Dr. Charles L. Epstein - The Phase Retrieval Problem in Coherent Diffraction Imaging

March 5, 2020 from 3pm-5:30pm in Toomey 295.  Light refreshments will be provided.

The flyer for the event may be found here: Flyer

Abstract for Lecture 1 (3pm, intended for a general audience):

In several high-resolution imaging modalities that use coherent x-rays to illuminate the sample, the measured data can be interpreted as the modulus of the Fourier transform of a function describing the unknown object. The Fourier transform is complex-valued and the phase cannot be measured directly. I will briefly explain the physics that underlies these facts. In order to reconstruct the object from such measurements we must “retrieve” the unmeasured phase of the Fourier transform. To do this requires some auxiliary information about the object, such as its general size and shape. This is a notoriously difficult problem. I will discuss the underlying geometric reasons for these difficulties, algorithms for recovering the phase and approaches to improving their performance.

Abstract for Lecture 2 (4:30pm):

In the first talk we introduced a geometric formulation of the phase retrieval problem in coherent diffraction imaging (CDI). In that talk we showed how this problem can be formulated as a search for the intersection between two subsets A, B of a very high dimensional space. A is always a torus and B is a subset defined by the auxiliary information, which makes this a non-linear problem. In the second talk we explain why A and B often fail to meet transversely and the effect that this has on the performance of standard algorithms to find the intersections of A and B. This is illustrated with some simple model problems. We then examine the linearization of the maps used to define these algorithms at fixed points, where they display some rather surprising properties. We close our discussion with an entirely different approach to the phase retrieval problem.

About the Speaker:

Dr. Charles L. Epstein is an analyst and applied mathematician with research interests in partial differential equations, medical imaging, mathematical biology, and scientific computation.  He received his PhD in Mathematics in 1983 from the Courant Institute, New York University, and completed an NSF Postdoctoral Fellowship at Princeton University in 1986.  Since that time, he has been affiliated with the University of Pennsylvania, where he presently holds the positions of Thomas A. Scott Professor of Mathematics and Graduate Group Chair for Applied Mathematics and Computational Science.  During his tenure at the University of Pennsylvania, he has also been affiliated with the Department of Radiology in the School of Medicine, as well as the Graduate Groups in Bioengineering and Genomics and Computational Biology. He is the author of over 75 peer-reviewed publications as well as a textbook on the Mathematics of Medical Imaging. His research has been supported by the Sloan Foundation, the NSF, the NIH, and DARPA. In 2014, he became a Fellow of the American Mathematical Society “for contributions to analysis, geometry, and applied mathematics including medical imaging, as well as for service to the profession”.

W. Tom Ingram
Professor Emeritus
Department of Mathematics 
Missouri S&T

Talk title: Applications of pure mathematics (April 27, 2017, flyer)

Talk title:  Continuous images of continua (April 28, 2017, flyer)

Applications of Pure Mathematics

Thursday, April 27, 2017
3:30-4:45 PM
Rolla Building G5

Many mathematicians pursue their art out of curiosity, not because they have some predetermined application in mind.  That does not necessarily mean that they are not interested in applications to “real world” problems – it is just not their primary motivation.  In this talk, intended for a campus-wide audience of students and faculty, I will discuss a research topic in my field of mathematics, topology, that has, to my surprise, been found to have applications outside mathematics

This talk is intended for a general audience.

Continuous Images of Continua

Friday, April 28, 2017
4:00-5:15 PM
Rolla Building G5

Interest in continuous images of topological objects has permeated the study of topology essentially from the beginning of the subject. Early on it was shown that every compact metric space is a continuous image of the Cantor set and continuous images of the interval were characterized as compact, connected and locally connected metric spaces. In this talk we discuss a relatively new area of topology, inverse limits with set-valued functions, and some relationships it has with continuous images of continua. 

Professor Ingram came to Missouri S&T (then UMR) from the University of Houston in 1989 as Professor and chair of the Department of Mathematics and Statistics.  He served as chair until 1998 and continued as professor until his retirement in December 2002.  He spent the following year as visiting professor at Baylor University. Since then he has remained very mathematically active, publishing numerous research articles and two Springer texts.  Much of his work since retiring concerns generalized inverse limit spaces with set valued bonding functions that he, together with William S. Mahavier, introduced in 2006.  This new area of topology has generated a great deal of interest with hundreds of articles published since its introduction.  The Ingram Lecture Series is possible because of a generous donation by Tom to the Missouri S&T Mathematics and Statistics Department upon his retirement from S&T.

Jie Shen

Professor, Department of Mathematics 
Director, Center for Computational and Applied Mathematics
Purdue University

Talk title: Fast spectral methods: algorithms, analysis and applications (March 6, 2017, flyer)

Talk title:  Phase-field models for multiphase complex fluids: modeling, numerical analysis and simulations (March 7, 2017, flyer)

Fast Spectral Methods: Algorithms, Analysis and Applications

Monday, March 6, 2017
4:10-5:20 PM
Butler-Carlton Hall 318

In recent years, spectral methods have become a major tool for computational scientists and engineers because of their superior accuracy and efficiency when properly implemented.  In this talk, I shall present essential ingredients to construct fast spectral algorithms and to carry out their error analysis. Particular emphasis will be given for problems with weak singularities for which direct application of spectral methods is not effective

This talk is intended for a general audience.

Phase-field Models for Multiphase Complex Fluids:  Modeling, Numerical Analysis and Simulations

Tuesday, March 7, 2017
4:10-5:20 PM
Butler-Carlton Hall 121

I shall present some recent work on phase-field model for multiphase incompressible flows. We shall pay particular attention to situations with large density ratios and with non-Newtonian components as they lead to formidable challenges in both analysis and simulation.

I shall also present unconditionally energy stable, decoupled numerical schemes which only require solving a sequence of linear elliptic equations at each time step for solving this coupled nonlinear system, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.

Professor Jie Shen received his B.S. in Computational Mathematics from Peking University in 1982, and his Ph.D in Numerical Analysis from Universite de Paris-Sud at Orsay in 1987. Before joining the Purdue Faculty in Fall 2002, he served as Professor of Mathematics at Penn State University and University of Central Florida.  Since Jan. 2012 he serves as the Director of Center for Computational and Applied Mathematics at Purdue University.

He is a recipient of the Fulbright award in 2008 and the Inaugural Research Award of the College of Science at Purdue University in 2013, and an elected Fellow of AMS. He serves on editorial boards for several leading international research journals, and has authored/coauthored over 160 peer-reviewed research articles and two books. His main research interests are numerical analysis, spectral methods and scientific computing with applications in computational fluid dynamics and materials science.

Ian H. Sloan, The University of New South Wales

Talk title: Imagining and calculating in many dimensions  (March 4, 2013, flyer)

Talk title: Lifting the curse of dimensionality: numerical integration in very high dimensions (March 5, 2013, flyer)

John A. Burns, Virginia Tech

Talk title: Science, engineering and mathematical challenges in designing net zero energy buildings (April 14, 2011, flyer)

Talk title: Control of infinite dimensional systems with applications to energy efficient buildings (April 15, 2011, flyer)

Robert L. Devaney, Boston University

Talk title: Chaos games and fractal images (April 15, 2005)

Talk title: Julia and Fatou, Cantor and Sierpinski (and Ingram): crazy topology in complex dynamics  (April 15, 2005)