulations is infeasible even on the most powerful computers because of the immensely long simulation timescales needed. Recently a technique for spatial discretization of the molecular state space designed to help overcome such problems, so-called Markov State Models (MSMs), has attracted a lot of attention. We review the theoretical background and algorithmic realization of MSMs and illustrate their use by some numerical examples. Furthermore we introduce a novel approach to using MSMs for the efficient solution of optimal control problems that appear in applications where one desires to optimize molecular properties by means of external controls.}, language = {en} } @misc{Schiela2007, author = {Schiela, Anton}, title = {An Interior Point Method in Function Space for the Efficient Solution of State Constrained Optimal Control Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10471}, number = {07-44}, year = {2007}, abstract = {We propose and analyse an interior point path-following method in function space for state constrained optimal control. Our emphasis is on proving convergence in function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.}, language = {en} } @misc{Schiela2007, author = {Schiela, Anton}, title = {Optimality Conditions for Convex State Constrained Optimal Control Problems with Discontinuous States}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10342}, number = {07-35}, year = {2007}, abstract = {We discuss first order optimality conditions for state constrained optimal control problems. Our concern is the treatment of problems, where the solution of the state equation is not known to be continuous, as in the case of boundary control in three space dimensions or optimal control with parabolic partial differential equations. We show existence of measure valued Lagrangian multipliers, which have just enough additional regularity to be applicable to all possibly discontinuous solutions of the state equation.}, language = {en} } @article{BorndoerferDaneckerWeiser2021, author = {Bornd{\"o}rfer, Ralf and Danecker, Fabian and Weiser, Martin}, title = {A Discrete-Continuous Algorithm for Free Flight Planning}, volume = {14}, journal = {Algorithms}, number = {1}, publisher = {MDPI}, issn = {1438-0064}, doi = {10.3390/a14010004}, pages = {4}, year = {2021}, abstract = {We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.}, language = {en} }