Research

I am presently working on four different projects:

• on "Microwave Imaging for detecting anomalies in living tissue using Adaptive Eigenspace Inversion"
  with Andrew Austin (University of Bristol, UK) and Mark Eltom (Vine Life Ltd)

• on "Adaptive Eigenspace Inversion: theory and Bayesian extension"
  with Bamdad Hosseini (University of Washington, Seattle WA, USA), Melissa Tacy (University of Auckland) and Marcus Grote (University of Basel, Switzerland)

• on "Imaging a kiwifruit"
  with Evert Duran Quintero and Kasper van Wijk (Physics, University of Auckland, NZ) 

• on "Inverse scattering problem with uncertain source term"
  with Ru Nicholson (Engineering Science, University of Auckland, NZ) 

I have been awarded the NZMS Early Career Research Award at the occasion of the Joint NZMS-AustMS-AMS meeting held in Auckland in December 2024.

I was asked to make a short video of presentation: watch the video.

Since February 2023, I received a Marsden Fast-Start (UOA2225) grant to work on Adaptive Eigenspace Inversion with Bamdad Hosseini (University of Washington, Seattle, WA USA). The funding also includes an MSc student, Tingwei Cheng, and a PhD student, Nasrin Nikbakht. Another collaborator on this project is Marcus J Grote (University of Basel, Basel, Switzerland).

See the links below from the Royal Society's website featuring my work:

• Is that buried treasure or a fish? Improving the maths that underlie high resolution imaging

• Using mathematics to get the big picture

EXTENDED IMAGE VOLUMES

We propose a low-rank representation of omnidirectional subsurface extended image volumes (EIVs), which are usually described as large dense matrices. Our approach combines probing techniques and randomized SVD algorithms and proves to be cost-effective. We indeed build the EIV without explicitly calculating the forward and adjoint wavefields, therefore the customary loop over sources is avoided. Moreover, storage issues are overcome thanks to the resulting low-rank factorization of the EIV. Then, information of the extended image volume can be extracted by evaluating a diagonal-estimator-based formula using a small number of probing vectors (relatively to the number of sources) that actually span the range of the transposition of the EIV. Our approach is developed for both the time-dependent and the time-harmonic wave equations.

Joint work with R. Kumar, M. Wang and F. J. Herrmann (2018)

LOCAL LIMIT OF NONLOCAL CONSERVATION LAWS

We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments suggest convergence in the local limit. However, recent analytic results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii) convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytic results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that the numerical viscosity of Lax-Friedrichs type schemes jeopardizes the reliability of the numerical scheme and erroneously detects convergence in cases where convergence is ruled out by analytic results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide more reliable results. 

Joint work with M. Colombo, G. Crippa and L.V. Spinolo (2018) 

INVERSE PROBLEMS: ADAPTIVE EIGENSPACE INVERSION METHOD (AEI)

We propose a new approach to solve the inverse scattering problem: the aim is to recover the location, the shape and the physical properties of an unknown obstacle surrounded by a known ambient medium. Our approach works directly with the wave equation in the time-dependent domain. The Adaptive Inversion (AI) method was initially proposed for the time-dependent viscoelasticity equation by M. de Buhan and A. Osses: The originality of this method comes from the parametrization of the problem. Instead of looking for the value of the unknown parameter at each node of the mesh, it projects the parameter into a basis composed by eigenvectors of the Laplacian operator. Then, the AI method uses an iterative process to adapt the mesh and the basis of eigenfunctions from the previous approximation to improve the reconstruction.

Joint work with M. de Buhan (2013)

A nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization.

 Joint work with M.J. Grote and U. Nahum (2016)

Parameter estimation is a fundamental task in engineering and science such as medical imaging and seismology. The aim of imaging is to infer characteristics of a medium from indirect measurements. Imaging problems are often cast as PDE constrained optimization problems that are typically ill-posed in the sense of Hadamard. Therefore, regularization is needed to define a well-posed problem and stabilize the solution. In this work, we analyse and extend a novel regularization technique called Adaptive Eigenspace Inversion (AEI). We will establish a rigorous theory of AEI that explains its link with well-known regularisation methods like total variation, and develop a probabilistic analogue of the method.

Joint work with B. Hosseini (2019)

WAVE SPLITTING AND SOURCE SEPARATION

We propose a new method for source separation in the time-dependent domain. The method uses absorbing boundary conditions to filter the signals. The novelty of this technique leads in the fact it is local, deterministic, derived directly for the time-dependent wave equation and does not rely on the frequency spectrum of the signal.

Joint work with M.J. Grote, F. Nataf and F. Assous (2016)

TIME-REVERSED ABSORBING CONDITIONS (TRAC)

We introduce time reversed absorbing conditions (TRAC) in time reversal methods. They enable one to “recreate the past” without knowing the source which has emitted the signals that are back-propagated. We present two applications in inverse problems : the  reduction of the size of the computational domain and the determination, from boundary measurements, of the location and volume of an unknown inclusion. The method does not rely on any a priori knowledge of the physical properties of the inclusion. Numerical tests with the wave equation illustrate the efficiency of the method. This technique is fairly insensitive with respect to noise in the data. In particular the TRAC method is applied to the differentiation between a single inclusion and a two close inclusion case.

Joint work with F. Nataf and F. Assous (2012)

Click on the picture to be redirected to the movie.

I prepared my PhD at the Laboratory Jacques-Louis Lions, Paris, France, under supervision of Frédéric Nataf.

I defended my PhD thesis on July 2nd 2012 in the presence of the jury

• Frédéric Nataf (supervisor), LJLL, CNRS, Paris, France ;

• Franck Assous (co-advisor), Ariel University Center, Israel ;

• Marc Bonnet (reviewer), ENSTA Paritech, CNRS, Paris, France ;

• Chrysoula Tsogka (reviewer), University of Crete, Heraklion, Greece ;

• Yvon Maday (president), LJLL, Université Pierre et Marie Curie, Paris, France ;

• Marcus Grote (jury), Basel Universität, Basel, Switzerland ;

• Jean-Paul Montagner (jury), IPG, Université Paris-Diderot, Paris, France.

My thesis work is about:

Signals reconstruction and objects identification by the TRAC method in time reversal.

Manuscript (in French) available on TEL.