TY - GEN A1 - Wolf, Thomas A1 - Efimovskaya, Olga T1 - On integrability of the Kontsevich non-abelian ODE system N2 - We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich. We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries. T3 - ZIB-Report - 11-24 KW - integrability, Lax pairs, noncommutative ODE, Laurent ODE Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13113 ER - TY - GEN A1 - Wolf, Thomas A1 - Schrüfer, Eberhard A1 - Webster, Kenneth T1 - Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs N2 - The paper reports on a computer algebra program {\sc LSSS} (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M.\ Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14. T3 - ZIB-Report - 12-32 KW - linear algebraic systems, integrability, Lax pairs, noncommutative ODE, Laurent ODE Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16102 SN - 1438-0064 ER - TY - GEN A1 - Vierhaus, Ingmar A1 - Fügenschuh, Armin A1 - Gottwald, Robert Lion A1 - Grösser, Stefan N. T1 - Modern Nonlinear Optimization Techniques for an Optimal Control of System Dynamics Models N2 - We study System Dynamics models with several free parameters that can be altered by the user. We assume that the user's goal is to achieve a certain dynamic behavior of the model by varying these parameters. In order to the find best possible combination of parameter settings, several automatic parameter tuning methods are described in the literature and readily available within existing System Dynamic software packages. We give a survey on the available techniques in the market and describe their theoretical background. Some of these methods are already six decades old, and meanwhile newer and more powerful optimization methods have emerged in the mathematical literature. One major obstacle for their direct use are tabled data in System Dynamics models, which are usually interpreted as piecewise linear functions. However, modern optimization methods usually require smooth functions which are twice continuously differentiable. We overcome this problem by a smooth spline interpolation of the tabled data. We use a test set of three complex System Dynamic models from the literature, describe their individual transition into optimization problems, and demonstrate the applicability of modern optimization algorithms to these System Dynamics Optimization problems. T3 - ZIB-Report - 14-08 KW - System Dynamics KW - Optimal Control KW - Nonlinear Optimization KW - Spline Interpolation Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-48159 SN - 1438-0064 ER - TY - GEN A1 - Fügenschuh, Armin A1 - Vierhaus, Ingmar T1 - A Global Approach to the Optimal Control of System Dynamics Models N2 - The System Dynamics (SD) methodology is a framework for modeling and simulating the dynamic behavior of socioeconomic systems. Characteristic for the description of such systems is the occurrence of feedback loops together with stocks and flows. The mathematical equations that describe the system are usually ordinary differential equations and nonlinear algebraic constraints. Therefore seemingly simple systems can show a nonintuitive, unpredictable behavior over time. Controlling a dynamical system means to specify potential interventions from outside that should keep the system on the desired track, and to define an evaluation schema to compare different controls among each other, so that a "best" control can be defined in a meaningful way. The central question is how to compute such globally optimal control for a given SD model, that allows the transition of the system into a desired state with minimum effort. We propose a mixed-integer nonlinear programming (MINLP) reformulation of the System Dynamics Optimization (SDO) problem. MINLP problems can be solved by linear programming based branch-and-bound approach. We demonstrate that standard MINLP solvers are not able to solve SDO problem. To overcome this obstacle, we introduce a special-tailored bound propagation method. We apply our new method to a predator-prey model with additional hunting activity as control, and to a mini-world model with the consumption level as control. Numerical results for these test cases are presented. T3 - ZIB-Report - 13-28 KW - System Dynamics KW - Global Optimal Control KW - Mixed-Integer Nonlinear Optimization KW - Bounds Strengthening Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-18600 SN - 1438-0064 ER - TY - GEN A1 - Quer, Jannes A1 - Donati, Luca A1 - Keller, Bettina A1 - Weber, Marcus T1 - An automatic adaptive importance sampling algorithm for molecular dynamics in reaction coordinates N2 - In this article we propose an adaptive importance sampling scheme for dynamical quantities of high dimensional complex systems which are metastable. The main idea of this article is to combine a method coming from Molecular Dynamics Simulation, Metadynamics, with a theorem from stochastic analysis, Girsanov's theorem. The proposed algorithm has two advantages compared to a standard estimator