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# Departmental Colloquia

Our department is proud to host weekly colloquium talks featuring research by leading mathematicians from around the world. Most colloquia are held on Fridays at 4pm in Parker Hall, Room 250 (unless otherwise advertised) with refreshments preceding at 3:30pm in Parker Hall, Room 244.

**DMS Colloquium: Dr. Zuofeng Shang**

**Sep 20, 2019 04:00 PM**

Speaker: **Dr. Zuofeng Shang**, Department of Mathematics and Statistics at Indiana University – Purdue University Indianapolis

Title: TBA

Abstract: TBA

Faculty host: Guanqun Cao

**DMS Colloquium: Virginia Vassilevska Williams**

**Sep 27, 2019 04:00 PM**

Speaker: **Virginia Vassilevska Williams** (MIT)

Title: TBA

**DMS Colloquium: Beatrice Riviere**

**Sep 06, 2019 04:00 PM**

Speaker: **Béatrice Rivière**, Rice University

Title: High order discontinuous Galerkin methods for solving the miscible displacement problem in porous media

Abstract: The accurate prediction of flow and transport in porous media is essential in optimizing the clean-up of contaminated groundwater or the production of hydrocarbons from oil reservoirs. In the miscible displacement problem, a solvent is injected and mixes with the resident fluid (contaminant or oil). The fluid mixture then propagates through the set of connected pores. At the Darcy scale, the mathematical model is a system of partial differential equations coupling flow and transport. This talk presents high order interior penalty discontinuous Galerkin (IPDG) methods and hybridizable discontinuous Galerkin (HDG) methods for solving the nonlinear system of equations. HDG methods retain the positive features of IPDG, but the number of globally coupled degrees of freedom for high order HDG is significantly smaller. The proposed numerical methods are shown to be accurate on coarse meshes when the polynomial degree increases. The numerical approximations of the solvent concentration exhibit sharp fronts even in highly heterogeneous media. Finally, the discontinuous Galerkin methods in space can accurately model viscous fingering. Viscous fingering in porous media may occur when a fluid with low viscosity is used to displace a fluid with high viscosity. For this type of flow instability, a tiny perturbation can be amplified exponentially, which triggers a finger-like pattern in the fluid concentration profile during the fluid displacement. Simulations in two and in three dimensions show the growth and propagation of fingers for large mobility ratios and large Peclet numbers. Results are compared with those obtained by using a generic cell-centered finite volume method.

**Short Bio-sketch**

Dr. Rivière has worked extensively of the development and analysis of numerical methods applied to problems in porous media and in fluid mechanics. Her current research deals with the development of high-order methods in time and in space for multiphase multicomponent flows (in rigid and deformable media); the modeling of pore scale flows for immiscible and miscible components; the numerical model of oxygen transport in a network of blood vessels; the analysis of PDE-based neural networks for image segmentation and the design of iterative solvers.

Dr. Rivière is an associate editor for the SIAM Journal on Numerical Analysis, for the SIAM Journal on Scientific Computing, for Results in Applied Mathematics, and a member of the editorial board for Advances in Water Resources. She has graduated a total of fourteen Ph.D. students, with eight working in academia and five in industry.

Faculty host: Thi-Thao-Phuong Hoang

**DMS Colloquium: Wei Cai**

**Aug 30, 2019 04:00 PM**

Speaker: **Wei Cai**, Southern Methodist University

Title: Algorithms for wave scattering of random media: Fast multipole method in layered media and a phase shift deep neural network for wideband learning

Abstract: In this talk, we will present two algorithms and numerical results for solving electromagnetic wave scattering of random meta-materials. Firstly, a fast multipole method for 3-D Helmholtz equation for layered media will be presented based on new multipole expansion (ME) and multipole to local translation (M2L) operators for layered media Green's functions. Secondly, a parallel phase shift deep neural network (PhaseDNN) is proposed for wideband data learning. In order to achieve uniform convergence for low to high frequency content of data, phase shifts are used to convert high frequency learning to low frequency learning. Due to the fast learning of many DNNs in the low frequency range, PhaseDNN is able to learn wideband data uniformly in all frequencies.

