# Applied Mathematics

## 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.

## Bayesian study design for nonlinear systems: an animal disease transmission experiment case study

Experimental design is a branch of statistics focused upon designing experimental studies in a way that maximizes the amount of salient information produced by the experiment. It is a topic which has been well studied in the context of linear systems. However, many physical, biological, economic, financial and engineering systems of interest are inherently non-linear in nature. Experimental design for non-linear models is complicated by the fact that the optimal design depends upon the parameters that we are using the experiment to estimate. A Bayesian, often simulation-based, framework is a natural setting for such design problems. We will illustrate the use of such a framework by considering the design of an animal disease transmission experiment where the underlying goal is to identify some characteristics of the disease dynamics (e.g. a vaccine effect, or the infectious period).

## Optimal Strategic Sizing of Energy Storage Facilities in Restructured Electricity Markets

In this seminar we will discuss a new model for strategic investment model for a merchant energy storage facility. The facility's actions impact market-clearing outcomes, and thus it is a price-maker facility. We consider the uncertainties associated with other generation units offering strategies and future load levels in the proposed model. Thestrategic investment decisions include the sizes of charging device,discharging device, and energy reservoir. The proposed model is astochastic bi-level optimization problem where planning and operation decisions of the energy storage facility are made in the upper level, and market clearing is modeled in the lower level under different operating conditions. To make the proposed model computationally tractable, an iterative solution technique based on Benders¹ decomposition is implemented. This provides a master problem and a set of subproblems for each scenario. Each subproblem is recast as a Mathematical Programs with Equilibrium Constraints (MPEC). Numerical results based on real-lifemarket data from Alberta's electricity market will be provided.