Support
Support from NSF grant DMS-1712790 (formerly 1634544, formerly 1514820) is gratefully acknowledged.Research Topics
My broad research interests are in Inverse problems and their applications to imaging sciences (e.g., medical and geophysical imaging). The analysis of an inverse problem divides into the following aspects: injectivity and stability assessments (i.e. is the unknown uniquely characterized by the data ? if yes, in what function spaces is the inverse continuous and what is its modulus of continuity ?); derivation of inversion formulas and their numerical implementation; strategies for dealing with and reducing noise in corrupted data. The main fields involved in my work are Analysis of PDE's (which is also the arXiv tag most often used for inverse problems papers, as "inverse problems" is not itself an arXiv classification), differential geometry, applied functional analysis and numerical methods.
I. Inverse problems in Boltzmann transport and applications to medical imaging
- Reconstruction of the scattering coefficient from angularly averaged boundary measurements in the stationary and time-harmonic setting, with applications in Optical Tomography.
- Numerical work (in Matlab and Python): coding a forward transport solver with image rotation techniques and controlled transverse diffusion to tackle the so-called "ray effect". Application to reconstruction of optical parameters in the stationary setting.
- Reconstruction of both attenuation and isotropic scattering from knowledge of ballistic and single scattering measurements, with applications to SPECT.
II. Theoretical and numerical aspects of coupled-physics (a.k.a. hybrid) medical imaging methods
- In particular, inverse conductivity problem with internal measurements of power density and current density type, with applications to some coupled-physics medical imaging modalities (UMEIT, UMOT, ImpACT, CDII). I am concerned with the derivation of reconstruction algorithms and their numerical validation in isotropic and anisotropic settings. Languages used for numerical work: MatLab and FreeFEM++.
- Reconstruction of isotropic and anisotropic elasticity tensors from internal displacement fields, with application in transient elastography.
III. Integral geometry, geodesic X-ray transforms, tensor tomography
- Numerical implementation of reconstruction algorithms for functions and solenoidal vector fields from knowledge of their ray transform. Theoretical and numerical generalization to other types of integrands (symmetric differentials and their transverse derivative).
- In two dimensions, reconstruction of solenoidal tensors of higher order.
- Study of non-simple metrics (e.g. metrics with conjugate points) and the detrimental effects of caustics on the qualitative properties of the X-ray transform. Study of cases with trapping and nontrivial topology.
- Inversion in the attenuated case.
Talks
(Hover on the title for some of the later abstracts)- Attenuated tensor tomography applied to the source reconstruction problem in transport
- The geodesic X-ray transform on surfaces with conjugate points (x)or trapping
- Efficient tensor tomography in fan-beam coordinates
- ANR Problèmes inverses, ENS Ulm, France, 05/04/2016 (in French)
- IPMS conference 2016, Fethiye, Turkey, 05/25/2016
- Toward imaging modalities with high(er) resolution
- Northeastern University, 01/19/2016
- University of Utah, 02/05/2016
- University of North Carolina at Charlotte, 02/09/2016
- Reconstruction methods for coupled-physics inverse problems
- Applied Mathematics Seminar, University of Utah, 11/30/2015
- Geodesic X-ray transforms on surfaces and tensor tomography.
- Spectral and Scattering Theory Seminar, Purdue University, 11/05/2015
- Applied and Interdisciplinary Mathematics Seminar, Northeastern University, 11/17/2015
- Applied Mathematics Colloquium, University of Utah, 02/05/2016
- Colorado School of Mines, 03/03/2016
- Inverse Problems Seminar, Colorado State University, 04/28/2016
- Inversions of ray transforms on simple surfaces
- "Geometric Inverse problems" conference, Institut Henri Poincaré, 06/09/2015
- Inversion of the attenuated geodesic X-ray transform on surfaces
- Applied Inverse Problems Conference 2015, Helsinki, 05/29/2015
- Joint Mathematical Meetings, Seattle, 01/08/2016
- Coupled-physics Inverse Problems for the System of Elasticity
- Applied Inverse Problems Conference 2015, Helsinki, 05/25/2015
- Four lectures on Integral Geometry on Riemannian surfaces with boundary
- Inverse Problems Seminar, U. of Washington, 05/13-14-15-18/2015
- Numerical Methods for geodesic X-ray transforms and applications to open theoretical questions
- Numerical Analysis Research Club, U. of Washington, 11/13/2014.
- Recent progress on the explicit inversion of geodesic X-ray transforms
- Geometric Analysis and PDE seminar, U. of Cambridge, 05/05/2014.
- Inverse Problems seminar, U. of Manchester, 04/29/2014
- Demi-journée problèmes inverses, Maison Jean Kuntzmann, Grenoble, 04/17/2014
- Inverse Problems and Imaging conference, Institut Henri Poincaré, Paris, 04/10/2014
- Research Seminar, Purdue University, Indiana, 01/17/2014.
- Inverse Problems and Geometry conference, Banff, Alberta, 09/17/2013.
- Coupled-physics methods for inverse conductivity
- Applied Inverse Problems Conference, KAIST (Daejeon, Korea), 07/01/2013.
- Inverse anisotropic diffusion from power density functionals in two dimensions
- Pontificia Universidad Católica de Chile, 09/03/2014, Santiago Chile.
- University of Helsinki, 06/10/2013.
- PASI Inverse Problems and PDE Control, 01/24/2012, Santiago Chile.
- The inverse conductivity problem with power densities in general dimension
- Inverse Problems conference in honor of Gunther Uhlmann, 06/19/2012, UC Irvine.
- Inverse Diffusion with power density measurements
- APAM Research Seminar, 03/04/2011, Columbia University.
- An accurate solver for forward and inverse transport
- Inverse Transport and Tomography conference, 05/2010, Banff, Alberta.