Overview of the short course:

This short course discusses a new approach for deriving stochastic fluid equations which describe the slow large-scale characteristics of GFD without having to resolve the small fast scales accurately via very costly high-resolution direct numerical simulations. Instead, we discuss parametrising the small fast scales by using a new approach based on the concept of stochastic transport, rather than stochastic diffusion.

Stochastic advection by Lie transport (SALT) -- Darryl D Holm

In this course, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the \emph{stochastic Euler--Poincar\'{e}} and \emph{stochastic Navier-Stokes--Poincar\'{e}} equations respectively. The stochastic Euler--Poincar\'{e} equations were previously derived from a stochastic variational principle by Holm in

(Holm, D. D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471.2176 (2015): 20140963)

which we will briefly review.

Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Stochastic modeling under location uncertainty (LU)-- Etienne M\'emin

In this lecture, we will describe a formalism to systematically derive large-scale stochastic representations of fluid flows dynamics which take into account the inherent uncertainty attached to the flow evolution. The uncertainty introduced here is described through a random field, and aims at representing principally the small-scale effects that are neglected in the large-scale evolution model. The resulting large-scale dynamics is built from a stochastic representation of the Reynolds transport theorem. This formalism enables, in the very same way as in the deterministic case, a physically relevant derivation (i.e. from the usual conservation law) of the sought evolution laws. We will in particular show how to derive stochastic representations of geophysical flow dynamics and reduced order stochastic dynamical systems. We will give several examples of computational simulations obtained from such systems and how they can be used in different contexts.

Particle Filters for Data Assimilation -- Dan Crisan

Particle filters are a set of probabilistic algorithms used to solve filtering problems arising in signal processing and Bayesian statistical inference. Their area of applicability is currently being extended to solve high dimensional problems such as those encountered in data assimilation problems for numerical weather prediction. The lecture will contain an elementary introduction to particle filters with emphasis on their applicability to such problems. I will discuss the specific difficulties encountered when applying particle filters to high dimensional problems as well as procedures required for their successful implementation. I will cover model reduction (high to low resolution), tempering, jittering, uncertainly quantification and initialization.

We will explain how stochastic transport rather than diffusion provides the balance between spread and accuracy that is needed for data assimilation method using particle filtering to be successful.

The running example covered in the lectures will be an application to a partially observed solution of a damped and driven incompressible 2D Euler equation with stochastic advection by Lie transport (SALT).

**Public information**:

Stochastic advection by Lie transport (SALT) -- Darryl D Holm

In this section, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler--Poincare and stochastic Navier-Stokes--Poincare equations respectively. The stochastic Euler--Poincare equations were previously derived from a stochastic variational principle by Holm in (Holm, D. D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471.2176 (2015): 20140963) which we will briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Stochastic modeling under location uncertainty -- Etienne Memin

In this section, we will describe a formalism to systematically derive large-scale stochastic representations of fluid flows dynamics which take into account the inherent uncertainty attached to the flow evolution. The uncertainty introduced here is described through a random field, and aims at representing principally the small-scale effects that are neglected in the large-scale evolution model. The resulting large-scale dynamics is built from a stochastic representation of the Reynolds transport theorem. This formalism enables, in the very same way as in the deterministic case, a physically relevant derivation (i.e. from the usual conservation law) of the sought evolution laws. We will in particular show how to derive stochastic representations of geophysical flow dynamics and reduced order stochastic dynamical systems. We will give several examples of computational simulations obtained from such systems and how they can be used in different contexts.

Particle Filters for Data Assimilation -- Dan Crisan

Particle filters are a set of probabilistic algorithms used to solve filtering problems arising in signal processing and Bayesian statistical inference. Their area of applicability is currently being extended to solve high dimensional problems such as those encountered in data assimilation problems for numerical weather prediction. The section will contain a brief introduction to particle filters with emphasis on their applicability to such problems. I will discuss the specific difficulties encountered when applying particle filters to high dimensional problems as well as procedures required for their successful implementation. I will cover model reduction (high to low resolution), tempering, jittering, uncertainly quantification and initialization.

I will also explain how stochastic transport rather than diffusion provides the balance between spread and accuracy that is needed for data assimilation method using particle filtering to be successful. Two examples will be covered in the lectures: one to a partially observed solution of a damped and driven incompressible 2D Euler equation and one to a partially observed solution of a two-layer quasi-geostrophic equation both with stochastic advection by Lie transport (SALT).