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We will explore connections between the structure of high-dimensional convex polytopes and information acquisition for compressible signals. A classical result in the field of convex polytopes is that if N points are distributed Gaussian iid at random in dimension n<<N, then only order (log N)^n of the points are vertices of their convex hull. Recent results show that provided n grows slowly with N, then with high probability all of the points are vertices of its convex hull. More surprisingly, a rich "neighborliness" structure emerges in the faces of the convex hull. One implication of this phenomenon is that an N-vector with k non-zeros can be recovered computationally efficiently from only n random projections with n=2e k log(N/n). Alternatively, the best k-term approximation of a signal in any basis can be recovered from 2e k log(N/n) non-adaptive measurements, which is within a log factor of the optimal rate achievable for adaptive sampling. Additional implications for randomized error correcting codes will be presented.
This work was joint with David L. Donoho.
Spitalfields Day: Aspects of Geometry
Meeting to mark Sir Roger Penrose's 75th Birthday
Meeting to mark Sir Roger Penrose's 75th Birthday
Meeting to mark Sir Roger Penrose's 75th Birthday
In this talk we present different strategies for regularization of the pure Newton method
(minimization problems)and of the Gauss-Newton method (systems of nonlinear equations).
For these schemes, we prove general convergence results. We establish also the global and
local worst-case complexity bounds. It is shown that the corresponding search directions can
be computed by a standard linear algebra technique.
We present a novel enhanced finite element method for the Darcy problem starting from the non stable
continuous $P_1 / P_0$ finite element spaces enriched with multiscale functions. The method is a departure
from the standard mixed method framework used in these applications. The methods are derived in a Petrov-Galerkin
framework where both velocity and pressure trial spaces are enriched with functions based on residuals of strong
equations in each element and edge partition. The strategy leads to enhanced velocity space with an element of
the lowest order Raviart-Thomas space and to a stable weak formulation preserving local mass conservation.
Numerical tests validate the method.
Jointly with Gabriel R Barrenechea, Universidad de Concepcion &
Frederic G C Valentin, LNCC
Strong horizontal gradients of density are responsible for the occurence of a large number of (often catastrophic) flows, such as katabatic winds, dust storms, pyroclastic flows and powder-snow avalanches. For a large number of applications, the overall density contrast in the flow remains small and simulations are carried in the Boussinesq limit, where density variations only appear in the body-force term. However, pyroclastic flows and powder-snow avalanches involve much larger density contrasts, which implies that the inhomogeneous Navier-Stokes equations need to be solved, along with a closure equation describing the mass diffusion. We propose a Lagrange-Galerkin numerical scheme to solve this system, and prove optimal error bounds subject to constraints on the order of the discretization and the time-stepping. Simulations of physical relevance are then shown.