**Short Bio-sketch**

Prof. Wei Cai obtained his B.S. and M.S. in Mathematics from the University of Science and Technology of China in 1982 and 1985, respectively, and his Ph.D. in Applied Mathematics at Brown University in 1989. Currently, he is the Clements Chair professor of applied mathematics at SMU. Before he joined SMU, he was an assistant and then associate professor at the University of California at Santa Barbara during 1995-96, and a full Professor at the University of North Carolina after 1999. He also taught and conducted research at Peking University, Fudan University, Shanghai Jiaotong University. His research interest focuses on the development of deterministic and stochastic numerical methods for studying electromagnetic and quantum phenomena with applications in meta-materials, nano-photonics, nano-electronics, biological systems, and quantum systems. He has published over 110 refereed articles in top international journals and is the author of the book "Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering, and Electron Transport" published by Cambridge University Press, 2013.

Faculty hosts: Junshan Lin and Yanzhao Cao

**DMS Colloquium: Matthias Heikenschloss**

**Apr 26, 2019 04:00 PM**

Speaker: **Matthias Heikenschloss**, Rice University

Title: Risk averse PDE constrained optimization

**DMS Colloquium: Frédéric Holweck**

**Apr 19, 2019 04:00 PM**

Speaker: **Frédéric Holweck**, Université de Technologie de Belfort-Montbéliard (France)

Title: Projective duality and quantum information

Abstract: Quantum Information is a nascent science which intends to use the properties of quantum physics to produce new computational paradigms. Quantum phenomena, like entanglement, are non-classical resources that need to be classified. Interestingly in the early 2000’s the old idea of projective duality regained new interest in the quantum physics literature as a potential tool for studying entanglement.

In this talk, after explaining the motivations from quantum physics, I will recall some classical notions regarding projective duality and introduce more recent results obtained with Luke Oeding about the calculation of hyperdeterminants from the \(E_8\)-discriminant.

Faculty host: Luke Oeding

**DMS Colloquium: Youssef Marzouk**

**Apr 12, 2019 04:00 PM**

Speaker: **Youssef Marzouk**, MIT

Title: Nonlinear filtering and smoothing with transport maps

Abstract: Bayesian inference for non-Gaussian state-space models is a ubiquitous problem, arising in applications from geophysical data assimilation to mathematical finance. We will present a broad introduction to these problems and then focus on high dimensional models with challenging nonlinear dynamics and sparse observations in space and time. While the ensemble Kalman filter (EnKF) yields robust ensemble approximations of the filtering distribution in this setting, it is limited by linear forecast-to-analysis transformations. To generalize the EnKF, we propose a methodology that transforms the non-Gaussian forecast ensemble at each assimilation step into samples from the current filtering distribution via a sequence of local nonlinear couplings. These couplings are based on transport maps that can be computed quickly using convex optimization, and that can be enriched in complexity to reduce the intrinsic bias of the EnKF. We discuss the low-dimensional structure inherited by the transport maps from the filtering problem, including decay of correlations, conditional independence, and local likelihoods. We then exploit this structure to regularize the estimation of the maps in high dimensions and with a limited ensemble size.

We also present variational methods---again based on transport maps---for smoothing and sequential parameter estimation in non-Gaussian state-space models. These methods rely on results linking the Markov properties of a target measure to the existence of low-dimensional couplings, induced by transport maps that are decomposable. The resulting algorithms can be understood as a generalization, to the non-Gaussian case, of the square-root Rauch--Tung--Striebel Gaussian smoother.

This is joint work with Ricardo Baptista, Daniele Bigoni, and Alessio Spantini.

Faculty hosts: Yanzhao Cao and Xiaoying Han

**DMS Colloquium: Emanuele Ventura**

**Apr 05, 2019 04:00 PM**

Speaker: **Emanuele Ventura**, Postdoc Texas A&M; Ph.D., Aalto University (Helsinki, Finland) 2017

Title: Tensors and their symmetry groups

Abstract: Tensors (multi-dimensional matrices) appear in many areas of pure and applied mathematics. I will discuss their use in algebraic complexity theory. Matrix multiplication is a tensor and its complexity is encoded in its tensor rank. To analyze the complexity of the matrix multiplication tensor, Strassen introduced a class of tensors that vastly generalize it, the tight tensors. These tensors have continuous symmetries. Pushing Strassen’s ideas forward, with A. Conner, F. Gesmundo, and J.M. Landsberg, we investigate tensors with large symmetry groups and classify them under a natural genericity assumption. Our study provides new paths towards upper bounds on the complexity of matrix multiplication.

Last Updated: 09/11/2015