George Papanicolaou

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We derive and analyze transport equations for the energy density of waves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization e ects, coupling of di erent types of waves, etc. We also show that di usive behavior occurs on long time and distance scales and(More)
The statistical properties of acoustic signals re ected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analysis of stochastic equations and geometrical acoustics is used to arrive at a set of transport equations that characterize multiply scattered signals observed at(More)
We consider a system of interacting diffusions. The variables are to be thought of as charges at sites indexed by a periodic one-dimensional lattice. The diffusion preserves the total charge and the interaction is of nearest neighbor type. With the appropriate scaling of lattice spacing and time, a nonlinear diffusion equation is derived for the time(More)
We propose to apply a technique called time-reversal to UWB communications. In time-reversal a signal is precoded such that it focuses both in time and in space at a particular receiver. Spatial focusing reduces interference to other co-existing systems. Due to temporal focusing, the received power is concentrated within a few taps and the task of equalizer(More)
In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black-Scholes formula gives the price of call options when the underlying is a geometric(More)
The phenomenon of super-resolution in time-reversal acoustics is analyzed theoretically and with numerical simulations. A signal that is recorded and then retransmitted by an array of transducers, propagates back though the medium, and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the(More)
We present a general method for estimating the location of small, well-separated scatterers in a randomly inhomogeneous environment using an active sensor array. The main features of this method are (i) an arrival time analysis of the echo received from the scatterers, (ii) a singular value decomposition of the array response matrix in the frequency domain,(More)
2 Locally Stationary Processes 2 2.1 Time-varying spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Locally stationary processes depending on a parameter . . . . . . . . . . . . 7 2.3 Local Cosine Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Pseudo-di erential Covariance Operators . . . . . . . . . . . .(More)