George Papanicolaou

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

We propose an illumination strategy for interferometric imaging that allows for robust depth recovery from intensity-only measurements. For an array with colocated sources and receivers, we show that all the possible interferometric data for multiple sources, receivers and frequencies can be recovered from intensity-only measurements provided that we have sufficient source location and frequency illumination diversity. There is no need for phase reconstruction in this approach. Using interferometric imaging methods we show that in homogeneous media there is no loss of resolution when imaging with intensities-only. If in these imaging methods we reduce incoherence by restricting the multifrequency interferometric data to nearby array elements and nearby frequencies we obtain robust images in weakly inhomogeneous background media with a somewhat reduced resolution.

Josselin Garnier, George Papanicolaou, Tzu-Wei Yang

We consider an one-dimensional conservation law with random space-time forcing and calculate using large deviations the exponentially small probabilities of anomalous shock profile displacements. Under suitable hypotheses on the spatial support and structure of random forces, we analyze the scaling behavior of the rate function, which is the exponential decay rate of the displacement probabilities. For small displacements we show that the rate function is bounded above and below by the square of the displacement divided by time. For large displacements the corresponding bounds for the rate function are proportional to the displacement. We calculate numerically the rate function under different conditions and show that the theoretical analysis of scaling behavior is confirmed. We also apply a large-deviation-based importance sampling Monte Carlo strategy to estimate the displacement probabilities. We use a biased distribution centered on the forcing that gives the most probable transition path for the anomalous shock profile, which is the minimizer of the rate function. The numerical simulations indicate that this strategy is much more effective and robust than basic Monte Carlo.

George Papanicolaou, Nicholas West, Tzu-Wei Yang

We consider a reduced order model of an air-breathing hypersonic engine with a time-dependent stochastic inflow that may cause the failure of the engine. The probability of failure is analyzed by the Freidlin-Wentzell theory, the large deviation principle for finite dimensional stochastic differential equations. We compute the asymptotic failure probability by numerically solving the constrained optimization related to the large deviation problem. A large-deviation-based importance sampling suggested by the most probable inflow perturbation is also implemented to compute the probability of failure of the engine. The numerical simulations show that the importance sampling method is much more efficient than the basic Monte Carlo method.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

We propose an approach for imaging in scattering media when large and diverse data sets are available. It has two steps. Using a dictionary learning algorithm the first step estimates the true Green's function vectors as columns in an unordered sensing matrix. The array data comes from many sparse sets of sources whose location and strength are not known to us. In the second step, the columns of the estimated sensing matrix are ordered for imaging using Multi-Dimensional Scaling with connectivity information derived from cross-correlations of its columns, as in time reversal. For these two steps to work together we need data from large arrays of receivers so the columns of the sensing matrix are incoherent for the first step, as well as from sub-arrays so that they are coherent enough to obtain the connectivity needed in the second step. Through simulation experiments, we show that the proposed approach is able to provide images in complex media whose resolution is that of a homogeneous medium.