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

[ Schedule ]   [ Abstracts ]

Seminar in Inverse Problems   [Spring 2017]

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 seminar addresses fundamental topics in inverse problems in a variety of applications.

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



Speaker: Ken McLaughlin, CSU



Time and location: Thursday, 2 pm, Weber 223





Title: Inverse problems and integrable PDEs

Abstract: I will discuss several nonlinear PDEs that are integrable: the Korteweg - de Vries equation, the nonlinear Schroedinger equation, and the Davey-Stewartson equation. They are solved by methods referred to as inverse spectral / inverse scattering methods. I hope to explain this in a manner accessible to all. My interest is in understanding by rigorous analysis singular behaviors of the solutions of these equations. I seek to characterize phenomena such as the resolution into solitons, the formation of shocks, and the onset and propagation of wild oscillations. These phenomena are ubiquitous in nonlinear dynamical systems and PDEs, but analytical results are usually only possible in integrable examples. Open research directions will be discussed along the way.




Fall 2017 Schedule

Aug. 24 Electrical impedance tomography imaging via the Radon transform Samuli Siltanen, University of Helsinki
Sept. 7 no seminar - JM at IMA
Sept. 20 The enclosure method for the anisotropic PDEs Yi-Hsuan Lin, University of Washington
Oct. 12 Forward and inverse problems in geodynamic modelling: Part I -- Evolution of island chains in the South Atlantic and Indian Ocean Rene Gassmoeller, CSU
Oct. 26 Forward and inverse problems in geodynamic modelling: Part II -- Thermochemical Convection Juliane Dannberg, CSU
Nov. 2 no seminar - Math Day
Nov. 9 Monte Carlo methods for radiative transfer with singular kernels Olivier Pinaud, CSU
Nov. 16 Inverse problems and integrable PDEs Kenneth McLaughlin, CSU




Abstracts

Samuli Siltanen, University of Helsinki   Aug. 24
Title: Electrical impedance tomography imaging via the Radon transform

Abstract: In Electrical Impedance Tomography (EIT) one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful tool for both analysis and practical reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam tomography of the conductivity. One of the consequences of this ``nonlinear Fourier slice theorem'' is a novel capability to recover inclusions within inclusions in EIT.

Yi-Hsuan, University of Washington   Sept. 21
Title: The enclosure method for the anisotropic PDEs

Abstract: We develop an enclosure-type reconstruction scheme to identify penetrable obstacle for the acoustic waves and the electromagnetic field with anisotropic medium in 3-dimensional space. The main difficulty in treating this problem lies in the fact that there are so far no complex geometrical optics (CGO) solutions (available for the anisotropic elliptic equation and anisotropic Maxwell's equation. Instead, we derive and use another type of special solutions called oscillating-decaying solutions. To justify this scheme, we use Meyers' L^{p} type estimates for both systems, to compare the integrals coming from oscillating-decaying solutions and those from the reflected solutions.

Rene Gassmoeller, CSU   Oct. 12
Title: Forward and inverse problems in geodynamic modelling: Part I -- Evolution of island chains in the South Atlantic and Indian Ocean

Abstract: Earth's surface shows many features whose genesis can only be understood through their connection with processes in Earth's deep interior. Because the Earth's interior is mostly inaccessible to direct observations, numerical modeling provides an excellent tool to study these processes. However, it remains a fundamental challenge to relate the complex dynamics of the Earth's interior to observations: Observational data are limited to either a snapshot of the present-day state, derived from seismic tomographic models, or the evolution of the Earth's surface through time, provided by the rock record. In addition, already solving the forward problem poses a computational challenge: The high resolution required to resolve the convection of rocks in the Earth's interior, together with the large number of time steps needed to capture the evolution of the Earth, lead to forward models that can cost millions of CPU hours. Both points listed above make it difficult to use the available data to invert for the Earth's temperature and chemical structure, and how it evolved through time. Here, I present solution strategies the geodynamics community applies to this problem, showcasing the example of how the Tristan-Gough island chain in the South Atlantic, and the Reunion island chain in the Indian Ocean formed and interacted with mid-ocean ridges.

Juliane Dannberg, CSU   Oct. 26
Title: Forward and inverse problems in geodynamic modelling: Part II -- Thermochemical Convection

Abstract: Earth's surface shows many features whose genesis can only be understood through their connection with processes in Earth's deep interior. Because the Earth's interior is mostly inaccessible to direct observations, numerical modeling provides an excellent tool to study these processes. However, it remains a fundamental challenge to relate the complex dynamics of the Earth's interior to observations: Observational data are limited to either a snapshot of the present-day state, derived from seismic tomographic models, or the evolution of the Earth's surface through time, provided by the rock record. In addition, already solving the forward problem poses a computational challenge: The high resolution required to resolve the convection of rocks in the Earth's interior, together with the large number of time steps needed to capture the evolution of the Earth, lead to forward models that can cost millions of CPU hours. Both points listed above make it difficult to use the available data to invert for the Earth's temperature and chemical structure, and how it evolved through time. Here, I present solution strategies the geodynamics community applies to this problem, showcasing the example of how hot, rising material in the Earth's mantle can explain geochemical observations of the Hawaiian Islands.

Olivier Pinaud, CSU   Nov. 9
Title: Monte Carlo methods for radiative transfer with singular kernels

Abstract: Radiative transfer equations (RTE) arise in various applications such as light propagation, neutron transport, and gas dynamics. The solution depending on both the position and the momentum, the large dimension of the space of variables makes it difficult to use standard grid based methods such as finite elements. Monte Carlo (MC) methods offer then an efficient alternative requiring a minimal coding effort. In the context of wave propagation in random media with long-range dependence that we are interested in, it turns out that some coefficients in the RTE are singular. This makes the simulation of the stochastic processes inherent to MC methods more difficult than in the classical smooth case, and we will present efficient techniques to overcome this issue. The talk is meant to be educational and a large part of it will be dedicated to the basics of MC methods.

Ken McLaughlin, CSU   Nov. 16
Title: Inverse problems and integrable PDEs

Abstract: I will discuss several nonlinear PDEs that are integrable: the Korteweg - de Vries equation, the nonlinear Schroedinger equation, and the Davey-Stewartson equation. They are solved by methods referred to as inverse spectral / inverse scattering methods. I hope to explain this in a manner accessible to all. My interest is in understanding by rigorous analysis singular behaviors of the solutions of these equations. I seek to characterize phenomena such as the resolution into solitons, the formation of shocks, and the onset and propagation of wild oscillations. These phenomena are ubiquitous in nonlinear dynamical systems and PDEs, but analytical results are usually only possible in integrable examples. Open research directions will be discussed along the way.



Past Seminars

[Spring 2015] [Fall 2014]