Continuous Optimization, Variational Analysis,

 and Inverse Problems


Job Openings: 

Research Projects:

The unifying focus of our work is algorithms, and our approach is best described by the experimental mathematics philosophy discussed below. Our research in variational analysis, optimization and inverse problems is motivated by applications in three areas: image processing, inverse scattering, and computational quantum chemistry. I have grouped my publications into two principal mathematical disciplines, variational analysis/optimization and scattering theory. This grouping represents the distinct phases and interests during my research career, though the two areas overlap to some degree. A third category, experimental and scientific mathematics, characterizes my broader vision of mathematics, the practice of mathematical research and the role these have in the empirical sciences. Within each discipline area I have grouped papers by application and research thread. I describe the content of each paper and how it relates to my own research in the cases where the paper was a collaborative effort. Almost all of my collaborative papers follow the standard practice in mathematics of listing authors alphabetically to reflect the fact that these are equally shared efforts. In the cases where alphabetical ordering was not followed, the lead author is listed first.

See below for software  industry collaboration, and my research collaborators.  This work has been supported in part by grants from the National Science Foundation (grant Numbers 0712796 and 0852454) and more recently from the German Research Foundation (Nahost Kooperation, Graduiertenkolleg 2088, Collaborative Research Center 755 and Graduiertenkolleg 1023), the Bundesministerium fuer Bildung und Forschung, the German Israeli Foundation and the Australian Research Council.



Active Research Projects:
  1. Mathematics of atomic orbital tomography   with Stefan Mathias (CRC1456 TPB01)

  2. Stochastic computed tomography: theory and algorithms for single-shot X-FEL imaging   with Helmut Grubmueller (CRC1456 TPC02)

  3. Nonconvex Quadratic Composite Minimization: Theory and Algorithms, with Marc Teboulle and Shoham Sabach.  (DFG, 2020-2022)

  4. Probabilistic Analysis and Stochastic Algorithms in Fixed Point Theory  with Anja Sturm (GRK2088 TPB5)

  5. Discrete Topological Optimization Techniques for the Statistical Analysis of Tree Structures  with Stephan Huckemann (GRK2088 TPA4)

A selection of expired research projects:


Conferences:  I have been involved in the organization of the following events:




Experimental and Scientific Mathematics

ramanujan continued fraction




Variational Analysis and Optimization



General





Nonconvex Feasibility

Projection schema




Phase Retrieval

Fourier magnitude constraint




Inverse Scattering Theory

1 direction, 10 frequencies
    This work has developed from a two-year postdoctoral research position with Rainer Kress' group at the University of Goettingen. My work has focused mainly on noniterative integral equation techniques for obtaining qualitative information about scattering obstacles from far field measurements. This work is in collaboration with Roland Potthast of the University of Goettingen and more recently with Anthony Devaney of Northeastern University, Tilo Arens of Karlsruhe Institute of Technilogy and Armin Lechleiter of University of Bremen. I have also published two independent works in this area.

    • `` MUSIC for extended scatterers as an instance of the Factorization Method.'', with Tilo Arens and Armin Lechleiter. SIAM J. Appl. Math. 70(4):1283-1304 (2009).
    • Recent work of Luke and Devaney showed that there exists an implementation of a modified linear sampling method that is equivalent to a MUSIC algorithm for scattering from sound soft obstacles. The correspondence is independent of the size of the scatterer or the wavelength of the incident field. As the proof was not constructive, an explicit implementation could not be justified. In the present work, we show that MUSIC is an instance of the factorization method applied to any nonabsorbing scatterer, thus providing a justification for the MUSIC algorithm at arbitrary illuminating frequency for arbitrary nonabsorbing scatterers. These results are also extended to scattering from cracks. With explicit constructions in hand, we are also able to provide error and stability estimates for practical implementations in noisy environments with limited data and to explain a curious behavior of the factorization method in the case of noisy data.

    • "Identifying scattering obstacles by the construction of nonscattering waves'', with Tony Devaney. SIAM J. Appl. Math. (2007).
    • I used the linear sampling method, together with the point source method of Potthast to prove the existence of a wave with arbitrarily small scattered field on the exterior of a Dirichlet obstacle. We use such an incident field to implement a MUSIC-type algorithm for determining the shape and location of extended scatterers without restrictions on the size of the obstacles or the frequency of the incident field. Though our numerical experiments demonstrated a ``proof of concept'' the theory for how one generates such a field was not resolved in this paper with Devaney. Together with Tilo Arens and Armin Lechleiter of the University of Karlsruhe, I have since come up with a constructive proof that also yields a theoretical justification of the numerical experiments in the paper with Devaney.

    • `` The Point Source Method for Inverse Scattering in the Time Domain'', with Roland Potthast. Mathematical Methods in the Applied Sciences 29(3): 1501--1521(2006).
    • Our goal here is twofold: first, to establish conditions on the time-dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi-frequency data.

    • `` Image synthesis for inverse obstacle scattering using the eigenfunction expansion theorem'', Computing 75(2-3):181-196(2005).
    • Potthast's point source method and its relatives determine the boundary of an unknown obstacle by reconstructing the surrounding scattered field. In the case of sound soft obstacles, the boundary is usually found as the minimum contour of the total field. In this work we derived a different approach for imaging the boundary from the reconstructed fields based on a generalization of the eigenfunction expansion theorem that is an extension to scattering obstacles of work by Rose and Cheney (1988). The aim of this alternative approach is the construction of higher contrast images than is currently obtained with the minimum contour approach.

    • ``Multifrequency inverse obstacle scattering: the point source method and generalized filtered backprojection'' Mathematics and Computers in Simulation 66:297-314(2004).
    • In this work I present Potthast's point source method for multifrequency data as a nonlinear generalization of the filtered backprojection algorithm and compare this generalization to the usual filtered backprojection technique based on the physical optics approximation. The practical importance of the paper is to show how one should average multifrquency data for reconstructions. This paper should get more attention as essentially single frequency, time-harmonic inverse scattering techniques are extended to time-domain inverse scattering.

    •  ``The no response test - a sampling method for inverse scattering problems ", D. R. Luke and R. Potthast. SIAM J. App. Math. 63(4):1292-1312 (2003).
    • The main theoretical interest of the paper is that this is a technique for determining the shape of scatterers from a single incident wave.

    • ``The Point Source Method in Acoustic Scattering :  numerical reconstruction of the scattered field from far field measurements of inhomogeneous media'', D. R. Luke and R. Potthast.  Proceedings of the IEEE 2002 International Conference on Acoustics, Speech and Signal Processing,  pp.IV-3541-IV-3544 (Orlando, FL, May 13-17, 2002).
    • This is an application of Potthast's point source method to inhomogeneous media.


top



Software



  

Industry Cooperation



Research Collaborators

top
 

 (main page)