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: Paul Sava, Colorado School of Mines



Time and location: Thursday, 2 pm, Weber 223





Title: Wavefield tomography of comet interiors

Abstract: Fundamental answers about small planetary bodies (comest and asteroids) origin and evolution hinge on our ability to image in detail their interior structure in 3D and at high resolution. The interior structure is not easily accessible without redundant data, e.g., radar reflections observed from multiple viewpoints, as in medical tomography. The Comet Radar Explorer mission aims to acquire a dense network of in-phase radar echoes from orbit to obtain a high resolution 3D image of comet Tempel 2's interior. Full wavefield inversion facilitate high quality imaging of the comet interiors, particularly if the comet nucleus is characterized by complex structure and large contrasts of physical properties. Knowledge of the comet shape and all-around orbital radar acquisition enable accurate and computationally-efficient 3D tomography. Using realistic numerical experiments, I will argue that (1) interior comet imaging is intrinsically 3D and conventional SAR processing does not satisfy imaging/resolution requirements; (2) interior tomography can be performed progressively with data acquired through successive orbits around the comet; (3) exploiting the known exterior shape of the comet facilitates cost-effective wavefield tomography; (4) multiscale wavefield tomography provides detailed information about the 3D internal physical properties; (5) imaging resolution exceeds the wavelength limits corresponding to low-frequency radar waves; (6) monostatic tomography yields 3D models comparable with what could be accomplished by more complex systems.




Spring 2018 Schedule

Jan. 18 Computational Uncertainty Quantification for Inverse Problems Johnathan Bardsley, University of Montana
Feb. 1 Electrical Impedance Tomography (EIT) Instrumentation: A Journey Through Design, Development, and Applications Tushar Kanti Bera, Dept. of Medical Imaging, University of Arizona
Mar. 8 Improving Electrical Impedance Tomography with Robust D-bar Methods Sarah Hamilton, Marquette University
Mar. 29 TBA Randy Bartels, Colorado State University
April 12 Wavefield tomography of comet interiors Paul Sava, Colorado School of Mines


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.

Johnathan Bardsley, University of Montana   Jan. 18
Title: Computational Uncertainty Quantification for Inverse Problems

Abstract: In this talk, I will begin with a review of the basic characteristics of inverse problems before moving into Bayesian methods for inverse problems. The connection between the choice of the regularization function in classical inverse problems and the choice of the prior probability density function (or simply the prior) in Bayesian inverse problems is well-known. Less well-known is that in imaging, Gaussian priors (quadratic regularization functions) can be derived from pixel-level statistical assumptions about the unknown image using Gaussian Markov random fields (GMRFs). With a GMRF prior in hand, uncertainty quantification is then performed by sampling from the posterior density function (or simply the posterior). When both the measurement error and prior variances are known, and the inverse problem is linear, each posterior sample requires the solution of a large linear system. When the inverse problem is nonlinear, however, a Markov chain Monte Carlo method is needed to compute posterior samples; we use Metropolis-Hastings (MH) with a proposal defined by an optimized-based sampling method called randomize-then-optimize (RTO). In cases where both the measurement error and prior variances are unknown, a so-called hierarchical model assumes hyper-prior probabilities on these scalar parameters, adding another layer of complexity to the posterior and to the resulting sampling problem. We end the talk by presenting a Gibbs sampler for hierarchal linear inverse problems and then discuss its extension - using RTO-MH - to hierarchal nonlinear inverse problems.

