## 37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]

### Refine

#### Document Type

- ZIB-Report (9)

#### Language

- English (9)

#### Has Fulltext

- yes (9)

#### Is part of the Bibliography

- no (9)

#### Keywords

- System Dynamics (2)
- Adaptive Importance Sampling (1)
- Bounds Strengthening (1)
- Computation error (1)
- Girsanov (1)
- Global Optimal Control (1)
- Markov operator (1)
- Measurable state space (1)
- Metadynamics (1)
- Metastability (1)

An automatic adaptive importance sampling algorithm for molecular dynamics in reaction coordinates
(2017)

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.

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.

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.

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.

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.

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.

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.

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.

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.