Research links at Columbia University:

Below are brief descriptions of some research areas.

Ultrasound tomography enjoys high resolution capabilities. However, because sound speeds vary little between healthy and non-healthy tissues, it is low-contrast.

One way to combine large contrast and high resolution is to use the

Mathematically, these inverse problems involve parameter reconstructions from knowledge of

For recent results on QPAT and the related quantitative Thermo-acoustic Tomography (QTAT), see the papers Inverse Diffusion Theory of Photoacoustics (with G. Uhlmann),

More generally, QPAT, QTAT, as well as the elasticity-based modalities Transient Elastography (TE) and Magnetic Resonance Elastography (MRE) all involve reconstructions of parameters in the elliptic equation \[\nabla\cdot a\nabla u+b\cdot\nabla u+cu=0\] from knowledge of internal functionals of the form \[H(x)=u(x) \quad \mbox{ or } \quad H(x) = \Gamma(x) u(x) \] for some typically unknown function \(\Gamma(x)\). The analysis of what can and cannot be reconstructed in the possibly

Another hybrid imaging methodology is

Recent results on the problem are reported in Cauchy problem for Ultrasound Modulated EIT ,

A recent

The

What the random fluctuations look like and what they depend upon is analyzed in a series of papers Central limits and homogenization in random media, (long version with more detailed proofs arXiv:0710.0363 ),

These papers show that the size of the random fluctuations of the elliptic solution strongly depend on whether the random parameters are short-range or long-range. The random fluctuations also need not be asymptotically Gaussian as is shown in Random Homogenization and Convergence to Integrals with respect to the Rosenblatt Process (with Y. Gu), submitted.

Consider the parabolic Anderson model \[ \partial_t u_\varepsilon + (-\Delta)^{\frac m2} u_\varepsilon + \frac{1}{\varepsilon^\alpha}q(\frac x\varepsilon) u_\varepsilon =0\] for \(\alpha\) chosen so that the

In dimensions \(d\geq m\), we obtain in the limit a homogenized, deterministic equation as is shown in: Homogenization with large spatial random potential,

In smaller dimensions \(d < m\), the solution to the equation with heterogeneous coefficients does not converge to a deterministic equation but rather to a stochastic partial differential equation (SPDE) with multiplicative noise as is shown in: Convergence to SPDEs in Stratonovich form,

The generalization to long-range, possibly time-dependent, potentials is analyzed in: Convergence to Homogenized or Stochastic Partial Differential Equations,

There are several applications to the macroscopic modeling of the random fluctuations of PDE solutions. One such application is the derivation of physics-based noise models in Inverse Problems; see below.

An other application pertains to the testing of numerical multi-scale algorithms. We can then ask ourselves whether in a well-understood environment, the random fluctuations generated by the algorithm are accurate discretizations of the random fluctuations of the continuous model; see Corrector theory for MsFEM and HMM in random media (with W. Jing),

Here is a two-hour talk given at the end of a course on "Waves in Random Media", two tutorial lectures ( Lecture 1; Lecture 2) given in September of 2005 at the CIRM, Luminy, France, and a fairly long presentation given in the fall of 2003 at IPAM, on wave propagation in random media and time reversal.

Here is a link to Lecture Notes for a course on waves in random media given at Columbia in the fall of 2005.

Inverse transport theories strongly depend on the amount of scattering in the forward transport equation. In highly scattering environments, inverse transport is replaced by inverse diffusion problems, which are well-studied. In the absence of scattering, inverse transport is an integral geometry problem, which is also well understood. Inverse transport bridges the gap between the two regimes. See the paper Inverse Transport Theory and applications (review paper),

A major difficulty in inverse transport theory is to understand which optical properties of the underlying medium may be stably reconstructed. This crucially depends on the source (isotropic versus angularly resolved) and on the measurements (time dependent or not, angularly resolved or angularly averaged). Here is a sequence of recent papers on this topic. Time-dependent angularly averaged inverse transport (with A. Jollivet),

Here is a sequence of three talks ( Lecture 1, Lecture 2, Lecture 3) given on the subject at the University of Washington during the Summer School organized in August 2005 by Gunther Uhlmann.

SPECT (Single Photon Emission Computed [or Computerized] Tomography) is an inverse source problem. One is interested in reconstructing the spatial distribution of a source of radiation from boundary measurements. Mathematically, because the photons are absorbed by human tissues before they can be measured, one has to invert an

Ray transforms also appear in geophysical imaging. The main difference with respect to most imaging techniques is that rays are now curved. In the simplified (yet practically useful) case where the rays are the geodesics of a hyperbolic metric, I have extended the explicit formulas obtained in Euclidean geometry to the scalar and vectorial source problem in hyperbolic geometry; see the paper: Ray Transforms in Hyperbolic Geometry.

Ray transforms also find applications in the reconstruction of concentration profiles in the atmosphere. In the non-scattering framework, frequency-dependent radiation emitted by different gases may be measured by satellites. In the one-dimensional setting, there is only one ray (radiation moves up). The z-dependent concentration profiles may thus be reconstructed from the frequency dependency in the measurements. Unlike the ray transform problems mentioned above, the concentration profile reconstruction is a severely ill-posed problem (so that noise is severely amplified during the reconstruction process). With Kui Ren, we show this and provide methods to recover thin layers with sharp contrast (dust layers or ozone layers) in the following paper Atmospheric Concentration Profile Reconstructions from Radiation Measurements.

Here is a presentation I gave at IPAM in the fall of 2003 on SPECT problems.

Motivated by works by S. Arridge, I have worked on the modeling of photon propagation in domains that are highly scattering everywhere except in small-volume non-scattering layers. The understanding of such domains is important if photons are to be used to image properties of the human head.

With Kui Ren, we have shown that a macroscopic (hence computationally cheap)

Inverse Problems in OT are known to be extremely ill-posed, with an error in the reconstruction of the diffusion and absorption properties logarithmic in the error level on the measured data (in the sense that an accuracy of 10E-10 in the data may result in an accuracy of 1/10 in the reconstruction). Here is a way to somewhat overcome this stability constraint by using asymptotic expansions of small volume inclusions. Another promising approach to increase the amount of available data is to use time harmonic sources. The following paper in collaboration with with K. Ren and A. Hielscher deals with this issue: Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer. In that paper, we show that increasing the source modulation frequency improves the reconstruction of the optial coefficients. Theoretical explanations based on careful stationary phase analyses can be found in the two papers: Angular average of time-harmonic transport solutions (with A.Jollivet, I.Langmore, and F.Monard)