Applied Mathematics Seminars Autumn 2011

All seminars (unless otherwise stated) will take place on Mondays at 3.00 pm in Room 505 which is located on the 5th floor of the Mathematics Department. See Where to Find Us for further details. There will be tea afterwards in room 606.

If you require any more information on the Applied seminars please contact Professor Yaroslav Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

03 October 2011 

Kirill Cherednichenko - University of Cardiff 

High-frequency spectral analysis of thin periodic acoustic strips

Abstract:

I shall discuss the asymptotic behaviour of the high-frequency spectrum of the wave equation with periodic coefficients in a ``thin'' elastic strip $\Sigma_\eta=(0,1)\times(-\eta/2,\eta/2),$ $\eta>0.$ The main geometric assumption is that the structure period is of the order of magnitude of the strip thickness $\eta$ and is chosen in such a way that $\eta^{-1}$ is a large positive integer. On the boundary $\partial\Sigma_\eta$ I set Dirichlet (clamped) or Neumann (traction-free) boundary conditions. Aiming to describe sequences of eigenvalues of order $\eta^{-2}$ in the above problem, which correspond to oscillations of high frequencies of order $\eta^{-1},$ I study an appropriately rescaled limit of the spectrum. Using a suitable notion of two-scale convergence for bounded operators acting on two-scale spaces, I show that the limit spectrumconsists of two parts: the Bloch (or band) spectrum and the `boundary'' spectrum. The latter corresponds to sequences of eigenvectors concentrating on the

verticalboundaries of $\Sigma_\eta$, and is characterised by a problem set in asemi-infinite periodic strip with either clamped or stress-free boundary conditions.

10 October 2011 

Paul Milewski - University of Bath 

Stability results in two layer shallow water

Abstract:

The simplest model for waves in stratified flows is that of two-layers of immiscible constant density fluid bounded by horizontal walls. We consider this problem in three-situations, with emphasis on the question of stability (or well-posedness) of the solutions: the hydrostatic Boussinesq case, the hydrostatic non-Boussinesq case and the weakly non-hydrostatic case.

17 October 2011

Nico Gray - University of Manchester 

Particle size segregation and spontaneous levee formation in geophysical mass flows

Abstract:

Hazardous geophysical mass flows, such as snow avalanches, debris-flows and pyroclastic flows, often spontaneously develop large particle rich levees that channelize the flow and enhance their run-out. The results of large scale experiments at the United States Geological Survey (USGS) debris-flow flume are used to show that the levee formation process is driven by a subtle segregation-mobility feedback effect. Simple models for particle segregation and the depth-averaged motion of granular avalanches are described and one of the first attempts is made to couple these two types of models together. This process proves to be non-trivial, yielding considerable complexity as well as pathologies that require additional physics to be included.

24 October 2011

Guillaume Bal - University of Columbia, USA

Coupled-physics Inverse Problems and Internal Functionals

Abstract:

Hybrid (coupled-physics) inverse problems aim at combining the high contrast of one imaging modality (such as e.g. Electrical Impedance Tomography or Optical Tomography in medical imaging) with the high resolution of another modality (such as e.g. based on ultrasound or magnetic resonance). Mathematically, these problems often take the form of inverse problems with internal information. This talk will review several results of uniqueness and stability obtained recently for such inverse problems.

31 October 2011

Martin Gander - University of Geneva 

Euler, Ritz, Galerkin, Courant: on the road to the finite element method

Abstract: 

The so-called Ritz-Galerkin method is one of the most fundamental tools of modern computing. Its origins lie in the variational calculus of Euler-Lagrange and in the thesis of Walther Ritz, who died just over 100 years ago at the age of 31 after a long struggle against tuberculosis. The thesis was submitted in 1902 in Goettingen, in a period of dramatic developments in Physics. Ritz tried to explain the phenomenon of Balmer series in spectroscopy using eigenvalue problems of partial differential equations on rectangular domains. While this physics of the model quickly turned out to be completely obsolete, his mathematics enabled him later to solve difficult problems in applied sciences. He thereby revolutionized the variational calculus and became one of the fathers of modern computational mathematics.

