# Applied Mathematics

## Models for the Spread of Cholera

There have been several recent outbreaks of cholera (for example, in Haiti and Yemen), which is a bacterial disease caused by the bacterium Vibrio cholerae. It can be transmitted to humans directly by person-to-person contact or indirectly via contaminated water. Random mixing cholera models from the literature are first formulated and briefly analyzed. Heterogeneities in person-to-person contact are introduced, by means of a multigroup model, and then by means of a contact network model. Utilizing an interplay of analysis and linear algebra, various control strategies for cholera are suggested by these models.

Pauline van den Driessche is a Professor Emeritus in the Department of Mathematics and Statistics at the University of Victoria. Her research focuses on aspects of stability in biological models and matrix analysis. Current research projects include disease transmission models that are appropriate for influenza, cholera and Zika. Most models include control strategies (e.g., vaccination for influenza) and aim to address questions relevant for public health. Sign pattern matrices occur in these models, and the possible inertias of such patterns is a current interest.

## Hybrid Krylov Subspace Iterative Methods for Inverse Problems

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).

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

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

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.

## Managing Patients with Chronic Conditions

Chronic disease management often involves sequential decisions that have long-term implications. Those decisions are based on high dimensional information, which pose a problem for traditional modeling paradigms. In some key instances, the disease dynamics might not be known, but instead are learned as new information becomes available. As a first step, we will describe some of the ongoing research modeling medical decisions of patients with chronic conditions. Key to the models developed is the incorporation of the individual patient's disease dynamics into the parameterization of the models of the disease state evolution. Model conception and validation is described, as well as the role of multidisciplinary collaborations in ensuring practical impact of this work.

## The Mathematics of Game Design

Mathematics is integral to every aspect of game development including character and level creation, movement, player input, NPC behaviour, physics simulations, and real-time rendering. Fortunately for game designers, most of this computation is conveniently supplied by software developers and/or handled by existing game engines. However, when designing a game, lots of systems and mechanics are dependent on numbers such as weapons ranges, jump heights, experience points, damage, rewards, currency, etc., many of which can have complex inter-relationships. Although much of the math may be basic, a good understanding of the underlying equations as well as the fields of logic, probability, and statistics can be incredibly beneficial to a designer, especially when it comes to game design and balancing. This lecture will give an overview of how even the most basic knowledge of these fields can benefit a game designer.

## PIMS-SFU 20th Anniversary Celebration: Nataša Pržulj - Data Driven Medicine

The Pacific Institute for the Mathematical Sciences (PIMS) was founded in 1996, and Simon Fraser University is a founding member. The members of PIMS now include all the major Canadian research universities west of Ontario, as well as universities in Washington and Oregon. Please join us to celebrate 20 years of productive collaboration, with a lecture by SFU alumna and professor at UCL Nataša Pržulj on Data Driven Medicine followed by a reception.

We are faced with a flood of molecular and clinical data. Various biomolecules interact in a cell to perform biological function, forming large, complex systems. Large amounts of patient-specific datasets are available, providing complementary information on the same disease type. The challenge is how to model and mine these complex data systems to answer fundamental questions, gain new insight into diseases and improve therapeutics. Just as computational approaches for analyzing genetic sequence data have revolutionized biological and medical understanding, the expectation is that analyses of networked “omics” and clinical data will have similar ground-breaking impacts. However, dealing with these data is nontrivial, since many questions we ask about them fall into the category of computationally intractable problems, necessitating the development of heuristic methods for finding approximate solutions.

We develop methods for extracting new biomedical knowledge from the wiring patterns of large networked biomedical data, linking network wiring patterns with function and translating the information hidden in the wiring patterns into everyday language. We introduce a versatile data fusion (integration) framework that can effectively integrate somatic mutation data, molecular interactions and drug chemical data to address three key challenges in cancer research: stratification of patients into groups having different clinical outcomes, prediction of driver genes whose mutations trigger the onset and development of cancers, and re-purposing of drugs for treating particular cancer patient groups. Our new methods stem from network science approaches coupled with graph-regularised non-negative matrix tri-factorization, a machine learning technique for co-clustering heterogeneous datasets.

## About Irreversibility in Rarefied Gas Dynamics

About Irreversibility in Rarefied Gas Dynamics

## 2016 Graduate Mathematical Modelling in Industry Workshop

This gallery contains photos from the 2016 Graduate Mathematical Modelling in Industry Workshop. See the event webpage for more information.

## A glamorous Hollywood star, a renegade composer, and the mathematical development of spread spectrum communications.

During World War II Hedy Lamarr, a striking Hollywood actress, together with George Antheil, a radical composer, invented and patented a secret signaling system for the remote control of torpedoes. The ideas in this patent have since developed into one of the ingredients in modern digital wireless communications. The unlikely biography of these two characters, along with some of the more modern developments in wireless communications will be described.