TY - GEN A1 - Winkelmann, Stefanie A1 - Schütte, Christof T1 - The spatiotemporal master equation: approximation of reaction-diffusion dynamics via Markov state modeling N2 - 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. T3 - ZIB-Report - 16-60 KW - reaction-diffusion KW - stochastic chemical kinetics KW - chemical master equation Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60999 SN - 1438-0064 ER - TY - GEN A1 - Winkelmann, Stefanie T1 - Markov Control with Rare State Observation: Average Optimality N2 - 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. T3 - ZIB-Report - 16-59 KW - Markov decision process KW - partial observability KW - average optimality KW - information costs Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60981 SN - 1438-0064 ER - TY - GEN A1 - Schwartz, Stephan A1 - Borndörfer, Ralf A1 - Bartz, Gerald T1 - The Graph Segmentation Problem N2 - We investigate a graph theoretical problem arising in the automatic billing of a network toll. Given a network and a family of user paths, we study the graph segmentation problem (GSP) to cover parts of the user paths by a set of disjoint segments. The GSP is shown to be NP-hard but for special cases it can be solved in polynomial time. We also show that the marginal utility of a segment is bounded. Computational results for real-world instances show that in practice the problem is more amenable than the theoretic bounds suggest. T3 - ZIB-Report - 16-53 KW - toll billing KW - graph segmentation KW - combinatorial optimization Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60750 SN - 1438-0064 ER - TY - GEN A1 - Noack, Matthias A1 - Wende, Florian A1 - Zitzlsberger, Georg A1 - Klemm, Michael A1 - Steinke, Thomas T1 - KART – A Runtime Compilation Library for Improving HPC Application Performance N2 - The effectiveness of ahead-of-time compiler optimization heavily depends on the amount of available information at compile time. Input-specific information that is only available at runtime cannot be used, although it often determines loop counts, branching predicates and paths, as well as memory-access patterns. It can also be crucial for generating efficient SIMD-vectorized code. This is especially relevant for the many-core architectures paving the way to exascale computing, which are more sensitive to code-optimization. We explore the design-space for using input-specific information at compile-time and present KART, a C++ library solution that allows developers to compile, link, and execute code (e.g., C, C++ , Fortran) at application runtime. Besides mere runtime compilation of performance-critical code, KART can be used to instantiate the same code multiple times using different inputs, compilers, and options. Other techniques like auto-tuning and code-generation can be integrated into a KART-enabled application instead of being scripted around it. We evaluate runtimes and compilation costs for different synthetic kernels, and show the effectiveness for two real-world applications, HEOM and a WSM6 proxy. T3 - ZIB-Report - 16-48 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60730 SN - 1438-0064 ER - TY - GEN A1 - Weiser, Martin A1 - Ghosh, Sunayana T1 - Theoretically optimal inexact SDC methods N2 - In several inital value problems with particularly expensive right hand side computation, there is a trade-off between accuracy and computational effort in evaluating the right hand sides. We consider inexact spectral deferred correction (SDC) methods for solving such non-stiff 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 an error model bounding the total error in terms of the right hand side 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 tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance. T3 - ZIB-Report - 16-52 KW - Spectral deferred correction, initial value problems, error propagation, adaptive control of tolerances Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-53140 SN - 1438-0064 ER - TY - GEN A1 - Pulaj, Jonad T1 - Cutting Planes for Families Implying Frankl's Conjecture N2 - We find previously unknown families which imply Frankl’s conjecture using an algorithmic framework. The conjecture states that for any non-empty union-closed (or Frankl) family there exists an element in at least half of the sets. Poonen’s Theorem characterizes the existence of weights which determine whether a given Frankl family implies the conjecture for all Frankl families which contain it. A Frankl family is Non–Frankl-Complete (Non–FC), if it does not imply the conjecture in its elements for some Frankl family that contains it. We design a cutting-plane method that computes the explicit weights which imply the existence conditions of Poonen’s Theorem. This method allows us to find a counterexample to a ten-year-old conjecture by R. Morris about the structure of generators for Non–FC-families. T3 - ZIB-Report - 16-51 KW - extremal combinatorics, extremal set theory, cutting plane, exact integer programming Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60626 SN - 1438-0064 ER - TY - GEN A1 - Sali, Adrian T1 - Coupling of Monodomain and Eikonal Models for Cardiac Electrophysiology N2 - 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. T3 - ZIB-Report - 16-50 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60508 SN - 1438-0064 ER - TY - GEN A1 - Fischer, Frank A1 - Grimm, Boris A1 - Klug, Torsten A1 - Schlechte, Thomas T1 - A Re-optimization Approach for Train Dispatching N2 - The Train Dispatching Problem (TDP) is to schedule trains through a network in a cost optimal way. Due to disturbances during operation existing track allocations often have to be re-scheduled and integrated into the timetable. This has to be done in seconds and with minimal timetable changes to guarantee smooth and conflict free operation. We present an integrated modeling approach for the re-optimization task using Mixed Integer Programming. Finally, we provide computational results for scenarios provided by the INFORMS RAS Problem Soling Competition 2012. T3 - ZIB-Report - 16-49 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60493 SN - 1438-0064 ER - TY - GEN A1 - Rehfeldt, Daniel A1 - Koch, Thorsten A1 - Maher, Stephen J. T1 - Reduction Techniques for the Prize-Collecting Steiner Tree Problem and the Maximum-Weight Connected Subgraph Problem N2 - The concept of reduction has frequently distinguished itself as a pivotal ingredient of exact solving approaches for the Steiner tree problem in graphs. In this paper we broaden the focus and consider reduction techniques for three Steiner problem variants that have been extensively discussed in the literature and entail various practical applications: The prize-collecting Steiner tree problem, the rooted prize-collecting Steiner tree problem and the maximum-weight connected subgraph problem. By introducing and subsequently deploying numerous new reduction methods, we are able to drastically decrease the size of a large number of benchmark instances, already solving more than 90 percent of them to optimality. Furthermore, we demonstrate the impact of these techniques on exact solving, using the example of the state-of-the-art Steiner problem solver SCIP-Jack. T3 - ZIB-Report - 16-47 KW - Steiner tree problems KW - reduction techniques KW - prize-collecting Steiner tree problem KW - maximum-weight connected subgraph problem Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60420 SN - 1438-0064 ER - TY - GEN A1 - Tesch, Alexander T1 - Exact Energetic Reasoning in O(n^2 log^2 n) N2 - In this paper, we address the Energetic Reasoning propagation rule for the Cumulative Scheduling Problem (CuSP). An energetic reasoning propagation algorithm is called exact, if it computes the maximum possible energetic reasoning propagation for all the jobs. The currently best known exact energetic reasoning algorithm has complexity O(n^3). In this paper, we present a new exact energetic reasoning propagation algorithm with improved complexity of O(n^2 \log^2 n). T3 - ZIB-Report - 16-46 KW - Cumulative Scheduling KW - Energetic Reasoning KW - Sweep Line KW - Constraint Programming Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60367 SN - 1438-0064 N1 - Preprint Version ER -