## Conference on Complex Systems, 19-22 September 2016, Amsterdam,

**Satellite session: Estimation of probability density functions in noisy complex flows,**

**Organizers:** Fred Wubs, Sven Baars, Henk Dijkstra

**Scheduled:** In the afternoon of September 20

**Keywords:** nonlinear dynamics, bifurcation analysis, high-performance computing, stochastic partial differential equations, uncertainty estimation.

**Background**
In the modeling of many fluid flows, such as those in the climate system, small-scale processes are often represented as noise. Due to nonlinear processes, many complex phenomena such as emergence of patterns and rapid transitions can occur in these flows and one would like to having estimates of the
full probability density function (PDF) of the flow. Another example appears in fluid flow measurements in geometries
that can only be molded to a certain accuracy to the one really desired. Here, one could approximate the
influence of this uncertainty by introducing noise to the geometry description.

The direct way to investigate the statistics of these flows is to use ensemble simulation techniques for many initial conditions to determine an estimate of the PDF of several observables of the flows. Other methods which have been suggested use some form of model reduction. For example, a stochastic Galerkin technique forms the basis of the Dynamical Orthogonal Field method whereas non-Markovian reduced models obtained by projecting on a basis of eigenvectors of the underlying deterministic system have also been used. The disadvantage of these methods is that it is cumbersome to vary the parameters, since for every new parameter one has to redo the whole simulation.

Meanwhile dynamical systems analysis (DSA) of these flows has developed into an interesting alternative to transient simulation for deterministic problems. In addition to the computation of trajectories for certain parameter values, DSA focusses on the direct computation of asymptotic sets (attracting forward and backward in time) versus parameters. The simplest of these sets are (stable and unstable) fixed points (steady states) and periodic orbits, which play a major role in organizing transition behavior in many fluid flows, such as Rayleigh-Benard-Marangoni convection and the Taylor-Couette flow. Though varying parameters is much easier than with transient approaches the computation of a PDF in a noisy flow is in general not an easy task. However, in a recently suggested deterministic-stochastic continuation method (Kuehn, 2011) the results from fixed points in a deterministic model can be used to obtain information on the PDF of the stochastically forced system under small noise. The price of this is that generalized Lyapunov equations have to be solved.

**Scope, objectives and benefits to participants** In this session we want to focus on
S(tochastic)PDEs describing fluid motions for which certain processes have been represented
stochastically or for which the forcing of the flow has stochastic properties. The aim is to bring
together researchers that study transitions of such flows and develop tools to do so. Since the
flow model is of high dimension these tools should be suited for high performance computers.
In this way we hope to pinpoint the state of the art, gain new insights, foster cooperations,
and to introduce junior researchers into the subject.

**Current contributors: ** Christian Kuehn, Valeria Simoncini, Sven Baars, Alexis Tantet, Daan Crommelin

**Submitting a contribution:**
There are still a few slots open in this session. If you like to contribute then, before July 5, send a half-page A4-size abstract in PDF to f.w.wubs at rug.nl. At last on July 10, you will be informed whether your contribution will be admitted.