@misc{Weber2017, author = {Weber, Marcus}, title = {Eigenvalues of non-reversible Markov chains - A case study}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-62191}, year = {2017}, abstract = {Finite reversible Markov chains are characterized by a transition matrix P that has real eigenvalues and pi-orthogonal eigenvectors, where pi is the stationary distribution of P. This means, that a transition matrix with complex eigenvalues corresponds to a non-reversible Markov chain. This observation leads to the question, whether the imaginary part of that eigendecomposition corresponds to or indicates the "pattern" of the nonreversibility. This article shows that the direct relation between imaginary parts of eigendecompositions and the non-reversibility of a transition matrix is not given. It is proposed to apply the Schur decomposition of P instead of the eigendecomposition in order to characterize its nonreversibility.}, language = {en} } @misc{Winkelmann2016, author = {Winkelmann, Stefanie}, title = {Markov Control with Rare State Observation: Average Optimality}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60981}, year = {2016}, abstract = {This paper investigates the criterion of long-term average costs for a Markov decision process (MDP) which is not permanently observable. Each observation of the process produces a fixed amount of \textit{information costs} which enter the considered performance criterion and preclude from arbitrarily frequent state testing. Choosing the \textit{rare} observation times is part of the control procedure. In contrast to the theory of partially observable Markov decision processes, we consider an arbitrary continuous-time Markov process on a finite state space without further restrictions on the dynamics or the type of interaction. Based on the original Markov control theory, we redefine the control model and the average cost criterion for the setting of information costs. We analyze the constant of average costs for the case of ergodic dynamics and present an optimality equation which characterizes the optimal choice of control actions and observation times. For this purpose, we construct an equivalent freely observable MDP and translate the well-known results from the original theory to the new setting.}, language = {en} } @misc{WinkelmannSchuette2016, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {The spatiotemporal master equation: approximation of reaction-diffusion dynamics via Markov state modeling}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60999}, year = {2016}, abstract = {Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest. We consider the spatiotemporal chemical master equation (ST-CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. The space of motion is decomposed into metastable compartments and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by the Gillespie method. We present the theory of Markov state models (MSM) as a theoretical foundation of this intuitive approach. By means of Markov state modeling, both the number and shape of compartments and the transition rates between them can be determined. We consider the ST-CME for two reaction-diffusion systems and compare it to more detailed models. Moreover, a rigorous formal justification of the ST-CME by Galerkin projection methods is presented.}, language = {en} } @article{GutscheMaeusleBurger2016, author = {Gutsche, Philipp and M{\"a}usle, Raquel and Burger, Sven}, title = {Locally Enhanced and Tunable Optical Chirality in Helical Metamaterials}, volume = {3}, journal = {Photonics}, arxiv = {http://arxiv.org/abs/1611.07748}, doi = {10.3390/photonics3040060}, pages = {60}, year = {2016}, language = {en} } @inproceedings{GutscheMaeusleBurger2016, author = {Gutsche, Philipp and M{\"a}usle, Raquel and Burger, Sven}, title = {Tailoring local optical chirality in helical metamaterials}, booktitle = {2016 10th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics}, doi = {10.1109/MetaMaterials.2016.7746440}, pages = {73 -- 75}, year = {2016}, language = {en} } @misc{WilsonHerrmannGutscheetal.2016, author = {Wilson, Jon and Herrmann, Sven and Gutsche, Philipp and Burger, Sven and McPeak, Kevin}, title = {The Chiral Coefficient: Rapid Optimization of Broadband Plasmonic Chirality}, journal = {2016 MRS Fall Meeting \& Exhibit}, year = {2016}, language = {en} } @misc{KlebanovSikorskiSchuetteetal.2016, author = {Klebanov, Ilja and Sikorski, Alexander and Sch{\"u}tte, Christof and R{\"o}blitz, Susanna}, title = {Empirical Bayes Methods, Reference Priors, Cross Entropy and the EM Algorithm}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61230}, year = {2016}, abstract = {When estimating a probability density within the empirical Bayes framework, the non-parametric maximum likelihood estimate (NPMLE) usually tends to overfit the data. This issue is usually taken care of by regularization - a penalization term is subtracted from the marginal log-likelihood before the maximization step, so that the estimate favors smooth solutions, resulting in the so-called maximum penalized likelihood estimation (MPLE). The majority of penalizations currently in use are rather arbitrary brute-force solutions, which lack invariance under transformation of the parameters(reparametrization) and measurements. This contradicts the principle that, if the underlying model has several equivalent formulations, the methods of inductive inference should lead to consistent results. Motivated by this principle and using an information-theoretic point of view, we suggest an entropy-based penalization term that guarantees this kind of invariance. The resulting density estimate can be seen as a generalization of reference priors. Using the reference prior as a hyperprior, on the other hand, is argued to be a poor choice for regularization. We also present an insightful connection between the NPMLE, the cross entropy and the principle of minimum discrimination information suggesting another method of inference that contains the doubly-smoothed maximum likelihood estimation as a special case.