2017 SIAM Pacific Northwest Regional Conference
Plenary Speakers
Department of Computer Science, University of British Columbia
Title of Presentation: Convex Duality in Computation
Abstract: Many modern applications rely on convex optimization, which offers a rich modeling paradigm as well as strong theoretical guarantees and computational advantages. Convex duality often plays a central role. I will discuss how different notions of duality are used to develop computationally efficient algorithms for a range of difficult problems.
Department of Applied Mathematics, University of Washington
Title of Presentation: Some Extensions of the Crouzeix-Palencia Result
Abstract: A longstanding conjecture in linear algebra and operator theory is Crouzeix's conjecture: For any square matrix A (or any linear operator on a Hilbert space) and any polynomial p (or any function analytic on the numerical range of A), the operator 2-norm of p(A) is less than or equal to 2 times the W(A)-norm of p. The W(A)-norm is defined to be the supremum of | p(z) | over z in the numerical range of A: W(A) := { <Aq,q> : <q,q> = 1 }. In 2007, Crouzeix proved such an inequality but with 2 replaced by 11.08. Recently, in summer 2016, Cesar Palencia announced that he had reduced the constant from 11.08 to 1 plus the square root of 2, approximately 2.415. Crouzeix and Palencia now have a joint paper [SIAM J. Matrix Anal. Appl. 38 (2017), pp. 649-655] with an elegant proof of this result.
In this talk, I will describe their result and show how their arguments can be extended to show that other regions in the complex plane that do not necessarily contain W(A) are K-spectral sets, for a value of K that may be close to 1 plus the square root of 2, or even close to 2. I will also show how their result can be used to give a new proof of von Neumann's inequality (that if the norm of A is less than or equal to 1 then the unit disk D is a spectral set for A) and of a result due to Okubo and Ando (that if W(A) is a subset of D, then D is a 2-spectral set for A). I will present an idea about how the constant 1 plus the square root of 2 might be reduced to 2.
Department of Mathematics, Oregon State University
Title of Presentation: "How Warm is it Getting?" and other Uncertainty Tales
Abstract: In the statistics community “Big Data” science is meant to suggest the combining of inferential and computational thinking. We also speak of big data in the geosciences. However, the problems we pursue are often extreme in the number of degrees of freedom, and in many instances, non-stationary in their statistics. This usually means that we are working with sparse observational data sets, even if the number of observations is large. The Bayesian framework is a natural inferential data assimilation strategy in geosciences, to some extent because the degrees of freedom in the problem vastly outnumber observations but more critically, because the models we use to represent nature have considerable predictive power.
Looking toward the future, we expect improvements in computational efficiency and finer resolutions in models, as well as improved field measurements. This will force us to contend with physics and statistics across scales and thus to think of ways to couple multiphysics and computational resolution, as well as to develop efficient methods for adaptive statistics and statistical marginalization.
How this coupling is exploited to improve estimates that combine model outcomes and data will be described in tracking hurricanes and improving the prediction of the time and place of coastal flooding due to ocean swells. Estimating the trend of Earth’s temperature from sparse multi-scale data will be used as an example of adaptivity in time series analysis.
Other open challenges in non-stationary big data problems will be described, where progress could result from “big data geoscience,” the tighter integration of geoscience, computation, and inference.
Department of Mathematics, Boise State University
Title of Presentation: Computing with functions in spherical and polar geometries
Abstract: Spherical and polar geometries are ubiquitous in computational science and engineering, arising in, for example, weather and climate forecasting, geophysics, and astrophysics. Central to many of these applications is the task of developing efficient and accurate approximations of functions defined on the surface of the unit sphere or on the disk. We present a new low rank method for this task by combining an iterative, structure-preserving variant of Gaussian elimination together with the classic double Fourier sphere method. The resulting scheme gives a compressed representation functions on the sphere or disk, ameliorates oversampling issues near the poles of the sphere or origin of the disk, and converges geometrically for sufficiently analytic functions. The low rank representation makes operations such as function evaluation, differentiation, and integration particularly efficient. We illustrate the applicability of our method to common computational tasks from data analysis, vector calculus, and the solution of partial differential equations, which are all implemented in the new Spherefun and Diskfun features of the Chebfun software system (www.chebfun.org). This is joint work with Prof. Alex Townsend and Heather Wilber (both at Cornell University).