Stochastic Superparameterization Through Local Data Generation
- Yoonsang Lee, Department of Mathematics, Dartmouth College
- Wednesday, September 11, 3:30 p.m., Location: Olney 430
Stochastic superparameterization is a class of multiscale methods that approximate large-scale dynamics of complex dynamical systems such as turbulent flows. Unresolved sub-grid scales are modeled by a cheap but robust stochastic system that mimics the true dynamics of the sub-grid scales, which is crucial to model non-trivial and non-equilibrium dynamics. In this talk, we propose a numerical procedure to estimate the modeling parameters, which avoids the use of climatological data.
Hardness Results for Sampling Connected Graph Partitions with Applications to Redistricting
- Daryl Deford, MIT
- Wednesday, September 18, 4 p.m., Location: Olney 430
The problem of constructing ”fair'' political districts and the related problem of detecting intentional gerrymandering has received a significant amount of attention in recent years. A key problem in this area is determining the expected properties of a representative districting plan as a function of the input geographic and demographic data. A natural approach is to generate a comparison ensemble of plans using MCMC and I will present successful applications of this approach in both court cases and legislative reform efforts. However, our recent work has demonstrated that the commonly used boundary-node flip proposal can mix poorly on real-world examples. In this talk, I will present some new proposal distributions for this setting and discuss some related open problems concerning mixing times and spanning trees. I will also discuss some generic hardness results for sampling problems on partitions of planar graphs.
Multiscale Convergence Properties for Spectral Approximations of a Model Kinetic Equation
- Zheng Chen, Department of Mathematics, UMass Dartmouth
- Thursday, October 3, 2 p.m., Location: Olney 430
We prove some convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form $N^{-q}$, where $N$ is the number of modes and $q$ depends on the regularity of the solution. However, in the multiscale setting, we show that the error estimate can be expressed in terms of the scaling parameter $\epsilon$, which measures the ratio of the mean-free-path to the domain in the system. In particular we show that the error in the spectral approximation is $\mathcal{O}(\epsilon^{N+1})$. More surprisingly, the coefficients of the expansion satisfy some super convergence properties. In particular, the error of the $\ell^{th}$ coefficient of the expansion scales like $\mathcal{O}(\epsilon^{2N})$ when $\ell =0$ and $\mathcal{O}(\epsilon^{2N+2-\ell})$ for all $1\leq \ell \leq N$. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. Numerical tests will also be presented to support the theoretical results.
Jump Process Approximation of Particle-Based Stochastic Reaction-Diffusion Models
- Samuel Isaacson, Department of Mathematics and Statistics, Boston University
- Wednesday, November 6, 4 p.m., Location: Olney 430
High resolution images of cells demonstrate the highly heterogeneous nature of the nuclear and cytosolic spaces. We are interested in understanding how this complex environment influences the dynamics of cellular processes. To investigate this question we have developed the convergent reaction-diffusion master equation (CRDME), a lattice particle-based stochastic reaction-diffusion method that can model the spatial transport and reactions of molecules within domains derived from imaging data. In this talk I will introduce the CRDME, and explain how it is similar in spirit to the popular reaction-diffusion master equation (RDME) model. The CRDME allows for the reuse of the many extensions of the RDME developed to facilitate modeling within biologically realistic domains, while eliminating one of the major challenges in using the RDME model.
Geometric Structure in Dependence Models and Applications
- Elisa Perrone, Department of Mathematical Sciences, UMass Lowell
- Wednesday, November 20, 4 p.m., Location: Olney 430
The growing availability of data makes it challenging yet crucial to model complex dependence traits. For example, hydrological and financial data typically display tail dependences, non- exchangeability, or stochastic monotonicity. Copulas serve as tools for capturing these complex traits and constructing accurate dependence models which resemble the underlying distributions of data. This talk inquires into the geometric properties of dependence models and copulas to address statistical challenges in several applications, such as hydrology and weather forecasting. In particular, we study the class of discrete copulas, i.e., restrictions of copulas to uniform grid domains, which admits representations as convex polytopes. In the first part of this talk, we give a geometric characterization of discrete copulas with desirable stochastic constraints in terms of the properties of their associated convex polytopes. In doing so, we draw connections to the popular Birkhoff polytopes, thereby unifying and extending results from both the statistics and the discrete geometry literature. We further consolidate the statistics/discrete geometry bridge by showing the significance of our geometric findings to construct entropy-copula models useful in hydrology. In the second part of this talk, we focus on weather forecasting problems. We show that discrete copulas are powerful tools for empirical modeling in applications. We discuss their use in the context of statistical postprocessing of ensemble weather forecasts, and present a case study for temperature forecasts in Austria.
Dimension of Sumsets of Restricted Digit Cantor Sets in the Integers
- Daniel Glasscock, Department of Mathematical Sciences, UMass Lowell
- Wednesday, December 4, 4 p.m., Location: Olney 430
Harry Furstenberg made a number of conjectures in the 60's and 70’s seeking to make precise the heuristic that there is no common structure between digit expansions of real numbers in different bases. Recent solutions to conjectures of his concerning the dimension of sumsets and intersections of p- and q- invariant sets now shed new light on old problems. In this talk, I will explain how to use tools from fractal geometry and uniform distribution to determine the dimension of sumsets of restricted digit Cantor sets with respect to different bases in the integers. This talk is based on joint work with Joel Moreira and Florian Richter.