arXiv subjects

Alexei Novikov

Publications and source records attributed to Alexei Novikov.

34 records · Page 2Linked to original sources

arXiv2016

Multifrequency interferometric imaging with intensity-only measurements

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.

math.OC↗

arXiv2016

A fractional kinetic process describing the intermediate time behaviour of cellular flows

This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result for the associated advection-diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion.

math.PR↗

arXiv2016

A survival guide for feeble fish

As avid anglers we were always interested in the survival chances of fish in turbulent oceans. This paper addresses this question mathematically. We show that a fish with bounded aquatic locomotion speed can reach any point in the ocean if the fluid velocity is incompressible, bounded, and has small mean drift.

math.OC↗

arXiv2016

Coherent imaging without phases

In this paper we consider narrow band, active array imaging of weak localized scatterers when only the intensities are recorded at an array with N transducers. We consider that the medium is homogeneous and, hence, wave propagation is fully coherent. This work is an extension of our previous paper, where we showed that using linear combinations of intensity-only measurements imaging of localized scatterers can be carried out efficiently using MUSIC or sparsity promoting optimization. Here we show the same strategy can be accomplished with only 3N-2 illuminations, therefore reducing enormously the data acquisition process. Furthermore, we show that in the paraxial regime one can form the images by using six illuminations only. In particular, this paraxial regime includes Fresnel and Fraunhofer diffraction. The key point of this work is that if one controls the illuminations, imaging with intensity-only can be easily reduced to a imaging with phases and, therefore, one can apply standard imaging techniques. Detailed numerical simulations illustrate the performance of the proposed imaging strategy with and without data noise.

math.OC↗

arXiv2015

Anomalous diffusion in fast cellular flows at intermediate time scales

It is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a Brownian motion. This paper studies the behaviour on "intermediate" time scales before diffusion sets in. Various heuristics suggest that an anomalous diffusive behaviour should be observed. We prove that the variance on intermediate time scales grows like $O(\sqrt{t})$. Hence, on these time scales the effective behaviour can not be purely diffusive, and is consistent with an anomalous diffusive behaviour.

math.AP↗

arXiv2014

Illumination strategies for intensity-only imaging

We propose a new strategy for narrow band, active array imaging of localized scat- terers when only the intensities are recorded and measured at the array. We consider a homogeneous medium so that wave propagation is fully coherent. We show that imaging with intensity-only measurements can be carried out using the time reversal operator of the imaging system, which can be obtained from intensity measurements using an appropriate illumination strategy and the polarization identity. Once the time reversal operator has been obtained, we show that the images can be formed using its singular value decomposition (SVD). We use two SVD-based methods to image the scatterers. The proposed approach is simple and efficient. It does not need prior information about the sought image, and guarantees exact recovery in the noise-free case. Furthermore, it is robust with respect to additive noise. Detailed numerical simulations illustrate the performance of the proposed imaging strategy when only the intensities are captured.

math.OC↗

arXiv2012

The regularizing effects of resetting in a particle system for the Burgers equation

We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330-345]. In this paper we study a particle system for the viscous Burgers equations using a Monte-Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with $\frac{1}{N}$ times the sum over these copies. A similar construction for the Navier-Stokes equations was studied by Mattingly and the first author of this paper [Iyer and Mattingly Nonlinearity 21 (2008) 2537-2553]. Surprisingly, for any finite N, the particle system for the Burgers equations shocks almost surely in finite time. In contrast to the full expected value, the empirical mean $\frac{1}{N}\sum_1^N$ does not regularize the system enough to ensure a time global solution. To avoid these shocks, we consider a resetting procedure, which at first sight should have no regularizing effect at all. However, we prove that this procedure prevents the formation of shocks for any $N\geq2$, and consequently as $N\to\infty$ we get convergence to the solution of the viscous Burgers equation on long time intervals.

math.PR↗

arXiv2012

A differential equations approach to $l_1$-minimization with applications to array imaging

