Inverse Problems Seminar, Fall 2012 - Department of Mathematics at Colorado State University

[ Schedule ]   [ Abstracts ]

Seminar in Inverse Problems   [Spring 2015]

Inverse problems is a field of mathematics comprised of many areas including analysis, modeling, PDE's and scientific computation. Inverse problems arise in abundance in engineering, biology, physics, geophysics and more. This weekly seminar addresses fundamental topics in inverse problems in a variety of applications.

Regular meeting times & location: Thursdays at 2 pm in Weber 223



Upcoming Speaker: Kui Ren, University of Texas at Austin




Speaker: Kui Ren, University of Texas at Austin
Title: Inverse Transport Problems with Internal Data and Applications

Abstract: We consider here some inverse coefficient problems for the transport equation with multiple internal data sets. Such problems find applications in recent hybrid imaging modalities such as (fluorescence) photoacoustic tomography. We will discuss some theoretical results on the uniqueness and stability of the inverse problems and propose some efficient reconstruction strategies which we demonstrate with numerical simulations.








Spring 2015 Schedule

Jan. 29 no seminar
Feb. 5 Regularization of the Inverse Scattering Problem using Shearlet Frames Gitta Kutyniok,Technical University, Berlin
Feb. 12 no seminar
Feb. 19 no seminar
Feb. 26 A Novel Regularization Technique for Electrical Impedance Tomography in an Open Half-Space Domain Ethan Murphy, Dartmouth University
March 5 Seismic Tomography; Past, Present, and Future Rick Aster, Dept. of Geosciences,CSU
March 12 Colored Buzz, or Glottal Inverse Filtering Samuli Siltanen, University of Helsinki, Finland
March 26 Seeing Beyond the Diffraction Limit Peijun Li, Purdue University
April 2 no seminar
April 9 Title: Inverse Transport Problems with Internal Data and Applications Kui Ren, University of Texas at Austin
April 16 no seminar
April 23 no seminar
April 30 TBA




Abstracts





Gitta Kutyniok, Technical University, Berlin   Feb. 5
Title:Regularization of the Inverse Scattering Problem using Shearlet Frames

Abstract: The scattering problem analyzes how incident waves, radiation, or particles, which are transmitted in a medium, are scattered at inhomogeneities of this medium. The associated inverse problem aims to determine characteristics of the inhomogeneities from the asymptotic behavior of such scattered waves. This problem appears in various flavors in different application areas, e.g. non-destructive testing, ultrasound tomography, and echolocation. In this talk, our focus will be on regularization techniques for the numerical solution of inverse scattering problems in two space dimensions. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions. Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for regularization. We then analyze two approaches, namely the nonlinear problem and a linearized problem obtained by the Born approximation technique. For the first class, we study the acoustic inverse scattering problem, whereas for the second class, we consider the inverse scattering problem of the Schrodinger equation. Besides a theoretical analysis of our numerical approaches, we will also provide numerical examples that highlight the effectiveness of our schemes. This is joint work with Volker Mehrmann and Philipp Petersen (TU Berlin).



Ethan Murphy, Dartmouth University   Feb. 26
Title: A Novel Regularization Technique for Electrical Impedance Tomography in an Open Half-Space Domain

Abstract: Electrical impedance tomography (EIT) aims to recover the electrical properties within a domain of interest from boundary measurements. Although uniqueness proves mainly concern closed domains, there is significant interest with EIT in open half-space domains. The application of this study is concerned with an end-fired probe to be used for near real-time reconstructions during radical prostatectomy surgery. The goal is to provide the surgeon feedback on cancerous or noncancerous tissues along the margins and thus improve surgery outcomes. The system has 17 electrodes on the face of a 10 mm diameter probe. A novel regularization technique using the standard Gauss-Newton reconstruction algorithm has been developed, which produces significantly images with improved spatial resolution and decreased artifacts in measured saline tank experiments. The method applied to excised prostate tissue will be shown and discussed.



Rick Aster, Dept. of Geosciences,CSU   March 5
Title: Seismic Tomography; Past, Present, and Future

Abstract: Seismic tomography has its roots in the earliest days of seismology, and is the principal methodology for interrogating the deep internal structure of the Earth (as well as the Sun and other starts) at the highest obtainable levels of resolution. I will present a historic and mathematical overview of the key concepts of seismic tomography and its specific methods and applications at a variety of scales. I will also summarize developments up to the current state-of-the art, and will then discuss new data collection and methodological developments that will advance the field into the next decade and beyond.



Samuli Siltanen, University of Helsinki, Finland   March 12
Title: Colored Buzz, or Glottal Inverse Filtering

Abstract: This talk reveals the deep connection between Stephen Hawking, birthday cakes, and the heavy metal group AC/DC.
Human speech is a sophisticated means of communication and plays an unparalleled role in today's society. Whether developing the latest voice recognition software for a smartphone or designing computers to aid people who have lost their voice through disease and illness, researchers are finding ways to map the precise mechanisms in the human vocal tract.
Through a number of practical demonstrations, it is shown how the human vocal folds and the mouth and lips combine to create the vowel sound. Furthermore, it is explained how a technique called glottal inverse filtering (GIF) can be used to determine the exact mechanisms behind vowel sounds from microphone recordings.
The main use of improved GIF is to provide disabled women and children with better computer-based speech prostheses. The higher fundamental frequency of women's and children's voices makes them more difficult for speech synthesis software. Using Bayesian inversion and Markov chain Monte Carlo (MCMC) method, we can significantly improve GIF results over traditional engineering approaches for challenging speech signals.



Peijun Li, Purdue University   March 26
Title: Seeing Beyond the Diffraction Limit

Abstract: In this talk, our recent progress on a class of inverse surface scattering problems will be discussed. I will present new approaches to achieve subwavelength resolution for these inverse problems. Based on transformed field expansions, the methods convert the problems with complex scattering surfaces into successive sequences of two-point boundary value problems, where explicit reconstruction formulas are made possible. The methods require only a single incident field and are realized by using the fast Fourier transform. The convergence and error estimates of the solutions for the model equations will be addressed. I will also highlight some ongoing projects in rough and random surface imaging.



Kui Ren, University of Texas at Austin   April 9
Title: Inverse Transport Problems with Internal Data and Applications

Abstract: We consider here some inverse coefficient problems for the transport equation with multiple internal data sets. Such problems find applications in recent hybrid imaging modalities such as (fluorescence) photoacoustic tomography. We will discuss some theoretical results on the uniqueness and stability of the inverse problems and propose some efficient reconstruction strategies which we demonstrate with numerical simulations.