When a stochastic process has a random environment, its trajectories are influenced by the behaviour of the environment. In this talk we discuss well known models in discrete and continous time, and we present some results that help to measure how the environment affects the movements of the process. Important tools we use are the Ito-McKean representation of diffusions and the Ito's excursion theory.

Zoom: https://arizona.zoom.us/j/88673934011

Abstract: New applications of scientific computing for solid and fluid mechanics problems include simulation of virtual materials in movie visual effects and virtual surgery. Both disciplines demand physically realistic dynamics for materials like water, smoke, fire, and soft tissues. New algorithms are required for each area. Teran will speak about the simulation techniques required in these fields and will share some recent results including: simulated surgical repair of biomechanical soft tissues; extreme deformation of elastic objects with contact; high resolution incompressible flow; and clothing and hair dynamics. He will also discuss a new algorithm used for simulating the dynamics of snow in Disney’s animated feature film, “Frozen”.

With vast amounts of data and the continuously growing body of applied mathematics and data science research, it is important to ensure that research is broadly approachable and accessible by the scientific community. In this talk, we will discuss what is Open Science, and important components of making your project more “open” as an applied mathematician. This includes tools and discussion of project management, workflow management, data management and lastly packaging it together in a documented and reproducible project.

Place: Zoom:   https://arizona.zoom.us/j/82075792519 
Password:  150721

How does a neural network approximate a given function? What kinds of functions can it approximate well/poorly? How does the optimization algorithm bias learning? What is the structure of the loss surface and Hessian and how does that impact generalization? Deep Learning has revolutionized many fields, and yet answers to fundamental questions like these remain elusive. Here we present a new emerging viewpoint -- the function space or spline perspective -- that has the power to answer these questions. We find that understanding neural nets is most easily done in the function space, via a novel spline parametrization. This change of coordinates sheds light on many perplexing phenomena, providing simple explanations for the necessity of overparameterization, the structure of loss surface and Hessian, the consequent difficulty of training, and, perhaps most importantly, the phenomenon and mechanism underlying implicit regularization.

Understanding the representation, learning dynamics and inductive bias of neural networks (NNs) is hindered by the opacity of the relationship between NN parameters and the function represented. As such, we propose reparametrizing ReLU NNs as continuous piecewise linear splines. Using this spline lens, we study learning dynamics in shallow univariate ReLU NNs, finding unexpected insights and explanations for several perplexing phenomena. We develop a surprisingly simple and transparent view of the structure of the loss surface, including its critical and fixed points, Hessian, and Hessian spectrum. We also show that standard weight initializations yield very flat functions upon initialization, and that this flatness, together with overparametrization and the initial weight scale, is responsible for the strength and type of implicit regularization, consistent with recent work. Our spline-based approach reproduces key implicit regularization results from recent work but in a far more intuitive and transparent manner. In addition to understanding, the spline lens suggests new kinds of data-dependent initializations and learning algorithms that combine gradient descent with other more global optimization algorithms.

We briefly discuss future work applying the spline lens to: neuronally consistent networks (with saturating activation functions, E/I Balance, and cell types) and to developing new experimental protocols that can test for and characterize implicit regularization in the brain. Going forward, we believe the spline lens will play a foundational role in efforts to understand and design artificial and real neural networks.

BIO:  Ankit B. Patel is currently an Assistant Professor at the Baylor College of Medicine in the Dept. of Neuroscience, and at Rice University in the Dept. of Electrical and Computer Engineering. Ankit is broadly interested in the intersection between (deep) machine learning and computational neuroscience, two areas essential for understanding and building truly intelligent systems, with a focus on the low-level mechanisms by which learned representations work. He works with neuroscientists to build a bridge between artificial and real neuronal networks, using theories and experiments about artificial nets to help understand and make testable predictions about real brain circuits.  Ankit returned to academia after spending 6 years in industry, building real-time inference systems trained on large-scale data for ballistic missile defense (MIT Lincoln Laboratory), and high-frequency trading. He received his graduate and undergraduate degrees in Computer Science and Applied Mathematics from Harvard University.

Hybrid: Math 501 and Zoom https://arizona.zoom.us/j/86997964863     Password:   Locute

We look forward to seeing your for the Math Department Holiday Lunch from 11:30-1pm on Thursday, December 9th.

We will celebrate the careers of:

Bruce Bayly

Jim Cushing

Theodore Laetsch

Vladimir Zakharov

Refreshments will be served

For details, see https://science.arizona.edu/academics/graduation-commencement 

Doors open at 1pm. COVID-19 protocols will be in place. 

Motivated by the aggressive push towards society-scale decarbonization, power grids  face pressure to pro-actively integrate large penetrations of renewable energy  and energy storage technologies, many of which vary in their techno-economic characteristics from the traditional centralized, economy-of-scale generation dogmas. Furthermore, electrification of traditionally fossil-dependent sectors (e.g. transportation, buildings and manufacturing) increases demand for electricity and alternative fuels, which further stresses power grids. Addressing these challenges require next-generation mathematical models and algorithms that help bridge the divide between these technologies and the current power grid practices.  In this presentation, we will adopt a bottom-up approach to describe how distributed generation resources can be used to assist power grids in accommodating renewable energy resources, and how to mitigate emerging communication and cybersecurity risks that limit the ability of these resource to provide grid support services at scale. We will focus on specific examples of high-rise buildings and electric vehicle charging stations to illustrate how these resources can be safely aggregated into grid-scale resources that can compete with traditional resources in electricity markets. Then, we will discuss a transition towards stochastic and risk-cognizant electricity market designs, and demonstrate the importance of internalizing variance of future market states and risk trading mechanisms to obtain efficient market outcomes.