Spring 2019

Date Speaker Topic
M Jan 21 No Seminar Martin Luther King Jr. Day
M Jan 28 Everyone Organizational Meeting
Abstract: Discuss plans for the semester.
M Feb 4 Daniel Sussman Multiple Network Inference: From Joint Embeddings to Graph Matching

Abstract: Statistical theory, computational methods, and empirical evidence abound for the study of individual networks. However, extending these ideas to the multiple-network framework remains a relatively under-explored area. Individuals today interact with each other through numerous modalities including online social networks, telecommunications, face-to-face interactions, financial transactions, and the sharing and distribution of goods and services. Individually these networks may hide important activities that are only revealed when the networks are studied jointly. In this talk, we’ll explore statistical and computational methods to study multiple networks, including a tool to borrow strength across networks via joint embeddings and a tool to confront the challenges of entity resolution across networks via graph matching.

M Feb 11 Maurice Fabien Multigrid techniques for high-order discontinuous Galerkin methods with application to porous media flows.

Abstract:

Accurate and reliable numerical simulations are important for assessing the performance and cost of enhanced oil recovery techniques, and examining the behavior of subsurface flows. To that end, we present a hybridizable discontinuous Galerkin method which generates high fidelity simulations of flow and transport in porous media. The proposed method is high-order accurate, conserves mass locally, and the number of globally coupled degrees of freedom is significantly reduced compared to standard discontinuous Galerkin methods. However, a significant challenge occurs when we consider solving the linear systems generated by high-order discontinuous Galerkin methods. High-order methods give rise to less sparse linear systems, with worse condition numbers when compared to linear systems that arise from their low-order counterparts. Physical properties from the underlying application need to be taken into consideration, as they also impact the conditioning of the linear system. In order to make the simulation process practical, fast linear solvers are required. We detail an efficient multigrid technique for discontinuous Galerkin discretizations, and demonstrate its robustness for heterogeneous flow and transport problems in porous media.

M Feb 18 No Seminar President’s Day
M Feb 25
Abstract:
M Mar 4 Carmen Rodrigo Robust multigrid solver for mixed-dimensional models for flow in fractured porous media
Abstract: The numerical simulation of models that involve coupled physical processes in a mixed-dimensional geometry is  a  challenging  task  which  is  getting increasing attention in recent years. The essential role played by these models in different applications demands the design of efficient methods for solving the corresponding flow models. In this talk, we describe the development of a robust multigrid solver for a flow model in fractured porous media, where the governing equations comprise a system of mixed-dimensional partial differential equations. Two-dimensional arbitrary fracture networks with vertical and/or horizontal possibly intersecting fractures are considered. The key point in the design of the multigrid solver is to combine two-dimensional multigrid components (smoother and inter-grid transfer operators) in the porous matrix with their one-dimensional counterparts within the fractures, giving rise to a mixed-dimensional multigrid method. Numerical experiments demonstrate the robustness of the monolithic mixed-dimensional multigrid method with respect to the permeability of the fractures, the grid-size and the number of fractures in the network.
M Mar 11 Zhiping Mao Efficient and Accurate Single/Multi-domain Spectral Methods for Fractional Diffusion Equations
Abstract:
Fractional diffusion equations (FDEs) are emerging as a powerful tool for modeling challenging phenomena including anomalous transport, and long range time memory or spatial interactions. However, FDEs present new mathematical and numerical difficulties that were not encountered in the context of integer-order PDEs, such as non-locality and boundary singularities, and so require rigorous modeling, numerical and mathematical analysis. In this talk, I will talk about efficient and accurate Efficient and accurate single/multi-domain spectral methods for FDEs. Due to the non-local feature of the fractional derivatives, local methods such as FDM and FEM loss a big advantage that they enjoy for usual PDEs. On the other hand, the main disadvantage of global methods such as spectral methods is no longer an issue for fractional PDEs. Another issue for numerically solving FDEs is the boundary singularities. By analyzing the explicit boundary behavior, I first present spectral (Petro-Galerkin and collocation) methods (of exponential convergence) for the two-side FDEs and give rigorous analysis. In order to obtain high accuracy for more general FDEs, I develop a multi-domain spectral Galerkin method as well as a efficient solver for FDEs generalizing high accuracy by using h/p refinements. I also develop a multi-domain spectral collocation method and minimize the jump in (integer) fluxes using a penalty method.
M Mar 18 No Seminar Spring Break
M Mar 25 Anna Little Theoretical Guarantees and Exact Cluster Recovery for Classical Multidimensional Scaling

