PDE & Applied math seminar
Wednesday 10:00 -10:50 am, on zoom
Organizers : Weitao Chen / Heyrim Cho / Yat Tin Chow / Qixuan Wang / Jia Guo / Mykhailo Potomkin
In spring 2021, the PDE & Applied math seminar will be launched through zoom. If you're interested, please contact Dr. Heyrim Cho (heyrimc@ucr.edu) or Dr. Qixuan Wang (qixuanw@ucr.edu) for the zoom information.
Spring 2021 Schedule
Mar 31 10:00 (Wed) Organization meeting
Apr 7 10:00 (Wed) Xingjie Li (University of North Carolina, at Charlotte) (QW,HC)
Apr 14 10:00 (Wed) Franziska Weber (Carnegie Mellon University)
Apr 21 10:00 (Wed) Denis Tsygankov (Georgia Tech) (QW)
Apr 21 12:00 (Wed) Rishi Sonthalia (University of Michigan) (joint with Topology seminar)
Apr 28 10:00 (Wed) Yang Yang (Michigan Technological University) (HC)
May 5 10:00 (Wed) Kevin Sanft (University of North Carolina Asheville) (QW)
May 12 10:00 (Wed) Yeonjong Shin (Brown University) (HC)
May 19 10:00 (Wed) Zhongqiang Zhang (Worcester Polytechnic Institute) (HC)
May 26 10:00 (Wed) Wei Guo (Texas Tech University) (WC)
June 2 10:00 (Wed) Steven Woodhouse (University College London) (QW)
Next talks:
June 2, 10:00-10:50am PST
Dr. Steven Woodhouse (University College London)
Title: Reconstruction and analysis of gene regulatory network models
Abstract: In this talk I will discuss published and ongoing work on applying methods from computer science to reconstruct and understand gene regulatory networks, with the aim of improving cancer therapy and regenerative medicine. Topics include using ideas from program synthesis to infer networks, and applying algebraic decision diagrams and game theory to model cancer evolution and therapy resistance.
Bio: I am currently a postdoctoral researcher in computational biology at the University College London, in Jasmin Fisher's lab. Previously I was a postdoctoral researcher at the University of Pennsylvania, where I worked on characterising and optimising chimeric antigen receptor (CAR) T cell therapy. Before that I worked at Microsoft Research, and I received my PhD at the University of Cambridge under Bertie Gottgens.
Previous talks:
Apr 7th, 10-10:50am PST
Dr. Xingjie Li (University of North Carolina, Charlotte)
Title: Coarse-graining of overdamped Langevin dynamics via the Mori-Zwanzig formalism
Abstract: 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. To illustrate the resulting equations and their use in deriving approximate models, some particular benchmark examples are studied both numerically and via a formal asymptotic expansion. The examples considered demonstrates the importance of memory effects and asymptotic estimates in determining the correct temporal behavior of such systems with entropic barriers.
This is a joint work with Dr. Thomas Hudson from the University of Warwick, UK.
Bio: Dr. Xingjie Helen Li graduated from University of Minnesota in 2012, did her postdoctoral at Brown University from 2012 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.
Apr 21st, 10-10:50am PST
Dr. Denis Tsygankov (Georgia Tech)
Title: Unveiling the development of co-existing wave domains of Rho activity in the cell cortex with a mass-conserved activator-substrate model
Abstract: In this talk, I present a simulation model that reproduces a complex self-organization phenomenon in Patiria miniata (starfish) oocytes, in which Rho GTPase activity partitions into multiple co-existing regions of coherent wave propagation – wave domains. The model reproduced the cell-level activity of this key cytoskeletal regulator with numerical precision (as measured by textural characteristics of the spatiotemporal patterns) and led to a number of intriguing observations. It revealed that the development of the wave domains is preceded by a distinct stage of low activity, which may not be readily observed in experiments but has specific characteristics that define the dynamics at the later stages of cell regulation. An automated tool for wave domain detection developed in this project also revealed an unexpectedly sharp reversal of pattern formation in the middle of anaphase in starfish oocytes. Finally, the model predicts that by perturbing the balance of the autocatalytic activation and the negative feedback in Rho signaling, the dynamics shifts to a different regime closely resembling Rho activity in Xenopus laevis (frog) oocytes.
