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Numerical methods for high-dimensional probability density function equations
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables
Efficient nonlinear data assimilation using synchronisation in a particle filter
Current data assimilation methods still face problems in strongly nonlinear cases. A
promising solution is a particle filter, which provides a representation of the state
probability density function (pdf) by a discrete set of particles. To allow a particle filter
to work in high-dimensional systems, the proposal density freedom is explored.We used
a proposal density from synchronisation theory, in which one tries to synchronise the
model with the true evolution of a system using one-way coupling, via the observations.
This is done by adding an extra term to the model equations that will control the
growth of instabilities transversal to the synchronisation manifold. In this paper, an
efficient ensemble-based synchronisation scheme is used as a proposal density in the
implicit equal-weights particle filter, a particle filter that avoids filter degeneracy by
construction. Tests using the Lorenz96 model for a 1000-dimensional system show
successful results, where particles efficiently follow the truth, both for observed and
unobserved variables. These first test show that the new method is comparable to
and slightly outperforms a well-tuned Local Ensemble Transform Kalman Filter. This
methodology is a promising solution for high-dimensional nonlinear problems in the
geosciences, such as numerical weather prediction
The instanton method and its numerical implementation in fluid mechanics
A precise characterization of structures occurring in turbulent fluid flows
at high Reynolds numbers is one of the last open problems of classical physics.
In this review we discuss recent developments related to the application of
instanton methods to turbulence. Instantons are saddle point configurations of
the underlying path integrals. They are equivalent to minimizers of the related
Freidlin-Wentzell action and known to be able to characterize rare events in
such systems. While there is an impressive body of work concerning their
analytical description, this review focuses on the question on how to compute
these minimizers numerically. In a short introduction we present the relevant
mathematical and physical background before we discuss the stochastic Burgers
equation in detail. We present algorithms to compute instantons numerically by
an efficient solution of the corresponding Euler-Lagrange equations. A second
focus is the discussion of a recently developed numerical filtering technique
that allows to extract instantons from direct numerical simulations. In the
following we present modifications of the algorithms to make them efficient
when applied to two- or three-dimensional fluid dynamical problems. We
illustrate these ideas using the two-dimensional Burgers equation and the
three-dimensional Navier-Stokes equations
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