University of Warwick

Abstract: Non-geometric rough paths arise
when one encounters stochastic integrals for which the the classical
integration by parts formula does not hold. We will introduce two notions of
non-geometric rough paths - one old (branched rough paths) and one new (quasi
geometric rough paths). The former (due to Gubinelli) assumes one knows nothing
about products of integrals, instead those products must be postulated as new
components of the rough path. The latter assumes one knows a bit about
products, namely that they satisfy a natural generalisation of the
"Ito" integration by parts formula. We will show why they are both
reasonable frameworks for a large class of integrals. Moreover, we will show
that Ito's formula can be derived in either framework and that this derivation
is completely algebraic. Finally, we will show that both types of non-geometric
rough path can be re-written as geometric rough paths living above an extended
version of the original path. This means that every non-geometric rough
differential equation can be re-written as a geometric rough differential
equation, hence generalising the Ito-Stratonovich correction formula.

Abstract: We consider the inverse problem of recovering u from a noisy, indirect observation We adopt a Bayesian approach, in which the aim is to determine the posterior distribution _y on the unknown u, given some prior information about u in the form of a prior distribution _0,together with the observation y. We are interested in the question of posterior consistency, which is the characterization of the behaviour of _y as more data become available. We work in a separable Hilbert space X, assuming a Gaussian prior _0 = N(0; _ 2C0). The theory is developed using two concrete problems: i) a family of linear inverse problems in which we want to _nd u from y where y = A

Unstable dynamical systems can be stabilized, and hence the solution

recovered from noisy data, provided two conditions hold. First, observe

enough of the system: the unstable modes. Second, weight the observed

data sufficiently over the model. In this talk I will illustrate this for the

3DVAR filter applied to three dissipative dynamical systems of increasing

dimension: the Lorenz 1963 model, the Lorenz 1996 model, and the 2D

Navier-Stokes equation.

I will report on joint work in progress with David Aldous, concerning a curious random metric space on the plane which can be constructed with the help of an improper Poisson line process.

We study the asymptotic behavior of deterministic dynamics of many interacting particles with random initial data in the limit where the number of particles tends to infinity. A famous example is hard sphere flow, we restrict our attention to the simpler case where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density $f_0(u,v)$ depending on $d$-dimensional position $u$ and velocity $v$. In the Boltzmann Grad scaling, we derive the validity of a Boltzmann equation without gain term for arbitrary long times, when we assume finiteness of moments up to order two and initial data that are $L^\infty$ in space. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods.

The Discontinuous Galerkin (DG) method has been used to solve a wide range of partial differential equations. Especially for advection dominated problems it has proven very reliable and accurate. But even for elliptic problems it has advantages over continuous finite element methods, especially when parallelization and local adaptivity are considered.

In this talk we will first present a variation of the compact DG method for elliptic problems with varying coefficients. For this method we can prove stability on general grids providing a computable bound for all free parameters. We developed this method to solve the compressible Navier-Stokes equations and demonstrated its efficiency in the case of meteorological problems using our implementation within the DUNE software framework, comparing it to the operational code COSMO used by the German weather service.

After introducing the notation and analysis for DG methods in Euclidean spaces, we will present a-priori error estimates for the DG method on surfaces. The surface finite-element method with continuous ansatz functions was analysed a few years ago by Dzuik/Elliot; we extend their results to the interior penalty DG method where the non-smooth approximation of the surface introduces some additional challenges.

This talk will consist of a pure PDE part, and an applied part. The unifying topic is mean curvature flow (MCF), and particularly mean curvature flow starting at cones. This latter subject originates from the abstract consideration of uniqueness questions for flows in the presence of singularities. Recently, this theory has found applications in several quite different areas, and I will explain the connections with Harnack estimates (which I will explain from scratch) and also with the study of the dynamics of charged fluid droplets.

There are essentially no prerequisites. It would help to be familiar with basic submanifold geometry (e.g. second fundamental form) and intuition concerning the heat equation, but I will try to explain everything and give the talk at colloquium level.

Joint work with Sebastian Helmensdorfer.

We establish a large deviations principle for the block sizes of a uniformly random non-crossing partition. As an application we obtain a variational formula for the maximum of the support of a compactly supported probability measure in terms of its free cumulants, provided these are all non-negative. This is useful in free probability theory, where sometimes the R-transform is known but cannot be inverted explicitly to yield the density.

There is a growing interest in the study of periodic phenomena in

large-scale nonlinear dynamical systems. Often the high-dimensional

system has only low-dimensional dynamics, e.g., many reaction-diffusion

systems or Navier-Stokes flows at low Reynolds number. We present an

approach that exploits this property in order to compute branches of

periodic solutions of the large system of ordinary differential

equations (ODEs) obtained after a space discretisation of the PDE. We

call our approach the Newton-Picard method. Our method is based on the

recursive projection method (RPM) of Shroff and Keller but extends this

method in many different ways. Our technique tries to combine the

performance of straightforward time integration with the advantages of

solving a nonlinear boundary value problem using Newton's method and a

direct solver. Time integration works well for very stable limit

cycles. Solving a boundary value problem is expensive, but works also

for unstable limit cycles.

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We will present some background material on RPM. Next we will explain

the basic features of the Newton-Picard method for single shooting. The

linearised system is solved by a combination of direct and iterative

techniques. First, we isolate the low-dimensional subspace of unstable

and weakly stable modes (using orthogonal subspace iteration) and

project the linearised system on this subspace and on its

(high-dimensional) orthogonal complement. In the high-dimensional

subspace we use iterative techniques such as Picard iteration or GMRES.

In the low-dimensional (but "hard") subspace, direct methods such as

Gaussian elimination or a least-squares are used. While computing the

projectors, we also obtain good estimates for the dominant,

stability-determining Floquet multipliers. We will present a framework

that allows us to monitor and steer the convergence behaviour of the

method.

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RPM and the Newton-Picard technique have been developed for PDEs that

reduce to large systems of ODEs after space discretisation. In fact,

both methods can be applied to any large system of ODEs. We will

indicate how these methods can be applied to the discretisation of the

Navier-Stokes equations for incompressible flow (which reduce to an

index-2 system of differential-algebraic equations after space

discretisation when written in terms of velocity and pressure.)

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The Newton-Picard method has already been extended to the computation

of bifurcation points on paths of periodic solutions and to multiple

shooting. Extension to certain collocation and finite difference

techniques is also possible.