The ``random sampling'' in the title has nothing whatsoever to
do with statistics. Instead, it refers to the perfect reconstruction of
a band-limited function from samples, a classical problem of
Fourier analysis and signal processing. With deterministic sampling,
almost everything is known in one dimension and almost nothing
is known in higher dimensions. It turns out that one loses very little in
efficiency by using random sampling, and in return one can use
probabilistic techniques to get some interesting theoretical results.
This is joint work with Karlheinz Gr\"ochenig.

Imaging of obscured targets in random media is a difficult and important problem. One of the central questions is that of stability which is particularly relevant to imaging in stochastic media. The main goal of the research is to develop a general criterion for multiple-frequency array imaging of multiple targets in stochastic media. An important feature of the cluttered media we considered here is that the coherent or mean signals do not vanish. It is called Rician fading medium in communication. Foldy-Lax formulas are used to simulate the exact wave propagations in the random media. In the talk, I will propose two models: passive array model and active array model. Then the stabilities of the imaging function with multiple point targets are given in both cases, followed by the numerical simulations to show the consistency with the analysis. If time permits, I will give some preliminary results about the stability conditions of the multiple extended targets, as well as the simulations.

We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.