George Papanicolaou

TRANSPORT EQUATIONS FOR ELASTIC AND OTHER WAVES IN RANDOM MEDIA

Abstract 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 effects, coupling of different types of waves, etc. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion coefficients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.

Super-resolution in time-reversal acoustics

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 time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution. A theoretical treatment of this phenomenon is given, and numerical simulations which confirm the theory are presented.

Imaging and time reversal in random media

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 (ATA) of the echo received from the scatterers, (ii) a singular value decomposition of the array response matrix in the frequency domain, and (iii) the construction of an objective function in the time domain that is statistically stable and peaks on the scatterers. By statistically stable we mean here that the objective function is self-averaging over individual realizations of the medium. This is a new approach to array imaging that is motivated by time reversal in random media, analysed in detail previously. It combines features from seismic imaging, like ATA, with frequency-domain signal subspace methodology like multiple signal classification. We illustrate the theory with numerical simulations for ultrasound.

Nonlinear Diffusion Limit for a System with Nearest Neighbor Interactions

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 evolution of the macroscopic charge density.

Application of time-reversal with MMSE equalizer to UWB communications

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 design becomes much simpler than without focusing. Furthermore, temporal focusing allows a large increase in transmission rate compared to schemes that let the impulse response ring out before the next symbol is sent. Our paper introduces time-reversal, investigates the benefit of temporal focusing, and examines the performance of an MMSE-TR equalizer in an UWB channel.

Bounds for Effective Parameters of Heterogeneous Media by Analytic Continuation

We give a mathematical formulation of a method for obtaining bounds on effective parameters developed by D. Bergman and G. W. Milton. This method, in contrast to others used before, does not rely on a variational principle, but exploits the properties of the effective parameter as an analytic function of the component parameters. The method is at present restricted to two-component media.

Effective equations for wave propagation in bubbly liquids

We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by using Foldys approximation in a nonlinear setting. We discuss in detail the range of validity of the effective equations as well as some of their properties.

Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension

The formation of singularities of self-focusing solutions of the nonlinear Schrodinger equation (NLS) in critical dimension is characterized by a delicate balance between the focusing nonlinearity ...

Convection enhanced diffusion for periodic flow

This paper studies the influence of convection by periodic or cellular flows on the effective diffusivity of a passive scalar transported by the fluid when the molecular diffusivity is small. The flows are generated by two-dimensional, steady, divergence-free, periodic velocity fields.

Multiscale stochastic volatility for equity, interest rate, and credit derivatives

Introduction 1. The Black-Scholes theory of derivative pricing 2. Introduction to stochastic volatility models 3. Volatility time scales 4. First order perturbation theory 5. Implied volatility formulas and calibration 6. Application to exotic derivatives 7. Application to American derivatives 8. Hedging strategies 9. Extensions 10. Around the Heston model 11. Other applications 12. Interest rate models 13. Credit risk I: structural models with stochastic volatility 14. Credit risk II: multiscale intensity-based models 15. Epilogue Bibliography Index.