Bio: Denis Tsygankov is an Assistant Professor in the Wallace H. Coulter Department of Biomedical Engineering at Georgia Institute of Technology and Emory School of Medicine. He graduated from the Moscow Institute of Physics and Technology with a B.S. and an M.S. in Applied Mathematics and Physics. He completed his Ph.D. in Physics under the supervision of Dr. Kurt Wiesenfeld at Georgia Tech. His dissertation was devoted to synchronization phenomena in complex dynamical systems. Next, Dr. Tsygankov conducted post-doctoral research on molecular motors with Dr. Michael Fisher at the University of Maryland, College Park. Subsequently, he continued post-doctoral training in Computational Cell Biology with Dr. Timothy Elston at the University of North Carolina, Chapel Hill. His current research interests are focused on developing computational models and computer vision techniques for integrative studies of complex biological processes (such as cell motility, polarization, and collective cell behavior) from the perspective of Systems Biology and Biomechanics.
Apr 21st, 12-12:50pm PST
Dr. Rishi Sonthalia (University of Michigan)
Title: Learning Optimal Metrics.
Abstract: Given a set of dissimilarity measures amongst data points, many machine learning problems are considerably ``easier'' if these dissimilarity measures adhere to a metric. Furthermore, learning the metric that is most ``consistent'' with the input dissimilarities or the metric that best captures the relevant geometric features of the data (e.g., the correlation structure in the data) is a key step in efficient, approximation algorithms for classification, clustering, regression, and feature selection. In practice, these metric learning problems are formulated as convex optimization problems subject to metric constraints, such as the triangle inequality, on all the output variables. Because of the large number of constraints, researchers have been forced to restrict either the kinds of metrics learned or the size of the problem that can be solved. In many cases, researchers have restricted themselves to learning (weighted) Euclidean or Mahalanobis metrics. This approach is, however, far from ideal as the inherent geometry of many data sets necessitates different types of metrics. Therefore, we need to develop optimization techniques that can optimize over the space of all metrics on a data set. In this talk, I will present many different aspects of the metric learning problem; the sparse version, the general convex version, and how such ideas can be used for dimensionality reduction in the presence of missing data.
Apr 28st, 10-10:50am PST
Dr. Yang Yang (Michigan Technological University)
Title: The Hybrid-dimensional Darcy’s Law: A Reinterpreted Discrete Fracture Model for Fracture and Barrier Networks on Non-conforming Meshes
Abstract: The discrete fracture model (DFM) has been widely used in the simulation of fluid flow in fractured porous media. Traditional DFM use the so-called hybrid-dimensional approach to treat fractures explicitly as low-dimensional entries (e.g. line entries in 2D media and face entries in 3D media) on the interfaces of matrix cells to avoid local grid refinements in fractured region and then couple the matrix and fracture flow systems together based on the principle of superposition with the fracture thickness used as the dimensional homogeneity factor. Because of this methodology, DFM is considered to be limited on conforming meshes and thus may raise difficulties in generating high qualified unstructured meshes due to the complexity of fracture’s geometrical morphology.
In the first part of the talk, we clarify that the discrete fracture model actually can be extended to non-conforming meshes without any essential changes. To show it clearly, we provide another perspective for DFM based on hybrid-dimensional representation of permeability tensor modified from the comb model to describe fractures as one-dimensional line Dirac-delta functions contained in permeability tensors. We claim that the proposed mode is not a new model, but a reinterpretation of the DFM (RDFM). A finite element RDFM scheme for single-phase flow on non-conforming meshes is then derived by applying Galerkin finite element method to it. Analytical analysis and numerical experiments show that the RDFM scheme automatically degenerates to the classical finite element DFM when the mesh is conforming with fractures. Moreover, the accuracy and efficiency of the model on non-conforming meshes is demonstrated by testing several benchmark problems. This model is also applicable to curved fracture with variable thickness.
In the second part, we extend the RDFM for flow simulation of fractured porous media containing flow blocking barriers on non-conforming meshes. The methodology of the approach is to modify the traditional Darcy’s law into the hybrid-dimensional Darcy’s law where fractures and barriers are represented as Dirac-delta functions contained in the permeability tensor and resistance tensor, respectively. This model is able to account for the influence of both highly conductive fractures and blocking barriers accurately on non-conforming meshes. The local discontinuous Galerkin (LDG) method is employed to accommodate the form of the hybrid-dimensional Darcy’s law and the nature of the pressure/flux discontinuity. The performance of the model is demonstrated by several numerical tests.