}, language = {en} } @misc{KlebanovSikorskiSchuetteetal.2016, author = {Klebanov, Ilja and Sikorski, Alexander and Sch{\"u}tte, Christof and R{\"o}blitz, Susanna}, title = {Empirical Bayes Methods for Prior Estimation in Systems Medicine}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/1612.01403}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61307}, year = {2016}, abstract = {One of the main goals of mathematical modelling in systems medicine related to medical applications is to obtain patient-specific parameterizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Therefore, before applying Bayes' rule separately to the data of each patient (which is typically performed using a non-informative prior), it is meaningful to use empirical Bayes methods in order to construct an informative prior from all available data. We compare the performance of four priors - a non-informative prior and priors chosen by nonparametric maximum likelihood estimation (NPMLE), by maximum penalized lilelihood estimation (MPLE) and by doubly-smoothed maximum likelihood estimation (DS-MLE) - by applying them to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modelling.}, language = {en} } @inproceedings{JaegerHammerschmidtKoeppeletal.2016, author = {J{\"a}ger, Klaus and Hammerschmidt, Martin and K{\"o}ppel, Grit and Burger, Sven and Becker, Christiane}, title = {On Accurate Simulations of Thin-Film Solar Cells With a Thick Glass Superstrate}, booktitle = {Light, Energy and the Environment 2016}, doi = {10.1364/PV.2016.PM3B.5}, pages = {PM3B.5}, year = {2016}, language = {en} } @inproceedings{GutscheMaeusleBurger2016, author = {Gutsche, Philipp and M{\"a}usle, Raquel and Burger, Sven}, title = {Circular polarization phenomena in chiral nano-optical devices}, booktitle = {Light, Energy and the Environment 2016}, doi = {10.1364/FTS.2016.JW4A.4}, pages = {JW4A.4}, year = {2016}, language = {en} } @article{HartmannSchuetteWeberetal.2017, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Weber, Marcus and Zhang, Wei}, title = {Importance sampling in path space for diffusion processes with slow-fast variables}, journal = {Probability Theory and Related Fields}, doi = {10.1007/s00440-017-0755-3}, pages = {1 -- 52}, year = {2017}, abstract = {Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.}, language = {en} } @misc{Sali2016, author = {Sali, Adrian}, title = {Coupling of Monodomain and Eikonal Models for Cardiac Electrophysiology}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60508}, year = {2016}, abstract = {The primary goal of this paper is to study the coupling of monodomain and eikonal models for the numerical simulation of cardiac electrophysiology. Eikonal models are nonlinear elliptic equations describing the excitation time of the cardiac tissue. They are often used as very fast approximations for monodomain or bidomain models - parabolic reaction-diffusion systems describing the excitation wavefront in terms of ionic currents. The excitation front is a thin region with high gradients, whereas excitation times vary over larger domains. Hence, eikonal equations can be solved on much coarser grids than monodomain equations. Moreover, as eikonal models are not time-dependent, no time integration is needed. Eikonal models are derived from monodomain models making additional assumptions and using certain approximations. While generally the approximation is rather good, several specific situations are not well captured by eikonal models. We consider coupling the two models, i.e. using the monodomain model in regions where more accurate results or the shape of the wavefront are needed, and the eikonal model in the remaining parts of the domain, where the excitation time is sufficient. Restricting the monodomain simulation to a small subdomain reduces the computational effort considerably. Numerical methods for the simulation of the individual models are presented, with the finite element method as the main ingredient. Coupling conditions as well as algorithms for implementing the coupling are explained. The approximation quality and efficiency of the coupled model is illustrated on simple geometries using an Aliev-Panfilov membrane model.