Stanford Applied Math Seminar

Wednesdays at 4:15 PM in 384H.

Spring 2016 Schedule

April 6
Graeme Milton, University of Utah
Extending the Theory of Composites to Other Areas of Science

Science is littered with examples where progress in one area has led to progress in other sometimes completely different areas thanks to similar underlying mathematical structures. Here we show that many of the linear wave-equations of physics can be reformulated in a way that is similar to the static or quasistatic equations of composites. The reformulation of Schr{\''o}dingers equation, for example, leads to new minimization variational principles. Desymmetrizing it leads to FFT methods for solving the multielectron Schrodinger equation which only requires FFT transforms in 2 spatial variables. Unlike density functional theory it is applicable even to excited states.

The theory of composites carries over to the response of inhomogeneous bodies, and integral representations and bounds for the dynamic response can be used in an inverse way to say something about what is inside the body from boundary measurements. Even the concept of functions can be generalized, based on the theory of composites. The new functions, superfunctions, have applications to accelerating FFT methods. A superfunction can be thought as a collection of subspaces, and there are associated rules for multiplication, division, addition, and subtraction of such superfunctions. This talk is based on a book I am editing, by the same title, and the work is coauthored with Maxence Cassier, Ornella Mattei, Mordehai Milgrom, and Aaron Welters.

April 13
Guillaume Bal, Columbia University
Boundary control in transport and diffusion equations

Consider a prescribed solution to a diffusion equation in a small domain embedded in a larger one. Can one (approximately) control such a solution from the boundary of the larger domain? The answer is positive and this form of Runge approximation is a corollary of the unique continuation property (UCP) that holds for such equations. Now consider a (phase space, kinetic) transport equation, which models a large class of scattering phenomena, and whose vanishing mean free path limit is the above diffusion model. This talk will present positive as well as negative results on the control of transport solutions from the boundary. In particular, we will show that internal transport solutions can indeed be controlled from the boundary of a larger domain under sufficient convexity conditions. Such results are not based on a UCP. In fact, UCP does not hold for any positive mean free path even though it does apply in the (diffusion) limit of vanishing mean free path. These controls find applications in inverse problems that model a large class of coupled-physics medical imaging modalities. The stability of the reconstructions is enhanced when the answer to the control problem is positive.

April 20
Messoud Efendiyev Mathematical modelling of biofilms: analysis and simulation
Coarse graining collective motion

In this talk, I will present new class of reaction -diffusion-transport equations which describe spatial spreading mechanisms of biomass. The feature of these equations is that they are comprising two kind of degeneracy : porous medium and fast diffusion. These classes of doubly-degenerate parabolic systems arising in the modelling of antibiotic disinfections of biofilms as well as biofilm growth in a porous media. Well-posedness, long-time dynamics of solutions in terms of global attractors and their Kolmogorov`s entropy as well as open questions will be discussed.

April 27
Doug James, Stanford

May 4
TBA, University of TBA

May 11
Torbjörn Lundh, Stanfordy

May 18
Benjamin Moll, Princeton

May 25,
Laurent Mertz, Universite Paris VI

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