ETH Zurich

Over million year time scales, the evolution and deformation of rocks on Earth can be described by the equations governing the motion of a very viscous, incompressible fluid. In this regime, the rocks within the crust and mantle lithosphere exhibit both brittle and ductile behaviour. Collectively, these rheologies result in an effective viscosity which is non-linear and may exhibit extremely large variations in space. In the context of geodynamics applications, we are interested in studying large deformation processes both prior and post to the onset of material failure.

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Here I introduce a hybrid finite element (FE) - Lagrangian marker discretisation which has been specifically designed to enable the numerical simulation of geodynamic processes. In this approach, a mixed FE formulation is used to discretise the incompressible Stokes equations, whilst the markers are used to discretise the material lithology.

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First I will show the a priori error estimates associated with this hybrid discretisation and demonstrate the convergence characteristics via several numerical examples. Then I will discuss several multi-level preconditioning strategies for the saddle point problem which are robust with respect to both large variations in viscosity and the underlying topological structure of the viscosity field.

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Finally, I will describe an extension of the multi-level preconditioning strategy that enables high-resolution, three-dimensional simulations to be performed with a small memory footprint and which is performant on multi-core, parallel architectures.

Crystalline solids are descibed by a material manifold endowed

with a certain structure which we call crystalline. This is characterized by

a canonical 1-form, the integral of which on a closed curve in the material manifold

represents, in the continuum limit, the sum of the Burgers vectors of all the dislocation lines

enclosed by the curve. In the case that the dislocation distribution is uniform, the material manifold

becomes a Lie group upon the choice of an identity element. In this talk crystalline solids

with uniform distributions of the two elementary kinds of dislocations, edge and screw dislocations,

shall be considered. These correspond to the two simplest non-Abelian Lie groups, the affine group

and the Heisenberg group respectively. The statics of a crystalline solid are described in terms of a

mapping from the material domain into Euclidean space. The equilibrium configurations correspond

to mappings which minimize a certain energy integral. The static problem is solved in the case of

a small density of dislocations.

Absence of arbitrage is a highly desirable feature in mathematical models of financial markets. In its pure form (whether as NFLVR or as the existence of a variant of an equivalent martingale measure R), it is qualitative and therefore robust towards equivalent changes of the underlying reference probability (the "real-world" measure P). But what happens if we look at more quantitative versions of absence of arbitrage, where we impose for instance some integrability on the density dR/dP? To which extent is such a property robust towards changes of P? We discuss these uestions and present some recent results.

The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton).

We will give a quick overview of the semigroup perspective on splitting schemes for S(P)DEs which present a robust, "easy to implement" numerical method for calculating the expected value of a certain payoff of a stochastic process driven by a S(P)DE. Having a high numerical order of convergence enables us to replace the Monte Carlo integration technique by alternative, faster techniques. The numerical order of splitting schemes for S(P)DEs is bounded by 2. The technique of combining several splittings using linear combinations which kills some additional terms in the error expansion and thus raises the order of the numerical method is called the extrapolation. In the presentation we will focus on a special extrapolation of the Lie-Trotter splitting: the symmetrically weighted sequential splitting, and its subsequent extrapolations. Using the semigroup technique their convergence will be investigated. At the end several applications to the S(P)DEs will be given.

We consider the approximation of the marginal distribution of solutions of stochastic partial differential equations by splitting schemes. We introduce a functional analytic framework based on weighted spaces where the Feller condition generalises. This allows us to apply the theory of strongly continuous semigroups. The possibility of achieving higher orders of convergence through cubature approximations is discussed.

Applications of these results to problems from mathematical finance (the Heath-Jarrow-Morton equation of interest rate theory) and fluid dynamics (the stochastic Navier-Stokes equations) are considered. Numerical experiments using Quasi-Monte Carlo simulation confirm the practicality of our algorithms.

Parts of this work are joint with J. Teichmann and D. Veluscek.

The vacant set is the set of vertices not visited by a random walk on a graph G before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the `thermodynamic limit'

when the graph grows), and the relation of the problem to the random interlacement percolation.  I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.

Semidefinite programming (SDP) techniques have been extremely successful

in many practical engineering design questions. In several of these

applications, the problem structure is invariant under the action of

some symmetry group, and this property is naturally inherited by the

underlying optimization. A natural question, therefore, is how to

exploit this information for faster, better conditioned, and more

reliable algorithms. To this effect, we study the associative algebra

associated with a given SDP, and show the striking advantages of a

careful use of symmetries. The results are motivated and illustrated

through applications of SDP and sum of squares techniques from networked

control theory, analysis and design of Markov chains, and quantum

information theory.

A random planar graph $P_n$ is a graph drawn uniformly at random from the class of all (labelled) planar graphs on $n$ vertices. In this talk we show that with probability $1-o(1)$ the number of vertices of degree $k$ in $P_n$ is very close to a quantity $d_k n$ that we determine explicitly. Here $k=k(n) \le c \log n$. In the talk our main emphasis will be on the techniques for proving such results. (Joint work with Kosta Panagiotou.)

Recently, Liang, Lyons and Qian developed a new methodology for the study of backward stochastic differential equations (BSDEs) on general filtered probability spaces. Their approach is based on the analysis of a particular class of functional differential equations, where the driver of the equation does not depend only on the present, but also on the terminal value of the solution.

The purpose of this work is to study fully coupled systems of forward functional differential equations, which are related to a broad class of fully coupled forward-backward stochastic dynamics with respect to general filtrations. In particular, these systems of functional differential equations have a more homogeneous structure with respect to the underlying forward-backward problems, allowing to partly avoid the conflicting nature between the forward and backward components.

Another advantage of the approach is that its generality allows to consider many other types of forward-backward equations not treated in the classical literature: this is shown with the help of several examples, which have interesting applications to mathematical finance and are related to parabolic integro-partial differential equations. In the second part of the talk, we introduce a numerical scheme for the approximation of decoupled systems, based on a time discretization combined with a local iteration approach.