}, language = {en} } @misc{Sali2016, type = {Master Thesis}, author = {Sali, Adrian}, title = {Coupling of Monodomain and Eikonal Models for Cardiac Electrophysiology}, pages = {63}, year = {2016}, abstract = {The primary goal of this paper is to study the coupling of monodomain and eikonal models for the numerical simulation of cardiac electrophysiology. Eikonal models are nonlinear elliptic equations describing the excitation time of the cardiac tissue. They are often used as very fast approximations for monodomain or bidomain models - parabolic reaction-diffusion systems describing the excitation wavefront in terms of ionic currents. The excitation front is a thin region with high gradients, whereas excitation times vary over larger domains. Hence, eikonal equations can be solved on much coarser grids than monodomain equations. Moreover, as eikonal models are not time-dependent, no time integration is needed. Eikonal models are derived from monodomain models making additional assumptions and using certain approximations. While generally the approximation is rather good, several specific situations are not well captured by eikonal models. We consider coupling the two models, i.e. using the monodomain model in regions where more accurate results or the shape of the wavefront are needed, and the eikonal model in the remaining parts of the domain, where the excitation time is sufficient. Restricting the monodomain simulation to a small subdomain reduces the computational effort considerably. Numerical methods for the simulation of the individual models are presented, with the finite element method as the main ingredient. Coupling conditions as well as algorithms for implementing the coupling are explained. The approximation quality and efficiency of the coupled model is illustrated on simple geometries using an Aliev-Panfilov membrane model.}, language = {en} } @misc{deLassonFrandsenBurgeretal.2016, author = {de Lasson, Jakob Rosenkrantz and Frandsen, Lars Hagedorn and Burger, Sven and Gutsche, Philipp and Kim, Oleksiy S. and Breinbjerg, Olav and Sigmund, Ole and Mork, Jesper and Gregersen, Niels}, title = {Comparison of four computational methods for computing Q factors and resonance wavelengths in photonic crystal membrane cavities}, journal = {Proceedings of the 7th International Conference on Metamaterials, Photonic Crystals and Plasmonics}, year = {2016}, language = {en} } @article{VillatoroZuehlkeRiebeetal.2016, author = {Villatoro, Jos{\´e} and Z{\"u}hlke, Martin and Riebe, Daniel and Beitz, Toralf and Weber, Marcus and Riedel, Jens and L{\"o}hmannsr{\"o}ben, Hans-Gerd}, title = {IR-MALDI ion mobility spectrometry: physical source characterization and application as HPLC detector}, journal = {International Journal for Ion Mobility Spectrometry}, number = {19/4}, doi = {10.1007/s12127-016-0208-1}, pages = {197 -- 297}, year = {2016}, language = {en} } @article{WeiserGhosh2018, author = {Weiser, Martin and Ghosh, Sunayana}, title = {Theoretically optimal inexact SDC methods}, journal = {Communications in Applied Mathematics and Computational Science}, number = {13-1}, doi = {10.2140/camcos.2018.13.53}, pages = {53 -- 86}, year = {2018}, abstract = {In several inital value problems with particularly expensive right hand side evaluation or implicit step computation, there is a trade-off between accuracy and computational effort. We consider inexact spectral deferred correction (SDC) methods for solving such initial value problems. SDC methods are interpreted as fixed point iterations and, due to their corrective iterative nature, allow to exploit the accuracy-work-tradeoff for a reduction of the total computational effort. On one hand we derive error models bounding the total error in terms of the evaluation errors. On the other hand, we define work models describing the computational effort in terms of the evaluation accuracy. Combining both, a theoretically optimal local tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance. The properties of optimal local tolerances and the predicted efficiency gain compared to simpler heuristics, and a reasonable practical performance, are illustrated on simple numerical examples.}, language = {en} } @article{LedentsovJrShchukinKroppetal.2017, author = {Ledentsov, Jr., Nikolay and Shchukin, Vitaly and Kropp, J{\"o}rg-R. and Zschiedrich, Lin and Schmidt, Frank and Ledentsov, Jr., Nikolay}, title = {Resonant cavity light-emitting diodes based on dielectric passive cavity structures}, volume = {10124}, journal = {Proc. SPIE}, doi = {10.1117/12.2253089}, pages = {101240O}, year = {2017}, language = {en} } @article{Lange2016, author = {Lange, Alexander}, title = {Reconstruction of disease transmission rates: applications to measles, dengue, and influenza}, volume = {400}, journal = {Journal of Theoretical Biology}, doi = {10.1016/j.jtbi.2016.04.017}, pages = {138 -- 153}, year = {2016}, language = {en} } @article{BauerGribovaLangeetal.2017, author = {Bauer, Gernot and Gribova, Nadezhda and Lange, Alexander and Holm, Christian and Gross, Joachim}, title = {Three-body effects in triplets of capped gold nanocrystals}, volume = {115}, journal = {Molecular Physics}, doi = {10.1080/00268976.2016.1213909}, pages = {1031 -- 1040}, year = {2017}, language = {en} } @inproceedings{LangePloentzkeSchaeferetal.2016, author = {Lange, Alexander and Pl{\"o}ntzke, Julia and Sch{\"a}fer, Stefan and R{\"o}blitz, Susanna}, title = {Follicular maturation in cows: mathematical models and data}, booktitle = {10. European Conference on Mathematical and Theoretical Biology}, year = {2016}, language = {en} }