Math 290 - Student Numerical Analysis Seminar - Fall 2011

We construct a sequence of row rotations which combine to give a hierarchy of isotropically random orthogonal transformations. This is done by decomposing such a transformation on an n-vector into two such transformations on n/2-vectors and combining the results. We can use this method to construct a random orthogonal transformation composed of n-1 rotations that is indistinguishable in the first row from a complete isotropically random orthogonal transformation. Using n(n-1)/2 rotations we can reconstruct a fully isotropic random orthogonal transformation. These rotations are calculated by numerically constructing and interpolating the appropriate cumulative density function of angles on spheres of various dimensions in order to map a flat random variables from [0,1] to an angle of rotation that has the correct probability density.

We present a novel framework for phase recovery from magnitude measurements via convex relaxation of a rank minimization problem. Under some sampling assumptions, the relaxation has been shown to recover the exact signal in the noiseless case with high probability. It is also provably stable in the presence of noise. Computational and theoretical aspects of this problem will be discussed.

The goal of this talk is to prove the result of Brascamp & Lieb that the logarithm of the eigenfunction with smallest eigenvalue for the operator $u \mapsto -\Delta u + V(x)u$ is concave, provided that $V$ is a convex function unbounded in every direction. The proof rests on the Trotter product formula and standard (though occasionally nontrivial) facts about log-concave functions. I'll state these mostly without proof, but there should be enough intuition that the main theorem makes sense.

The reference for this talk is H. J. Brascamp and E. H. Lieb, "On Extensions of the Brunn-Minkowski and Pr\'{e}kopa-Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation," in \textit{Journal of Functional Analysis} 22 (1976), pp. 366-389.