All seminars (unless otherwise stated) will take place on **Tuesdays at 3.00pm in ****Room ****Physics A1/3** (Physics Building,** **Gower Street). See the map for further details. There will be tea afterwards in Mathematics Room 606 (25 Gordon Street). If you require any more information on the Applied seminars please contact Prof Slava Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

## 4 October 2016

### Berangere Delourme (Paris 13 University, France)

##### Title: Homogenization of a thin perforated wall of finite length

##### Abstract:

This talk deals with the resolution of a scattering problem in a domain made of a thin and periodic layer of finite length placed into an homogeneous medium. The presence of this thin periodic layer of holes is responsible for the appearance of two different kinds of singular behaviors. First, a highly oscillatory boundary layer appears in the vicinity of the periodic layer. Additionaly, since the thin periodic layer has a finite length, corners singularities come up in the neighborhood of its extremities. Based on an approach mixing matched asymptotic expansions and (surface) periodic homogenization, we provide and justify a high order asymptotic expansion which takes into account these two phenomena. Numerical experiments are carried out to illustrate the method.

## 11 October 2016 - cancelled

### Prof Ivan Graham (University of Bath)

##### Title: High dimensional problems arising from PDE with random coefficients

##### Abstract:

We consider non-uniformly elliptic problems with coefficients given as lognormal random fields. We focus on the forward problem of assessing how uncertainty propagates from data to solution, which leads to very high-dimensional parametrised systems of PDEs. We combine the fast realisation of data via circulant embedding techniques with quasi-Monte Carlo methods for dealing with high dimension. We prove rates of convergence independent of dimension and illustrate the results on some problems motivated by flow in porous media.

This is joint work with Rob Scheichl (Bath) and Frances Kuo and Ian Sloan (New South Wales)

## 18 October 2016

### Dr Rhodri Nelson (Imperial College of London)

##### Title: Potential problems in multiply-connected domains

##### Abstract:

The talk will begin by briefly introducing the Schottky-Klein Prime (SKP) function and following this a new and efficient method for computing this function and its derivatives will be presented. This SKP function is then used as the basis of a calculus for solving potential problems in multiply-connected domains. Some examples, including the motion of vortex patches in multiply-connected domains and waterflooding in an oil resevoir, are then given.

## 25 October 2016

Please see the Departmental Colloquia webpage

## 1 November 2016

### Dr Hanne Kekkonen (Warwick University)

##### Title: Bayesian and frequentist uncertainty quantification for inverse problems

##### Abstract:

The aim of the talk is to discuss connections between inverse problems and the emerging field of uncertainty quantification. Uncertainty quantification is necessary in inverse problems to assess statistical reliability of the obtained solutions. Ill-posedness of the underlying model generates challenges that are not typically considered in classical statistics literature. In complex parameter spaces, such as those encountered in inverse problems, calculating frequentist confidence regions can be an almost impossible task, whereas Bayesian uncertainty quantification is often computationally cheap. The problem is that the theoretical and objective meaning of such posterior based inferences is largely unclear. On the other hand frequentist uncertainty quantification is well understood and studied in traditional statistics.

## 8 November 2016

### Dr. Davit Harutyunyan (Swiiss Federal Institute of Technology)

##### Title: Towards characterization of all 3 \times 3 extremal quasiconvex quadratic forms

##### Abstract:

A quasiconvex quardatic form defined on the set of all N times n matrices is en extremal if it looses its quasiconvexity whenever a rank-one form is subtracted from it. In the case N=2 or n=2 it is known that any quasi convex quadratic corm is polyconvex, thus the only extremals are the Null-Lagrangians. If n, N>=3, then the study of extremals was widely open and even there was no extremal known (other then the Null-Lagrangians) until 2013. We find a link between the extremality of quadratic forms and extremality of the acoustic tensor determinant of the form as a polynomial. These extremals are believed to derive a new complete theory of bounds on the effective properties of composites (as proposed by Milton, 2013) as did (sometimes providing not optimal bounds) the Null-Lagangians. We also believe they can help construct new examples of functions that are rank-one convex but not quasiconvex and thus providing a possible approach to Morrey's conjecture.

This is joint work with Grame Milton (University of Utah).

## 15 November 2016

### Prof Jonathan Healey (Keelee University)

##### Title: On the mechanisms, and control, of boundary layer instability

##### Abstract:

The stability of laminar boundary layers has been investigated intensively for a long time, and remains central to the fundamental problem of predicting and controlling transition to turbulence. And yet some aspects are not easily explained, like why does viscosity destabilize flows? We present a new interpretation of the Orr-Sommerfeld equation (which describes shear layer instabilities) as a coupled oscillator system, and show how it leads to suggestions for flow modifications with strong stabilizing properties, even when the modifications create inflexion points.

## 22 November 2016

### Dr. Bernhard Scheichl (Technische Universität Wien)

##### Title: Short-to Long-Scale Interaction in Weakly Viscous Super- to Transcritical Liquid-Layer Flows

##### Abstract:

We consider a thin liquid film past a horizontal plate under the action of gravity acting vertically, surface tension, and relatively low viscosity. A manifold of intriguing phenomena arise given the three disparate length scales involved: distance from layer by jet impingement driving the layer to the trailing edge of the plate (long), height of the film (short), and, under transcritical conditions, an intrinsic intermediate one. The steady free overfall serves as a paradigm for triggering the destabilising effect of viscosity on the short scale upstream. In supercritical flow, this culminates in a self-sustained, localised wave crest, governed by viscous-inviscid interaction and set apart from the edge. In the transcritical limit, a generic transonic-flow singularity provokes an interactive Korteweg-de-Vries regime. Here several limits and the role of isolated surface protuberances are addressed, where so-called "marginal states" associated with weak hydraulic jumps are identified.

## 6 December 2016

### Prof Ivan Graham (University of Bath)

##### Title: High dimensional problems arising from PDE with random coefficients

##### Abstract:

We consider non-uniformly elliptic problems with coefficients given as lognormal random fields. We focus on the forward problem of assessing how uncertainty propagates from data to solution, which leads to very

high-dimensional parametrised systems of PDEs. We combine the fast realisation of data via circulant embedding techniques with quasi-Monte Carlo methods for dealing with high dimension. We prove rates ofconvergence independent of dimension and illustrate the results on some problems motivated by flow in porous media.

This is joint work with Rob Scheichl (Bath) and Frances Kuo and Ian Sloan (New South Wales).