Geo Mathematical Imaging Group, Purdue University

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Research

Maarten de Hoop
Director, GMIG
mdehoop@purdue.edu

Dept of Mathematics
Purdue University
150 N Univ St
West Lafayette, IN 47907
ph. (765) 496-7678
Fax (765) 496-1169

Barbara Doremire
Assistant to the Director
bjd@purdue.edu

Research summary

GMIG faculty and their collaborators pioneered the notions of (i) extensions and annihilators in imaging and inverse scattering based on the single scattering approximation, (ii) partial reconstruction using curvelets, and (iii) ray-wave duality using multi-scale analysis. The extensions lead to microlocally invertible operators critical in the process of velocity inversion using annihilators in the presence of caustics, while they also lead to Fourier integral operators associated with canonical graphs, which are amenable to compression and fast algorithms again using multi-scale techniques and prolate sphedroidal wave functions.


Key results in 2011-2012 include the completion of a randomized sampling approach in structured direct approximate solvers which avoids the storage of any dense matrices, a massively parallel structured multifrontal solver for (multi-component) time-harmonic elastic waves in 3D anisotropic media implicitly allowing the relevant systems to be of non-principal type, the introduction of a projected nonlinear steepest descent method including frequency progression and an associated convergence criterion for full waveform inversion (FWI), and conditional regularity estimates leading to the notion of an internal multiple scattering series. From a data perspective, we present results on frequency extrapolation investigating whether it is possible to infer from finite-bandwidth data information about the wavefield at frequencies below, and also above, this bandwidth. We furthermore revisit sampling criteria and introduce a new nonlinear wavefield reconstruction across acquisition gaps using wave packets. Moreover, we relate the Dirichlet-to-Neumann map, used as the data in our convergence analysis of FWI via a single layer potential operator to "source blending", and justify the use of the Hilbert-Schmidt or the Schatten $p$-norm. Moreover, we obtained a direct reconstruction of a Riemannian metric (that is, a velocity field) using diffraction time expansions, which are closely related to the so-called shape operator, as the data. The reconstruction gives the metric tensor in Riemannian normal coordinates. We restrict ourselves to metrics which are conformal to the Euclidean metric (corresponding to isotropic velocity fields) and complete the reconstruction in Cartesian coordinates. We derive a closed system of differential equations the solution of which generates the relevant coordinate transform, which can be viewed as a generalization of the time-to-depth conversion step in Dix' method. Our framework for fast algorithms is founded on the principle of finite accuracy computation and compression with controlled error.