of dynamic quantities: firstly, it is possible to produce estimators with a lower variance and, secondly, we can speed up the sampling. One of the main problems for building importance sampling schemes for metastable systems is to find the metastable region in order to manipulate the potential accordingly. Our method circumvents this problem by using an assimilated version of the Metadynamics algorithm and thus creates a non-equilibrium dynamics which is used to sample the equilibrium quantities. T3 - ZIB-Report - 17-09 KW - Adaptive Importance Sampling KW - Molecular Dynamics KW - Metastability KW - Variance Reduction KW - Non Equilibrium Sampling KW - Metadynamics KW - Girsanov Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-62075 SN - 1438-0064 ER - TY - GEN A1 - Nielsen, Adam T1 - The Monte Carlo Computation Error of Transition Probabilities N2 - In many applications one is interested to compute transition probabilities of a Markov chain. This can be achieved by using Monte Carlo methods with local or global sampling points. In this article, we analyze the error by the difference in the $L^2$ norm between the true transition probabilities and the approximation achieved through a Monte Carlo method. We give a formula for the error for Markov chains with locally computed sampling points. Further, in the case of reversible Markov chains, we will deduce a formula for the error when sampling points are computed globally. We will see that in both cases the error itself can be approximated with Monte Carlo methods. As a consequence of the result, we will derive surprising properties of reversible Markov chains. T3 - ZIB-Report - 16-37 KW - Reversible Markov chain KW - Monte Carlo methods KW - Computation error KW - Measurable state space KW - Markov operator Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59933 SN - 1438-0064 ER - TY - GEN A1 - Anco, Stephen A1 - Mohiuddin, Mohammad A1 - Wolf, Thomas T1 - Travelling waves and conservation laws for complex mKdV-type equations N2 - Travelling waves and conservation laws are studied for a wide class of $U(1)$-invariant complex mKdV equations containing the two known integrable generalizations of the ordinary (real) mKdV equation. The main results on travelling waves include deriving new complex solitary waves and kinks that generalize the well-known mKdV $\sech$ and $\tanh$ solutions. The main results on conservation laws consist of explicitly finding all 1st order conserved densities that yield phase-invariant counterparts of the well-known mKdV conserved densities for momentum, energy, and Galilean energy, and a new conserved density describing the angular twist of complex kink solutions. T3 - ZIB-Report - 12-34 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16124 SN - 1438-0064 ER - TY - GEN A1 - Anco, Stephen A1 - MacNaughton, Steven A. A1 - Wolf, Thomas T1 - Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions N2 - Symmetries and conservation laws are studied for two classes of physically and analytically interesting radial wave equations with power nonlinearities in multi-dimensions. The results consist of two main classifications: all symmetries of point type and all conservation laws of a general energy-momentum type are explicitly determined, including those such as dilations, inversions, similarity energies and conformal energies that exist only for special powers or dimensions. In particular, all variational cases (when a Lagrangian formulation exists) and non-variational cases (when no Lagrangian exists) for these wave equations are considered. As main results, the classification yields generalized energies and radial momenta in certain non-variational cases, which are shown to arise from a new type of Morawetz dilation identity that produces conservation laws for each of the two classes of wave equations in a different way than Noether's theorem. T3 - ZIB-Report - 12-33 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16111 SN - 1438-0064 ER - TY - GEN A1 - Nielsen, Adam A1 - Fackeldey, Konstantin A1 - Weber, Marcus T1 - On a Generalized Transfer Operator N2 - We introduce a generalized operator for arbitrary stochastic processes by using a pre-kernel, which is a generalization of the Markov kernel. For deterministic processes, such an operator is already known as the Frobenius-Perron operator, which is defined for a large class of measures. For Markov processes, there exists transfer operators being only well defined for stationary measures in $L^2$. Our novel generalized transfer operator is well defined for arbitrary stochastic processes, in particular also for deterministic ones. We can show that this operator is acting on $L^1$. For stationary measures, this operator is also an endomorphism of $L^2$ and, therefore, allows for a mathematical analysis in Hilbert spaces. T3 - ZIB-Report - 13-74 KW - Transfer Operator KW - Pre Kernel KW - Perron Frobenius Generalization Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-43162 SN - 1438-0064 ER -