High-fidelity fast algorithms for inverse problems and imaging in three dimensions
Abstract
Acoustic and electromagnetic scattering problems cover a wide range of very important topics incomputational wave phenomena, with applications in medical imaging, non-destructive testing,ocean acoustics and climate science, and radar and sonar. In classical scattering problems, the goalis to determine the response of an inhomogeneity (e.g. an obstacle or material with spatiallyvarying sound-speed) to a known incoming wave. These problems are governed by linear partialdifferential equations (PDEs) whose physical and mathematical underpinnings are very wellunderstood. Furthermore, in the past decade, state of the art high-order accurate computationalmethods for solving such scattering problems have finally been developed (e.g. fast multipolemethod-accelerated boundary integral equation solvers).On the other hand, the inverse scattering problem is to determine the nature of theinhomogeneity from knowledge of both the incoming field and the scattered field. Example inversescattering problems include determining the location and size of a tumor in medical imaging,detecting defects in structural components in non-destructive testing, and identifying objects viasonar or radar. While arguably more important than forward scattering problems, the availablehigh-fidelity computational tools for solving inverse scattering problems is much less mature.Inverse problems are highly nonlinear by nature, and often quite ill-conditioned. Most high-orderaccurate solvers only address two-dimensional problems, and in some respects, the underlyingmathematics in various regimes is still being investigated.Our proposed work is aimed at the development of high resolution modern computationalmethods for inverse scattering problems in three dimensions by leveraging the recent advances inasymptotically fast integral equation solvers, for example fast multipole methods (FMMs), fastdirect solvers, and related algorithms. Efficient, first principles algorithms for inverse problems inthree dimensions has historically been hindered by the lack of efficient solvers for the forwardproblem. The landscape of this problem has changed only in the past few years, and it is nowpossible to bring to bear these fast solvers on long outstanding inverse problems. We intend tofocus on three specific inverse problems in three dimensions: (1) recovering the shape of theboundary of arbitrary impenetrable obstacles, (2) recovering a compactly supported spatiallyvaryingsound-speed profile in a volumetric region, and (3) recovering the shape and impedanceboundary condition along the surface of an obstacle.We plan to solve each of the above problems first in the time-harmonic acoustic scatteringregime, and then extend the results to the time-harmonic electromagnetic scattering regime. Eachclass of algorithms developed will rely on formulating the inverse scattering problem as anoptimization problem and stably computing minimizers via a Gauss-Newton method. The keymathematical idea to be used is known as recursive linearization, by which a sequence of inverseprobnverse problem isinitialized with the minimizer from a slightly lower-frequency inverse problem. Furthermore,analytic generalized shape and material gradients can be computed via solving an associatedscattering problem. Achieving these objectives will specifically leverage computational PDE andcomputational geometry developments made by ONR-funded awards over the last five years.This project is most relevant to ONRs Science and Technology Organization, Code 31. Inparticular, the work is most directly related to the effort of Division 311: The Applied andComputational Analysis Program.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- May 05, 2021
- Source ID
- N000142112383
Entities
People
- Michael O Neil
Organizations
- New York University
- Office of Naval Research
- United States Navy