David George
Research Mathematician, Cascades Volcano Observatory

US Geological Survey

Talk Title: Modeling hazardous granular-fluid flows with a two-phase hyperbolic system and software

Many natural hazards are characterized by free-surface gravity-driven flows that involve

mixtures of, and interactions between, granular materials and fluids. Examples include overland

floods that entrain material, water-saturated landslides, debris flows, earthen dam breaches,

landslide generated tsunamis, and cascading hazards that involve a sequence of these events.

Numerical models for these large-scale flows traditionally utilize depth-averaged single-phase

fluid models, sometimes embedded with elaborate and tunable rheological laws to account for

the presence of grains. However, these flows do not merely involve statically composed

granular-fluid mixtures  the macroscopic flow behavior strongly depends on microscale

interactions and coupled feedbacks between solid grain concentrations and fluid pressure.

We have developed a depth-averaged two-phase granular-fluid model to account for the

strongly coupled evolution of solid concentrations and fluid pressure in these flows, which

strongly affects slope-stability, flow mobility, and runout extent. The model is a non-

conservative hyperbolic system of five equations for depth, momentum, volume fractions, and

pore-fluid pressure. The model is implemented in the open-source software, D-Claw, belonging

to the Clawpack family of software.

I will provide an overview of important physical mechanisms influencing slope stability and flow

behavior and properties of our model designed to capture this behavior. Applications involving

simulation and hazard mitigation in the Pacific Northwest will be presented.

Anne Greenbaum
Emeritus Professor, Department of Applied Mathematics

University of Washington

Talk Title: Optimal Polynomial Approximation to Rational Matrix Functions Using

the Arnoldi Algorithm

Joint with:  Tyler Chen, Natalie Wellen

Given an n by n matrix A and an n-vector b, along with a rational  function R(z) := D(z )^{-1} N(z), we show how to find the optimal approximation to R(A) b from the Krylov space, span( b, Ab, ... , A^{k-1} b), using the basis vectors produced by the Arnoldi algorithm.  Here {\em optimal} is taken to mean optimal in the D(A )^{*} D(A)-norm. Similar to the case for linear systems, we show that for non-Hermitian problems, eigenvalues alone cannot provide information about the convergence behavior of this algorithm and we discuss other possible error bounds for highly nonnormal matrices.

Giovanni Paolini
Senior Applied Scientist, Amazon Web Services and CalTech

Talk Title: Mathematics in the age of generative AI

As generative AI technologies are revolutionizing industries and our daily lives, what is going to happen to the role of the mathematician? In this talk, I will highlight recent breakthroughs in deep learning and AI and explore how current and future advancements might alter the way we do mathematics.

Heather Wilber
Assistant Professor, Department of Applied Mathematics

University of Washington, Seattle

Talk Title: Three big ideas in computational rational approximation

In the past decade, new algorithmic tools for constructing and computing with rational approximations to functions have been introduced and applied across a wealth of applications, including signal processing, reduced order modeling, the solving of PDEs, and the design of algorithms for matrix computations. In this talk, we describe three foundational ideas in computational rational approximation. These ideas come with computational methods that often make working with rationals more practical and accessible than one might imagine. We share several applications, and highlight in particular work on discretization-oblivious methods for computing with functions of operators.