Bio: I received my PhD degree in 2013 from Brown with Prof. Chi-Wang Shu, then became an assistant professor at MTU. In 2017, I was promoted to associate professor with tenure. My research areas are: (1) Discontinuous Galerkin methods for convection-diffusion equations. (2) Numerical methods for partial differential equations with blow-up solutions. (3) Single and multiphase flows in fractured porous media (4) Numerical cosmology (5) Gaseous detonations
May 5th, 10-10:50am PST
Dr. Kevin Sanft (University of North Carolina Asheville)
Title: Algorithms for Simulating Large Spatial Discrete Stochastic Biochemical Models
Abstract: Discrete stochastic simulation can be computationally expensive, especially for spatially discretized reaction-diffusion processes. There are several Gillespie-type simulation algorithms, such as Elf’s Next Subvolume Method, that have different performance and scaling properties. We discuss the underlying relationships between these algorithms and how they can be formulated to have constant algorithmic complexity with respect to the size of the model. We then present some recent software for user-friendly model development and simulation of multiscale biochemical systems.
Short bio: Kevin Sanft is an assistant professor of computer science at the University of North Carolina Asheville (UNCA). His research interests span the broad fields of computational science and data science. Much of his work involves developing efficient algorithms and software for multiscale stochastic modeling and simulation. He was a developer of the StochKit2 software package and has recently contributed to the GillesPy2 Python library. Prior to UNCA, he held postdoc positions at the University of Minnesota working with Hans Other and at St. Olaf College. He received his Ph.D. from the University of California Santa Barbara, supervised by advisors Linda Petzold and Frank Doyle. Prior to graduate school, he worked on Parkinson’s disease research at the Mayo Clinic.
May 12th, 10-10:50am PST
Dr. Yeonjong Shin (Brown University)
Title: Towards a mathematical understanding of modern machine learning: theory, algorithms, and applications
Abstract: Modern machine learning (ML) has achieved unprecedented empirical success in many application areas. However, much of this success involves trial-and-error and numerous tricks. These result in a lack of robustness and reliability in ML. Foundational research is needed for the development of robust and reliable ML. This talk consists of two parts. The first part will present the first mathematical theory of physics informed neural networks (PINNs) - one of the most popular deep learning frameworks for solving PDEs. Linear second-order elliptic and parabolic PDEs are considered. I will show the consistency of PINNs by adapting the Schauder approach and the maximum principle. The second part will focus on some recent mathematical understanding and development of neural network training. Specifically, two ML phenomena are analyzed -- "Plateau Phenomenon" and "Dying ReLU." New algorithms are developed based on the insights gained from the mathematical analysis to improve neural network training.
Bio. Yeonjong Shin is a Prager Assistant Professor at the Division of Applied Mathematics, Brown University. He completed his PhD in mathematics at the Ohio State University in 2018, advised by Professor Dongbin Xiu. His research interests lie in mathematics of machine learning, scientific computing, approximation theory, and uncertainty quantification.
May 19th, 10-10:50am PST
Dr. Zhongqiang Zhang (Worcester Polytechnic Institute)
Title: Error estimates of residual minimization using neural networks for linear PDEs
Abstract: We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates for both continuous and discrete formulations of residual minimization in strong and weak forms. The formulations cover recently developed physics-informed neural networks based on strong and variational formulations. See the full text at https://arxiv.org/abs/2010.08019.
Bio: Zhongqiang Zhang (张中强) is an Associate Professor of Mathematics at Worcester Polytechnic Institute. His research interests include numerical methods for stochastic and integral differential equations, computational probability, and mathematics for machine learning. Before he joined in Worcester Polytechnic Institute in 2014, he received Ph.D. degrees in mathematics at Shanghai University in 2011 and in applied mathematics at Brown University in 2014. He co-authored a book with George Karniadakis on numerical methods for stochastic partial differential equations with white noise.
May 26th, 10:10:50am PST
Dr. Wei Guo (Texas Tech University)
Title: Low Rank Tensor Methods for Vlasov Simulations
Abstract: In this talk, we present a low-rank tensor approach for solving the Vlasov equation. Among many existing challenges for Vlasov simulations (e.g. multi-scale features, nonlinearity, formation of filamentation structures), the curse of dimensionality and the associated huge computational cost have been a long-standing key obstacle for realistic high-dimensional simulations. In this work we propose to overcome the curse of dimensionality by dynamically and adaptively exploring a low-rank tensor representation of Vlasov solutions in a general high-dimensional setting. In particular, we develop two different approaches: one is to directly solve the unknown function, and the other is to solve the underlying flow map, aiming to obtain a low-rank approximation with optimal complexity. The performance of both proposed algorithms are benchmarked for standard Vlasov-Poisson/Maxwell test problems.