51°µÍø

Fall, 2021

Department of Mathematics, 51°µÍø
Room: 126 Clements Hall (if not specified, Refreshment starts 15 minutes before talks)

[1] Speaker: Kui Ren, Department of Applied Physics & Applied Mathematics, Columbia University

Time: 3:45pm, Thursday, October 7, 2021

Venue: (zoom online, see link below)

Title: Revisiting the Classical Least-Squares Formulation for Computational Learning and Inversion 

Abstract: The classical least-squares formulation has provided a successful framework for the computational solutions of learning and inverse problems. In recent years, modifications and alternatives have been proposed to overcome some of the disadvantages of this classical formulation in dealing with new applications. We will present some recent progress on the understanding of a general weighted least-squares optimization framework in such a setting. The main part of the talk is based on joint works with Bjorn Engquist and Yunan Yang.

Biography: Kui Ren received his PhD in applied mathematics in 2006 from Columbia University. He then spent a year at the University of Chicago as a L. E. Dickson instructor before moving to University of Texas at Austin to become an assistant professor in mathematics in 2008. He returned to Columbia University in 2018 as a professor in applied mathematics. Kui Ren's recent research interests include inverse problems, mathematical imaging, random graphs, fast algorithms, kinetic modeling, and computational learning.

Zoom link: 

Passcode: caimeeting, zoom meeting id 512 004 3096

 

[2] Speaker: Gunnar Martinsson, Oden Institute, UT Austin

Time: 3:45pm, Thursday, October 28, 2021

Venue: Clements Hall 126

Title: Direct solvers for elliptic PDEs

Abstract: That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will describe some recent work in the field and will argue that direct solvers have several advantages, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments.

Biography: Gunnar Martinsson is a Professor of Mathematics at the University of Texas at Austin, where he also holds the W.A. "Tex" Moncrief, Jr. Chair in Simulation-Based Engineering and Sciences in the Oden Institute for Computational Engineering and Sciences. Prior to joining UT Austin, Martinsson served as a Professor of Mathematics at the University of Oxford, and he has previously held faculty positions at the University of Colorado at Boulder and at Yale University. He earned his Ph.D. in computational and applied mathematics in 2002 from UT Austin. He received a M.Sc. degree in Engineering Physics in 1996 and a "Licentiate" degree in Mathematics in 1998, both from the Chalmers University of Technology in Sweden. Martinsson was the recipient of the SIAM 2017 Germund Dahlquist Prize, and was named a Fellow of SIAM in 2021.

Martinsson's research concerns the development of faster algorithms for ubiquitous computational tasks in scientific computing and data sciences. Recent work has focused on randomized methods in linear algebra, fast solvers for elliptic PDEs, O(N) complexity direct solvers, structured matrix computations, and high order accurate methods for scattering and fluid problems

 

[3] Speaker: Alexandre Tartakovsky, Department of Civil and Environmental Engineering, UIUC

Time: 3:45pm, Thursday, November 11, 2021

Venue: Zoom

 

Passcode: caimeeting, zoom meeting id 512 004 3096

Title: Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems

Abstract: This talk will present a novel physics-informed machine learning approach for large-scale data assimilation and parameter estimation and demonstrate its application for estimating transmissivity and hydraulic head in the two-dimensional steady-state subsurface flow model of the Hanford Site given synthetic measurements of said variables. The synthetic data are generated using the hydraulic conductivity field and boundary conditions that were estimated in a previous Hanford Site calibration study. In our approach, we extend the physics-informed conditional Karhunen-Loeve expansion (PICKLE) method for modeling subsurface flow with unknown flux (Neumann) and varying head (Dirichlet) boundary conditions that are typical for large-scale systems like the Hanford Site. We demonstrate that overall, the PICKLE method is comparable in accuracy with the standard maximum a posteriori (MAP) parameter estimation method but is significantly faster than MAP for large-scale problems. Both methods use a mesh to discretize the computational domain. In MAP, the parameters and states are discretized on the mesh; therefore, the size of the MAP parameter estimation problem directly depends on the mesh size. In PICKLE, the mesh is used to evaluate the residuals of the governing equation, while the parameters and states are approximated by the truncated conditional Karhunen-Loeve expansions with the number of parameters controlled by the smoothness of the parameter and state fields, and not by the mesh size. For a considered example, we demonstrate that the computational cost of PICKLE increases near linearly with the problem size while that of MAP increases faster than the power of three of the problem size. We demonstrate that once trained for one set of the Dirichlet boundary condition, the PICKLE method provides accurate estimates of the hydraulic head for any values of the Dirichlet boundary conditions.

Biography:  Prof. Tartakovsky got his PhD in Hydrology, University of Arizona, 2002 and currently a full professor at Department of Civil and Environmental Engineering, UIUC. The main focus of  Prof. Tartakovsky's research is the theoretical and computational aspects of the modeling of flow and reactive transport in porous and fractured media under both saturated and unsaturated conditions. In his research in the subsurface flow and transport phenomena, he pursues a multi-disciplinary approach and relies on methods and techniques used in hydrology, mathematics, computational physics, and chemical engineering.

 

[4] Speaker: Xingjie Helen Li, Department of Mathematics, UNC Charlotte

Time: 3:45pm, Thursday, December 2, 2021

Venue: Clements Hall 126

Title:  Coarse – Graining of stochastic system

Abstract: Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. To achieve the efficiency, dimension reduction is often required in both space and time. In this talk, I will talk about our recent work on both spatial and temporal reductions.  For spatial dimension reduction, the Mori-Zwanzig formalism is applied to derive equations for the evolution of linear observables of the Langevin dynamics for both overdamped and general cases. For temporal dimension reduction, we introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. 

(This is a joint work with Dr. Thomas Hudson from the University of Warwick, UK; Dr. Fei Lu from the Johns Hopkins University and Dr Xiaofeng Felix Ye from SUNY at Albany).

Biography:  Dr. Xingjie Helen Li graduated from University of Minnesota in 2012, did her postdoctoral at Brown University from 20212 to 2015 and joined UNC Charlotte in 2015. Her research lies in the area of applied and computational mathematics, focusing on multiscale modeling and structure-preserving schemes. She has built interdisciplinary collaborations with mathematicians, physicists and engineers in the following areas and received the NSF-DMS early career award in 2019.