Polynomial embeddings for quadratic algebraic equations
In this talk we analyze the problem of vector reconstruction from magnitudes
of frame coefficients. We show how the system of quadratic equations can
be embedded in a set of higher order polynomial equations. In particular
we analyze the linear independence of the system of homogeneous polynomial
equations. We obtain characterizations for the range of tensor frame analysis operator
that provide necessary and sufficient conditions for linear independence.
Such polynomial embeddings provide closed form solutions for the
problem of reconstruction from magnitudes of frame coefficients, which in turn
can be used to construct unbiased estimators in the presence of noise.
Controlled Active Vision with Various Applications
In this talk, we will describe some of the key issues in controlled active vision, namely the utilization of visual information in a feedback loop. The applications range from visual tracking (e.g., laser tracking in turbulence, flying in formation of UAVs, etc.), nanoparticle flow control, and various medical imaging applications (image guided therapy and surgery).
Accordingly, we will describe several models of active contours for which both local (edge-based) and global (statistics-based) information may be included for various segmentation tasks. We will indicate how statistical estimation and prediction ideas (e.g., particle filtering) may be naturally combined with this methodology. A novel model of directional active contour models for path-planning will be considered. In addition to segmentation, the second key component of many active vision tasks is registration. The registration problem (especially in the elastic case) is still one of the great challenges in vision and image processing. Registration is the process of establishing a common geometric reference frame between two or more data sets obtained by possibly different imaging modalities. Registration has a substantial literature devoted to it, with numerous approaches ranging from optical flow to computational fluid dynamics.
For this purpose, we propose using ideas from optimal mass transport.
The mass transport problem was first formulated by Gaspar Monge in 1781, and concerned finding the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. This problem was given a modern formulation in the work of Kantorovich, and is now known as the "Monge/Kantorovich problem." The optimal mass transport approach has strong connections to optimal control, and can be the basis for a geometric observer theory for target tracking in which shape information is explicitly taken into account. Finally, we will describe how mass transport ideas may be utilized in order to formulate a novel distance on the space of probability measures with applications to spectral estimation and information geometry. The talk is designed to be accessible to a general audience with an interest in vision, control, and image processing. We will demonstrate our techniques on a wide variety of data sets both military and medical.
Exploring complex energy landscape is a challenging issue in many applications. Besides locating equilibrium states, it is often also important to identify transition states given by saddle points. In this talk, we will discuss existing and new algorithms for the computation of such transition states with a focus on the newly developed Shrinking Dimer Dynamics and present some related mathematical theory on stability and convergence. We will consider a number of applications including the study of frustrations of interacting particles and nucleation in solid state phase transformations.
Is adoption of new products affected by the social network?
Mathematical Marketing and Agent-Based approaches
The adoption of new products which mainly spread through word-of-mouth (such as fax machines, skype, facebook, Ipad, etc.) is one of the key problems in Marketing research. Ideally, given the sales data of the first few months, one would like to be able to predict both the future sales and the overall market potential.
In this talk I will first present the classic Bass model and the agent-based approach for the adoption of new products. Then, I will present some recent analytic results on the effect of the social network on the adoption of new products.
This is joint work with Ro'i Gibori and Eitan Muller
I describe a new approach to locating key material transport barriers in unsteady flows induced by two-dimensional, non-autonomous dynamical systems.
Seeking transport barriers as minimally stretching material lines, one obtains that such barriers must be shadowed by minimal geodesics under the metric induced by the Cauchy-Green strain tensor field associated with the flow map. As a result, snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations. Using this approach, hyperbolic barriers (generalized stable and unstable manifolds), elliptic barriers (generalized KAM curves) and parabolic barriers (generalized shear jets) can be found with high precision in temporally aperiodic flows defined over a finite time interval.
I illustrate the results by computing transport barriers in several unsteady flows arising in mechanics and fluid dynamics.
Standard Krylov subspace methods only allow the user to choose a single preconditioner, although in many situations there may be a number of possibilities. Here we describe an extension of GMRES, multi-preconditioned GMRES, which allows the use of more than one preconditioner. We give some theoretical results, propose a practical algorithm, and present numerical results from problems in domain decomposition and PDE-constrained optimization.
These numerical experiments illustrate the applicability and potential of the multi-preconditioned approach.
Dr. Anders C. Hansen, Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Generalized Sampling and Infinite-Dimensional Compressed Sensing
I will discuss a generalization of the Shannon Sampling Theorem that allows for reconstruction of signals in arbitrary bases. Not only can one reconstruct in arbitrary bases, but this can also be done in a completely stable way. When extra information is available, such as sparsity or compressibility of the signal in a particular bases, one may reduce the number of samples dramatically. This is done via Compressed Sensing techniques, however, the usual finite-dimensional framework is not sufficient. To overcome this obstacle I'll introduce the concept of Infinite-Dimensional Compressed Sensing.
Least squares methods are of common use
when one needs to approximate a function
based on its noiseless or noisy observation
at n scattered points by a simpler function
chosen in an m dimensional space with m
less than n. Depending on the context,
these points may be randomly drawned
according to some distribution, or
deterministically selected by the user.
