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Scientific

Hybrid Krylov Subspace Iterative Methods for Inverse Problems

Speaker: 
James Nagy
Date: 
Fri, May 5, 2017
Location: 
PIMS, University of Manitoba
Conference: 
Mathematical Imaging Science
Abstract: 
Inverse problems arise in many imaging applications, such as image reconstruction (e.g., computed tomography), image deblurring, and digital super-resolution. These inverse problems are very difficult to solve; in addition to being large scale, the underlying mathematical model is often ill-posed, which means that noise and other errors in the measured data can be highly magnified in computed solutions. Regularization methods are often used to overcome this difficulty. In this talk we describe hybrid Krylov subspace based regularization approaches that combine matrix factorization methods with iterative solvers. The methods are very efficient for large scale imaging problems, and can also incorporate methods to automatically estimate regularization parameters. We also show how the approaches can be adapted to enforce sparsity and nonnegative constraints. We will use many imaging examples that arise in medicine and astronomy to illustrate the performance of the methods, and at the same time demonstrate a new MATLAB software package that provides an easy to use interface to their implementations. This is joint work with Silvia Gazzola (University of Bath) and Per Christian Hansen (Technical University of Denmark).

Rufus Bowen Conference - Lunchtime Speeches

Speaker: 
Brian Marcus
Date: 
Wed, Aug 2, 2017
Location: 
PIMS, University of British Columbia
Conference: 
Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen
Abstract: 
These speeches were given during the remembrance lunch as part of the conference "Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen".

Rufus Bowen Conference - Lunchtime Slideshow

Speaker: 
Brian Marcus
Date: 
Wed, Aug 2, 2017
Location: 
PIMS, University of British Columbia
Conference: 
Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen
Abstract: 
This slideshow and the accompanying toasts were given during the remembrance lunch as part of the conference "Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen".

The Case for T-Product Tensor Decompositions: Compression, Analysis and Reconstruction of Image Data

Speaker: 
Misha Kilmer
Date: 
Fri, May 5, 2017
Location: 
PIMS, University of Manitoba
Conference: 
Mathematical Imaging Science
Abstract: 
Most problems in imaging science involve operators or data that are inherently multidimensional in nature, yet traditional approaches to modeling, analysis and compression of (sequences of) images involve matricization of the model or data. In this talk, we discuss ways in which multiway arrays, called tensors, can be leveraged in imaging science for tasks such as forward problem modeling, regularization and reconstruction, video analysis, and compression and recognition of facial image data. The unifying mathematical construct in our approaches to these problems is the t-product (Kilmer and Martin, LAA, 2011) and associated algebraic framework. We will see that the t-product permits the elegant extension of linear algebraic concepts and matrix algorithms to tensors, which in turn gives rise to new, highly parallelizable, algorithms for the imaging tasks noted above.

The Geometry of the Phase Retrieval Problem

Speaker: 
Charles Epstein
Date: 
Fri, May 5, 2017
Location: 
PIMS, University of Manitoba
Conference: 
Mathematical Imaging Science
Abstract: 
Phase retrieval is a problem that arises in a wide range of imaging applications, including x-ray crystallography, x-ray diffraction imaging and ptychography. The data in the phase retrieval problem are samples of the modulus of the Fourier transform of an unknown function. To reconstruct this function one must use auxiliary information to determine the unmeasured Fourier transform phases. There are many algorithms to accomplish task, but none work very well. In this talk we present an analysis of the geometry that underlies these failures and points to new approaches for solving this class of problems.

Quantum Graph Theory

Speaker: 
Vern I. Paulsen
Date: 
Thu, Mar 9, 2017
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
Many numerical invariants of a graph, such as the independence number, clique number and chromatic number, have game theoretic descriptions. In these games a referee poses questions to two collaborating non-communicating players and they return answers. Quantum graph theory is concerned with how these graph parameters change when the players are allowed to use the random outcomes of quantum experiments to determine their answers. In this talk I will explain these concepts, focusing on the chromatic number, survey some of what little is known about the quantum chromatic numbers of graphs, explain the connection between these ideas and famous open conjectures of A. Connes and B. Tsirelson, and introduce an algebra affiliated with a graph whose representation theory determines the values of these parameters. Biography: Vern Paulsen is a Professor of Pure Mathematics and the Institute for Quantum Computing at the University of Waterloo. He was a Professor of Mathematics and John and Rebecca Moores Chair at the University of Houston before moving to Waterloo in 2015. His primary research focus is on the theory of operator algebras and their applications in quantum information theory. He is the author of five research monographs and over 100 research articles. He received his PhD from the University of Michigan.

Limit Theorems for the Frontier of One-Dimensional Branching Diffusions

Speaker: 
Thomas Sellke
Date: 
Thu, Nov 24, 2016
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
This talk will discuss results in a joint paper of mine with Steve Lalley from 1992. Suppose a particle, starting at position 0, moves according to a diffusion process on the real line. Suppose also that this particle emits daughter particles according to a branching process whose instantaneous rate can depend on location, though not on time. The daughter particles move independently according to the same diffusion process, starting at their points of birth, and in turn emit their own daughters according to the same branching process. The simplest special case of this situation is standard branching Brownian motion, with the rate of reproduction not depending on location. Let R_t be the position of the right-most particle at time t, and let m_t be the median of R_t. In 1937, Kolmogorov, Petrovskii, and Piskunov showed that, for standard branching Brownian motion, (m_t)/ t converges to SQRT(2) and that (R_t - m_t) converges in distribution to a nondegenerate limiting distribution.It turns out that results like those proved by Kolmogorov, et al, hold in great generality for one-dimensional branching diffusions. If the branching diffusion is "recurrent" (in the sense that the initial position is re-visited at arbitrarily large times by _some_ particle), and if space is rescaled so that m_t grows linearly, then (R_t - m_t) converges in distribution to a location-mixture of extreme value distributions. We also have the Andy Warhol Theorem, according to which every particle ever born has a descendant in the lead at some point in the future.

Asynchronous Consensus

Speaker: 
Faith Ellen
Date: 
Thu, Oct 20, 2016
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
The consensus problem plays a central role in the theory of distributed computing. I will prove that consensus is impossible to solve in some asynchronous shared memory systems and I will present some algorithms for solving it in others, together with matching lower bounds on the amount of time and space needed. Consensus is universal: using consensus and read/write registers, I will show how to implement any shared object. The consensus hierarchy is used to classify the computational power of shared objects. I will conclude by discussing some limitations of this classification that have been recently discovered.

On De Giorgi Conjecture and Beyond

Speaker: 
Jun-Cheng Wei
Date: 
Thu, Sep 29, 2016
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
Classifying solutions is one of central themes in nonlinear partial differential equations. This is the content of various Liouville type theorems and more recently De Giorgi type conjectures. I will report recent progress towards De Giorgi's conjecture for Allen-Cahn equation and free boundary problems, and related issues in nonlinear Schroedinger equations.

Extrema of 2D Discrete Gaussian Free Field - Lecture 16

Speaker: 
Marek Biskup
Date: 
Sun, Jul 30, 2017
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-CRM Summer School in Probability 2017
Abstract: 
The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

 

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