Gibson Grd floor SR

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.

We discuss the problem of estimating a structured matrix with a large number of elements. A key motivation for this problem occurs in multi-task learning. In this case, the columns of the matrix correspond to the parameters of different regression or classification tasks, and there is structure due to relations between the tasks. We present a general method to learn the tasks' parameters as well as their structure. Our approach is based on solving a convex optimization problem, involving a data term and a penalty term. We highlight different types of penalty terms which are of practical and theoretical importance. They implement structural relations between the tasks and achieve a sparse representations of parameters. We address computational issues as well as the predictive performance of the method. Finally we discuss how these ideas can be extended to learn non-linear task functions by means of reproducing kernels.

Problems of packing shapes with maximal density, sometimes into a

container of restricted size, are classical in discrete

mathematics. We describe here the problem of packing a given set of

ellipsoids of different sizes into a finite container, in a way that

allows overlap but that minimizes the maximum overlap between adjacent

ellipsoids. We describe a bilevel optimization algorithm for finding

local solutions of this problem, both the general case and the simpler

special case in which the ellipsoids are spheres. Tools from conic

optimization, especially semidefinite programming, are key to the

algorithm. Finally, we describe the motivating application -

chromosome arrangement in cell nuclei - and compare the computational

results obtained with this approach to experimental observations.

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This talk represents joint work with Caroline Uhler (IST Austria).

{\bf This seminar is at ground floor!}

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An important question in geometry and analysis is to know when two $k-$forms

$f$ and $g$ are equivalent. The problem is therefore to find a map $\varphi$

such that%

\[

\varphi^{\ast}\left( g\right) =f.

\]

We will mostly discuss the symplectic case $k=2$ and the case of volume forms

$k=n.$ We will give some results when $3\leq k\leq n-2,$ the case $k=n-1$ will

also be considered.

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The results have been obtained in collaboration with S. Bandyopadhyay, G.

Csato and O. Kneuss and can be found, in part, in the book below.\bigskip

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Csato G., Dacorogna B. et Kneuss O., \emph{The pullback equation for

differential forms}, Birkha\"{u}ser, PNLDE Series, New York, \textbf{83} (2012).

I will discuss the motion of screw dislocations in an elastic body under antiplane shear. In this setting, dislocations are viewed as points in a two-dimensional domain where the strain field fails to be a gradient. The motion is determined by the Peach-Koehler force and the slip-planes in the material. This leads to a system of discontinuous ODE, where the vector field depends on the solution to an elliptic PDE with Neumann data. We show short-time existence of solutions; we also have uniqueness for a restricted class of domains. In general, global solutions do not exist because of collisions.

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of L\'evy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. We also provide here a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant. We find that the rate of convergence is uniformly with respect to the ``jump activity'' (e.g. characterised by the Blumenthal-Getoor index).

In this talk we will discuss the mathematical modelling of the performance of Lithium-ion batteries. A mathematical model based on a macro-homogeneous approach developed by John Neuman will be presented. The uniqueness and existence of solution of the corresponding problem will be also discussed.

The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves $W^*(X)$ on a smooth algebraic variety $X$ over a finite field, computing the cristalline cohomology of $X$. I am going to present a non-commutative generalization of this: even for a non-commutative ring $A$, one can define a functorial "Hochschild-Witt complex" with homology $WHH^*(A)$; if $A$ is commutative, then $WHH^i(A)=W^i(X)$, $X = Spec A$ (this is analogous to the isomorphism $HH^i(A)=H^i(X)$ discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

We describe our current efforts to develop finite volume

schemes for solving PDEs on logically Cartesian locally adapted

surfaces meshes. Our methods require an underlying smooth or

piecewise smooth grid transformation from a Cartesian computational

space to 3d surface meshes, but does not rely on analytic metric terms

to obtain second order accuracy. Our hyperbolic solvers are based on

Clawpack (R. J. LeVeque) and the parabolic solvers are based on a

diamond-cell approach (Y. Coudi\`ere, T. Gallou\"et, R. Herbin et

al). If time permits, I will also discuss Discrete Duality Finite

Volume methods for solving elliptic PDEs on surfaces.

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To do local adaption and time subcycling in regions requiring high

spatial resolution, we are developing ForestClaw, a hybrid adaptive

mesh refinement (AMR) code in which non-overlapping fixed-size

Cartesian grids are stored as leaves in a forest of quad- or

oct-trees. The tree-based code p4est (C. Burstedde) manages the

multi-block connectivity and is highly scalable in realistic

applications.

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I will present results from reaction-diffusion systems on surface

meshes, and test problems from the atmospheric sciences community.