Tushar Kanti Bera, Dept. of Medical Imaging, University of Arizona   Feb. 1
Title: Electrical Impedance Tomography (EIT) Instrumentation: A Journey Through Design, Development, and Applications

Abstract: The Electrical Impedance Tomography (EIT) is a computed tomographic technique which provides the images of the spatial distribution of the electrical conductivity or resistivity of a domain under test (DUT) from a set of voltage-current data measured at the boundary of the DUT surrounded by an array of surface electrodes. Due to several advantages, EIT is being applied in several fields of engineering and other applied sciences. In EIT, a PC based EIT instrumentation is required to generate boundary voltage-current data by injecting a constant amplitude low frequency alternating current signal to the DUT. Full set of boundary voltage-current data are sent to the PC and the impedance distribution of the domain are reconstructed using a computer program called image reconstruction algorithm. The reliability, efficiency and image quality depend on the boundary data accuracy which is highly influenced by the instrumentation and data acquisition system. The commercial EIT system manufacturers are limited in numbers and sometimes the commercial EIT systems are developed as application specific devices. Therefore, design and development of laboratory-based standalone EIT systems are found sometimes essential. As the impedance response of material in frequency domain provide a number of material information, multifrequency multifunction EIT (Mf-EIT) systems are proposed and studied by several research groups. In this talk, after introducing the EIT technology and its application, the designing and development of multifrequency instrumentation will be discussed in detail. The electrode switching and data acquisition procedures with different current patterns will be discussed thoroughly. The system interfacing, operation and control by the graphical user interface (GUI) will be summarized. The data processing with amplifiers and filter blocks will be conferred followed by a brief discussion on image reconstruction from the boundary data. The talk will be concluded by a conversation on the present challenges and limitations in EIT instrumentation development.

Sarah Hamilton, Marquette University   Mar. 8
Title:Improving Electrical Impedance Tomography with Robust D-bar Methods

Abstract: In Electrical Impedance Tomography (EIT), electrical measurements are taken on electrodes at the body's surface and a mathematical inverse problem is solved to recover the electrical conductivity/admittivity inside the body. As electrical properties are tissue dependent, EIT images may then be used to monitor heart and lung function in ICU patients, classify strokes (ischemic vs. hemorrhagic) and breast tumors (benign vs. malignant), and provide nondestructive evaluation of construction materials. The most common solution methods for EIT solve an optimization problem to minimize the error between measured and predicted data (e.g. voltage/current data) therefore requiring a finely-tuned forward model. By contrast, D-bar methods solve the inverse problem directly by using a tailor-made nonlinear Fourier transform of the measured boundary voltage/current allowing real-time imaging. Low-pass filtering of the transform data provides robustness to noise as well as incorrect boundary shape modeling for static as well as time-difference imaging. In this talk, we explore how the benefits of a priori knowldege can be incorporated into D-bar methods to provide reconstructed images with sharp boundaries important in medical imaging. Extensions to partial boundary data, super-resolution, as well as machine learning are discussed. Reconstructions from experimental EIT data are presented.

Paul Sava, Colorado School of Mines   Apr. 12
Title: Wavefield tomography of comet interiors

Abstract: Fundamental answers about small planetary bodies (comest and asteroids) origin and evolution hinge on our ability to image in detail their interior structure in 3D and at high resolution. The interior structure is not easily accessible without redundant data, e.g., radar reflections observed from multiple viewpoints, as in medical tomography. The Comet Radar Explorer mission aims to acquire a dense network of in-phase radar echoes from orbit to obtain a high resolution 3D image of comet Tempel 2's interior. Full wavefield inversion facilitate high quality imaging of the comet interiors, particularly if the comet nucleus is characterized by complex structure and large contrasts of physical properties. Knowledge of the comet shape and all-around orbital radar acquisition enable accurate and computationally-efficient 3D tomography. Using realistic numerical experiments, I will argue that (1) interior comet imaging is intrinsically 3D and conventional SAR processing does not satisfy imaging/resolution requirements; (2) interior tomography can be performed progressively with data acquired through successive orbits around the comet; (3) exploiting the known exterior shape of the comet facilitates cost-effective wavefield tomography; (4) multiscale wavefield tomography provides detailed information about the 3D internal physical properties; (5) imaging resolution exceeds the wavelength limits corresponding to low-frequency radar waves; (6) monostatic tomography yields 3D models comparable with what could be accomplished by more complex systems.



Past Seminars

[Spring 2015] [Fall 2014]