Broadly speaking, I have studied three main areas in the chronological order of my career: (i) turbulent transport and homogenization (ii) waves in random media (iii) imaging . The first two areas deal with the problem of how structures affect physical processes (transport of stuffs, the direct problem) while (iii) deals with the problem of how physical processes reveal structures (transport of information, the inverse problem) . My current research is focused on (iii) at the moment.
Compressive imaging is the idea of imaging sparse objects with comparably sparse measurement. The approach is based the compressed sensing (CS) theory originating in harmonic analysis, information theory and statistics. In the standard theory, problems of imaging are typically formulated in the continuum setting and results are usually qualitative. For example, the standard inverse scattering theory asserts the uniqueness of scatterer given the entirety of scattering data (i.e. all incident and scattering angles). But in reality one has only a fi- nite (sometimes small) number of scattering data, let alone the complete continuum set of scattering data!
From this perspective, CS is a natural, alternative route to inverse scattering and, more generally, imaging. As such, compressive imaging has seen a flurry of activities in the last few years. Few, though, really address the heart of the problem. A simple fact is that the striking results of CS demand equally striking assumptions on sensing matrices and ob jects of interest. Moreover, for problems of imaging physical constraints often result in sensing matrices not previously analyzed from the CS perspective. How does one conduct measurements and collect data in such a way that these desirable properties are provably met under all the constraints of wave physics? This question is typically evaded in most literature on this topic which is exactly the main focus of my research.
For example, in a series of papers I reformulate the far-field and near-field imaging prob- lems in the CS framework and put compressive imaging on a rigorous ground by proposing novel measurement schemes and establishing various desirable CS properties for the resulting measurement matrices depending on the imaging geometry and the physical constraints. The paper dealing with the single-input-multiple-output (SIMO) measurement schemes has been on the list of 10 most read papers of the journal Inverse Problems since its publication in February 2010.
Here is a brief summary of major findings reported in 5 recent papers.
Time reversal of waves has had a huge impact in optics (phase conjugation) and more recently in acoustics. It provides a critical technique for overcoming clutters and focusing wave energy in multiply scattering media .
I propose a time reversal multiple-input-multiple-output (MIMO) broadcast scheme to simultaenously transmit information through clutters to multiple users. The clutters are modelled as either the paraxial waves with random potentials or Rayleigh fading channels with or without multiple layers of randomly distributed pinholes. The issue is how to maximize the information rate without inter-user interferences. The communication scheme prescribes nearly minimum numbers of sufficiently separated transceivers and frequencies. I show rigorously that pinholes are detrimental to information transfer in the sense that they reduce the information rate exponentially as the number of layers increases. In the absence of pinholes, however, time reversal can achieve the physically possible optimal information rate to the same order of magnitude.
I (with Solna) prove that for waves in a fractal medium the time reversal resolution is a nonlinear (between linear and quadratic) function of the wavelength and independent of the aperture. Hence superresolution can be achieved by time reversal in rough media with subwavelength structures. We also prove a duality relationship between the forward propagation and time reversal. The duality has two aspects: First there is a fundamental uncertainty inequality between the wave spread and time-reversal resolution, analogous to the uncertainty principle in quantum mechanics. The inequality becomes an equality when the wave structure function is of the Gaussian form which means the media is smooth. Second, the time reversal resolution is identical to the coherence length.
Theory of radiative transfer (RT) was revived by the Nobel laureate Subrahmanyan Chandrasekhar in the middle of twentieth century and becomes an essential tool for analyzing wave propagation in random media such as stellar atmospheres.
My first main contribution is to introduce a novel martingale approach to wave analysis. Using the martingale method I derive rigorously several transport equations from self-averaging scaling limits. My second contribution is to introduce the two-frequency method to theory of radiative transfer. The goal of the two-frequency method is to describe the cross-frequency correlation which is lost in the standard RT theory. I extend the martingale method to the two-frequency analysis and derive two-frequency transport equations in various self-averaging scaling limits. I also find an exact solution to one of the transport equations and use it to estimate the information rate of the multiple-frequency time reversal scheme. Here the two-frequency method is an indispensible tool for the analysis.
The main issues I address in this work is competition or coordination between the nonlinearity to focus or defocus and the (weak) random potential to disperse. In particular, for white-noise potentials, I derived a variance (in)equality in a completely elementary way and derive the bound for wave spread and the condition for singularity formation. In the critical dimension, I obtain the exact law for wave spread which is surprising. I also derive similar variance (in)equality for nonlinear transport equations of various kinds.
Since my graduate school days, I had worked on this project for about ten years. I have the following main contributions
This is part of my effort of understanding how chaotic or random dynamics affect transport and mixing. But instead of the standard notion of mixing time which deals with the weaker notion of (reversible) mixing. I introduce a new notion of irreversible mixing (dissipation) time to account for the confluence of dynamics and external noise. I (with Wolowski and Nonnenmacher) analyze the dissipation time of chaotic systems and obtain results that inspire the recent research of fluid mixing in a new direction. The dissipation time is the time scale of noise-induced irreversibility and is stronly dependent on the underlying dynamics. We also extend our method and results to quantum chaotic systems. In the quantum context, the dissipation time is the time scale on which a small noise can make classically chaotic quantum systems behave like their classical counterparts. Here the dissipation time is related to the decoherence time.