The Ritz method was immediately recognized by Russian mathematicians as a fundamental contribution, and put to use in the computational simulation of beams and plates, which led to the seminal paper of Galerkin in 1915. In Europe however, especially in the mathematical center of that time in Goettingen, it received very little attention, even though Ritz obtained a price from the French Academy of Sciences, after having lost in the official competition for the Vaillant price in 1907 to Hadamard. It was only during the second world war, long after Ritz's death, in an address of Courant in front of the AMS, that the potential of Ritz's invention was fully recognized, and Courant presented what we now call the finite element method. This name was given to the method after Clough reinvented it in a seminal paper, working for Boeing.

We will see in this talk that the path leading to modern computational methods and theory was a long struggle over three centuries requiring the efforts of many great mathematicians.

07 November 2011

Igor Vigdorovich - Moscow St. University 

Self-similar turbulent boundary layer in pressure gradient. Four flow regimes

Abstract:

Self-similar flows in a turbulent boundary layer when the free-stream velocity is specified as a power function of the longitudinal coordinate are investigated. The self-similar formulation not only simplifies solving the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalized distance from the wall and the exponent in the law specifying the free-stream velocity distribution in the outer region and a universal function of the local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data. There exist four different self-similar flow regimes. Each regime is related to its similarity parameter, one of which is the well-known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and skin friction, which points out the possibility of hysteresis in near-separating flow. Separation occurs not at the minimal value of the exponent that corresponds to the strongest adverse pressure gradient but at a higher one whose dependence on Reynolds number is calculated in the paper. The results of the theory are in good agreement with experimental data.

14 November 2011

TBA

TBA

21 November 2011

S. Sitanen - University of Helsinki

Electrical impedance imaging using nonlinear Fourier transform 

Abstract:

The aim of electrical impedance tomography (EIT) is to reconstruct the inner structure of an unknown body from voltage-to-current measurements performed at the boundary of the body. EIT has applications in medical imaging, nondestructive testing, underground prospecting and process monitoring. The imaging task of EIT is nonlinear and an ill-posed inverse problem. A non-iterative EIT imaging algorithm is presented, based on the use of a nonlinear Fourier transform. Regularization of the method is provided by nonlinear low-pass filtering, where the cutoff frequency is explicitly determined from the noise amplitude in the measured data. Numerical examples are presented, suggesting that the method can be used for imaging

the heart and lungs of a living patient .

28 November 2011

S. Vacaru - University Alexandru Ioan Cuza (UAIC), Iasi Romania

Decoupling of Gravitational Field Equations, Off-Diagonal Solutions and Solitonic Hierarchies

Abstract:

We prove that there is a general decoupling property of Einstein equations and certain generalizations/modifications for (non) commutative metric-affine, extra dimension, string/brane, Horava-Lifshitz, Finsler etc models of gravity theories (see arXiv: 1108.2022 and references therein). This allows us to construct very general classes of generic off-diagonal solutions depending, in principle, on all spacetime

coordinates and various generating functions and (broking, or not) symmetry parameters. There are provided main theorems when such generalized metrics and connections can be encoded into bi-Hamilton

structures and associated solitonic hierarchies and characterized by corresponding solitonic symmetries. Finally, we analyze explicit examples of black ellipsoid, toroidal, nohomogeneous cosmology, wormholes etc

solutions characterized by generic off-diagonal metrics and discuss possible implications in modern gravity and particle physics.

05 December 2011

W Symes - Rice University, USA

Inverse problems for wave propagation in heterogeneous media

Abstract:

Inverse problems in wave propagation rely upon hyperbolic partial (integro-)differential systems to model physical wave motion. However, rocks, manufactured materials, and other natural and human-made wave propagation environments may exhibit spatial heterogeneity at a wide variety of scales. Therefore accuracy in modeling (hence in inversion) requires tha coefficient functions representing material parameter fields be permitted some degree of nonsmoothness. I will show how to formulate well-posed initial/boundary-value problems for hyperbolic systems with bounded and measureable coefficients, as instances of a class of abstract first-order evolution problems. This framework yields well-defined realizations of the mappings occurring in widely-used optimization formulations of inverse problems, and justifies the use of Newton's method and its relatives for their solution. The finite speed of propagation for waves in material models with bounded and measurable heterogeneity also follows from this framework. Another useful by-product is a mathematical foundation for (unphysical) hyperbolic systems with operator coefficients, which are crucial components of a class of seismic inversion algorithms.