In this talk, I shall analyze the
stability and approximation properties of
least squares method. This analysis
involves the relative size of m with
respect to n as well as the spatial
distribution of the samples.
Applications with be given for high
dimensional sparse polynomial approximation
to parametric-stochastic PDE's
and approximation of acoustic fields
by plane waves.
Scaling pattern with size by a morphogen-directed cell division rule
Morphogen gradients guide the patterning of tissues and organs during the development of multicellular organisms. Morphogen signaling is required also for proliferation of the patterned tissue, thereby coupling growth and patterning. Possible consequences of this coupling are poorly understood, as most previous studies analyzed the formation of morphogen gradients while neglecting tissue growth. We propose that morphogen-dependent coupling of growth and patterning may play a role in scaling the morphogen gradient with the size of the growing tissue, thus ensuring proportionate patterning in individuals of different sizes and during growth. We examine this proposition using a mathematical framework that accounts for this coupling. Growth affects the morphogen profile by diluting it and transporting it along the growing tissue, while morphogen signaling directs cell division according to some local division rule, which needs to be specified. We derive a simple division rule that leads to a growing tissue with three desirable properties: (1) morphogen gradient scales with tissue size, (2) growth of the tissue is spatially uniform and (3) the tissue size converges to a final, finite value. All three properties are obtained in a robust manner. We conclude, using extensive numerical simulations, that even in the absence of feedbacks, scaling and uniform growth can emerge naturally from the mutual dependence of patterning and growth on signaling by the same morphogen. Thus, local coupling of cell division with morphogen signaling establishes global scaling properties of the growing tissue. The lecture presents joint work with N. Barkai and D. Ben-Zvi (Weizmann Institute of Science, Israel)
Sea ice is a leading indicator of climate change, and a key component of Earth's climate system.
As a material, sea ice is a porous composite of pure ice with millimeter scale brine inclusions whose
volume fraction and connectedness vary significantly with temperature. Fluid flow through
the brine microstructure mediates a broad range of geophysical and biological processes. Sea ice
also displays composite structures over much larger scales. For example, the sea ice pack itself
is a fractal composite of ice floes and ocean, while surface ponds on melting Arctic sea ice
exhibit complex dynamics and self-similar geometrical configurations. I will discuss how methods
of homogenization and statistical physics are being used to study the effective properties of such
systems. This work helps improve our understanding of the role of sea ice in the climate system,
and the representation of sea ice in climate models.
Homogenization of Bound State of Bose-Einstein Gas
How do atoms of integer spin (Bosons) behave at very low temperatures?
This question was first addressed by Einstein in 1924, on the basis of earlier work by Bose on photons. Einstein predicted that non-interacting massive Bosons should condense to a macroscopic state at sufficiently low temperatures. This "Bose-Einstein condensation" is a quantum mechanical phenomenon, and was first observed experimentally in trapped dilute atomic gases in 1995.
The dilute-gas system is amenable to systematic theory due to the weak particle interactions.
In this talk, I will formally discuss implications of controlling the interactions between Bosons of a trapped gas via a spatially dependent "scattering length" with a periodic microstructure. The starting point is the many-body Hamiltonian of the system and the many-body Schroedinger equation. For the lowest bound state, I will focus on the derivation and predictions of effective equations of motion.
The main part of my talk will concern zero temperature; if time allows, I will discuss aspects of the more challenging problem with finite temperatures below the phase transition point.
Fast waveform extraction from gravitational perturbations
Numerical wave simulation on finite computational domains requires the introduction of fictitious boundaries at which one must specify boundary conditions. Such boundary conditions may stem from exact reduction of an infinite domain, allowing for radiation flux off the finite computational domain. For the ordinary wave equation such radiation boundary conditions specify a transparent or non-reflecting boundary. A related issue is the extraction from the finite computational domain of the asymptotic waveform (the "signal" which would reach infinity).
In the context of blackhole perturbation theory, we describe extraction of an asymptotic waveform from a time series recorded at a finite radial location. Our technique is easy to implement (from the user's standpoint), affords high accuracy, and works for both axial (Regge-Wheeler) and polar
(Zerilli) perturbations. It is based on Alpert, Greengard, and Hagstrom's treatment of non-reflecting boundary conditions for the ordinary wave equation. Much of the talk will be on the ordinary wave equation and accessible without a background in gravitational physics.
ENO interpolation is stable:
high resolution and the sign property
ENO is an adaptive procedure to recover piecewise-smooth data with high resolution. It is based on piecewise-polynomial interpolants with stencils which adapt themselves to the smoothness of the data. We prove that the ENO interpolation is stable: the jump of the ENO pointvalues at each cell interface has the same sign and in fact, the same size as the jump of the underlying data across that interface.
This sign property, which is shown to hold for ENO interpolation of arbitrary order of accuracy and on non-uniform meshes, manifests a remarkable rigidity of the piecewise-polynomial ENO procedure. Similar sign properties hold for the ENO reconstruction procedure from cell averages, which is used in high-resolution, entropy stable computations of nonlinear conservation laws.