Abstract:

Classical multidimensional scaling (CMDS) is a widely used dimension reduction technique in machine learning and statistics. However, few theoretical results characterizing its statistical performance under randomness exist. This work provides a theoretical framework for analyzing the quality of embedded samples produced
by CMDS. As an application, we study the performance of CMDS in the setting of clustering noisy data. Our results provide scaling conditions on the sample size, ambient dimensionality, between-class distance and noise level under which CMDS followed by a simple clustering algorithm can exactly recover the cluster labels of all samples with high probability. Numerical simulations confirm these scaling conditions are sharp in low, moderate, and high dimensional regimes.

! W Apr 3 ! Lisa Claus Algebraic Multigrid Smoothers for Maxwell's equations

Abstract:

Algebraic Multigrid (AMG) is used to speed up linear system solves in
a wide variety of applications. This talk is concentrated on expanding AMG’s applicability to important new classes of problems through algorithms that automatically construct advanced smoothing techniques when needed. As a relevant application, we consider a curl-curl problem (commonly referred to as the second-order definite Maxwell equations) that often arises in time-domain electromagnetic simulations. AMG algorithms are usually designed by first assuming that the smoother is a simple pointwise smoother, then great effort is put into constructing an interpolation and corresponding coarse-grid correction that complements the smoother and leads to fast O(N) convergence. The so-called weak approximation property and basic two-grid theory are used to guide algorithm development. However, for some classes of problems, pointwise smoothers are not sufficient for achieving the desired O(N) computational complexity.  In this talk, we use two-grid theory to motivate the development of new algorithms for automatically constructing more complex (non-pointwise) smoothers. These algorithms have their roots in smoothing techniques used
for geometric multigrid methods, the well known Arnold-Falk-Winther and Hiptmair smoothers. In addition, we introduce the idea of automatically constructing complementary interpolation operators. We use a Nédélec H(curl)-conforming finite element approach to discretize the problem. We also discuss future directions and the potential of these AMG smoothers in more general application settings.

M Apr 8
Abstract:
M Apr 15  No Seminar Patriot’s Day
M Apr 22 James Brannick Algebraic Multigrid: Theory and Practice

Abstract:

This talk focuses on developing a generalized bootstrap algebraic multigrid algorithm for solving sparse matrix equations. As a motivation of the proposed generalization, we consider an optimal form of classical algebraic multigrid interpolation that has as columns eigenvectors with small eigenvalues of the generalized eigen-problem involving the system matrix and its symmetrized smoother. We use this optimal form to design an algorithm for choosing and analyzing the suitability of the coarse grid. In addition, it provides insights into the design of the bootstrap algebraic multigrid setup algorithm that we propose, which uses as a main tool a multilevel eigensolver to compute approximations to these eigenvectors. A notable feature of the approach is that it allows for general block smoothers and, as such, is well suited for systems of partial differential equations. In addition, we combine the GAMG setup algorithm with a least-angle regression coarsening scheme that uses local regression to improve the choice of the coarse variables. These new algorithms and their performance are illustrated numerically for scalar diffusion problems with highly varying (discontinuous) diffusion coefficient, Maxwell equations and for the linear elasticity system of partial differential equations.

M Apr 29 Xu Zhang
Abstract: