Past

http://www.maths.ox.ac.uk/arg

One of the oldest approach in meshless methods for PDEs is the Interpolating Moving Least Squares (IMLS) technique developed in the 1980s. Although widely accepted by users working in fields as diverse as geoinformatics and crack dynamics I shall take a fresh look at this method and ask for the equivalent difference operators which are generated implicitly. As it turns out, these operators are optimal only in trivial cases and are "strange" in general. I shall try to exploit two different approaches for the computation of these operators.

On the other hand (and very different from IMLS), Total Variation Flow (TVF) PDEs are the most recent developments in image processing and have received much attention lately. Again I shall show that they are able to generate "strange" discrete operators and that they easily can behave badly although they may be properly implemented.

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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