We present an ordinary differential equations approach to the analysis of algorithms for constructing $l_1$ minimizing solutions to underdetermined linear systems of full rank. It involves a relaxed minimization problem whose minimum is independent of the relaxation parameter. An advantage of using the ordinary differential equations is that energy methods can be used to prove convergence. The connection to the discrete algorithms is provided by the Crandall-Liggett theory of monotone nonlinear semigroups. We illustrate the effectiveness of the discrete optimization algorithm in some sparse array imaging problems.

math.NA↗

arXiv2011

Blow-up of solutions to a p-Laplace equation

Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field blows up in the L-infinity norm as the distance between the conductors tends to zero. We give here a concise rigorous justification of the rate of this blow-up in terms of the distance between the conductors.

math.AP↗

arXiv2011

From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude $A$, in a two-dimensional domain with $L^2$ cells. For fixed $A$, and $L \to \infty$, the problem homogenizes, and has been well studied. Also well studied is the limit when $L$ is fixed, and $A \to \infty$. In this case the solution equilibrates along stream lines. In this paper, we show that if \textit{both} $A \to \infty$ and $L \to \infty$, then a transition between the homogenization and averaging regimes occurs at $A \approx L^4$. When $A\gg L^4$, the principal Dirichlet eigenvalue is approximately constant. On the other hand, when $A\ll L^4$, the principal eigenvalue behaves like ${\bar σ(A)}/L^2$, where $\bar σ(A) \approx \sqrt{A} I$ is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent $L^p \to L^\infty$ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.

math.AP↗

arXiv2010

Homogenization of the G-equation with incompressible random drift in two dimensions

We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These conditions are expressed in terms of travel times for the associated control problem. When the spatial dimension is equal to two and the fluid velocity is divergence-free, we verify that these conditions hold under suitable assumptions about the growth of the random stream function.

math.AP↗

arXiv2009

Exit times of diffusions with incompressible drift

Let $Ω\subset\mathbb R^n$ be a bounded domain and for $x\inΩ$ let $τ(x)$ be the expected exit time from $Ω$ of a diffusing particle starting at $x$ and advected by an incompressible flow $u$. We are interested in the question which flows maximize $\|τ\|_{L^\infty(Ω)}$, that is, they are most efficient in the creation of hotspots inside $Ω$. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow $u\equiv 0$ maximises $\|τ\|_{L^\infty(Ω)}$. We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, $\|τ\|_{L^\infty(Ω)}$ is maximized by the zero flow on the ball.

math.AP↗

arXiv2009

The explosion problem in a flow

We consider the explosion problem in an incompressible flow introduced in the paper of H. Berestycki, L. Kagan, G. Joulin and G. Sivashinsky. We use a novel $L^p-L^\infty$ estimate for elliptic advection-diffusion problems to show that the explosion threshold obeys a positive lower bound which is uniform in the advecting flow. We also identify the flows for which the explosion threshold tends to infinity as their amplitude grows and obtain an effective description of the explosion threshold in the strong flow asymptotics in a two-dimensional one-cell flow.

math.AP↗

arXiv2006

Fictitious Fluid Approach and Anomalous Blow-up of the Dissipation Rate in a 2D Model of Concentrated Suspensions

We present a two-dimensional (2D) mathematical model of a highly concentrated suspension or a thin film of the rigid inclusions in an incompressible Newtonian fluid. Our objectives are two-fold: (i) to obtain all singular terms in the asymptotics of the overall viscous dissipation rate as the interparticle distance parameter $δ$ tends to zero, (ii) to obtain a qualitative description of a microflow between neighboring inclusions in the suspension.

math.AP↗

arXiv2005

On a conjecture of Widom

We prove a conjecture of H.Widom stated in [W] (math/0108008) about the reality of eigenvalues of certain infinite matrices arising in asymptotic analysis of large Toeplitz determinants. As a byproduct we obtain a new proof of A.Okounkov's formula for the (determinantal) correlation functions of the Schur measures on partitions.

math.CO↗

arXiv2002

Content Distribution in Unicast Replica Meshes

We propose centralized algorithm of data distribution in the unicast p2p network. Good example of such networks are meshes of WWW and FTP mirrors. Simulation of data propogation for different network topologies is performed and it is shown that proposed method performs up to 200% better then common apporaches

cs.NI↗