Applied and Computational Mathematics Seminars
Sep 27, 2024 01:00 PM
328 Parker Hall
Sep 20, 2024 01:00 PM
328 Parker Hall
Speaker: Cao-Kha Doan (Auburn)
Title: Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations.
Abstract: In this work, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we reformulate the original equations into an equivalent system that incorporates the energy evolution process. First- and second-order dynamically regularized Lagrange multiplier (DRLM) schemes are derived based on the backward differentiation formulas and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting the accuracy and stability of the numerical solutions. Various numerical experiments including the Taylor-Green vortex problem, lid-driven cavity flow, and Kelvin-Helmholtz instability are carried out to demonstrate the performance of the DRLM schemes. Extension of the DRLM method to the Cahn-Hilliard-Navier-Stokes system will also be discussed.
DMS Applied and Computational Mathematics Seminar
Sep 06, 2024 01:00 PM
328 Parker Hall
DMS Applied and Computational Mathematics Seminar
Apr 26, 2024 02:00 PM
328 Parker Hall
Speaker: Yuanzhe Xi (Emory)
Title: Acceleration methods for scientific and data science applications
Abstract: There has been a surge of interest in recent years in general-purpose `acceleration' methods that take a sequence of vectors converging to the limit of a fixed point iteration, and produce from it a faster converging sequence. A prototype of these methods that attracted much attention recently is the Anderson Acceleration (AA) procedure. In this talk, we will discuss a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as with inexact Newton methods- depending on which variant is implemented. We will demonstrate the efficiency of the proposed method on a variety of problems from simulation experiments to deep learning applications.
This is joint work with Yousef Saad, Huan He, Ziyuan Tang, and Shifan Zhao.
DMS Applied and Computational Mathematics Seminar
Apr 19, 2024 02:00 PM
328 Parker Hall
Speaker: Akil Narayan (University of Utah)
Title: Model selection, combination, and management: Sowing with exploration and reaping with exploitation
Abstract: Modern simulation-based scientific models are complex and multi-faceted, involving computationally demanding physics-based modeling and discretization, reliable and robust data assimilation, and an accurate accounting for uncertainty in the face of unknown model parameter values and/or genuine stochasticity. To meet such demands, many simulations of real-world systems often involve combinations of homogenized, microscale, or reduced order model components that target accuracy of specific system subcomponents. To complicate matters, individual subcomponents may have several competing models whose query cost and accuracy payoff for system-wide prediction is opaque. For example, there may be numerous ways to identify macroscopic closure terms that are informed through analysis of a suite of microscale models.
We discuss how computational exploration-exploitation meta-algorithms from bandit learning can accomplish model selection and combination for computational budget allocation in scientific computing applications: An exploration phase is devoted to learning about model relationships and interactions, followed by an exploitation phase that uses information learned in exploration to make decisions about optimal model selection and subsequently provides full distributional information of stochastic outputs. This framework leads to flexible procedures, capable of managing disparate models and multi-modal data in adaptive and real-time scenarios. We also show how our exploration-exploitation approach can be used in complementary schemes for multilevel uncertainty quantification that exercise control variates and BLUE estimators. We will identify existing theoretical guarantees for such procedures along with promising and impactful directions for new analysis and algorithm development.
DMS Applied and Computational Mathematics Seminar
Apr 12, 2024 02:00 PM
ZOOM
Speaker: Yulong Lu (University of Minnesota, Twin Cities)
Title: Diffusion Models: Theory and Applications (in PDEs)
Abstract: Diffusion models, particularly score-based generative models (SGMs), have emerged as powerful tools in diverse machine learning applications, spanning from computer vision to modern language processing. In the first part of this talk, we delve into the generalization theory of SGMs, exploring their capacity for learning high-dimensional distributions. Our analysis shows that SGMs achieve a dimension-free generation error bound when applied to a class of sub-Gaussian distributions characterized by certain low-complexity structures. In the second part of the talk, we consider the application of diffusion models in solving partial differential equations (PDEs). Specifically, we present a physics-guided diffusion model for reconstructing high-fidelity solutions from their low-fidelity counterparts. This application showcases the adaptability of diffusion models and their potential to scientific computation.
DMS Applied and Computational Mathematics Seminar
Apr 05, 2024 02:00 PM
328 Parker Hall
Title: Deep filtering with adaptive learning rates
Abstract: Given the state of a system is not completely observable, filtering is concerned with state estimation based on partial observations of the system state. It enjoys many applications in the control of partially observed systems, target tracking, signal processing, statistics, and financial engineering. Devoted to the conditional distribution or density, the celebrated results of the Kushner equation and Duncan-Mortensen-Zakai equation produce nonparametric estimations of the conditional distribution/density. Approximating their solutions will suffer the curse of dimensionality. In this talk, we first introduce a filtering algorithm termed deep filtering based on the deep learning framework. Then we present our work on deep filtering with adaptive learning rates. Instead of approximating the conditional distribution or density, we focus on state estimation or conditional mean and convert the filtering problem to an optimization problem by finding the optimal weights of a deep neural network (DNN). This solves a long-standing (60-year-old) challenging problem in computational nonlinear filtering and has the potential to overcome the curse of dimensionality. We constructed a stochastic gradient-type procedure to approximate the weight parameters of DNN and developed another recursion to update the learning rate adaptively. We showed the convergence of the continuous time interpolated learning rate process using stochastic averaging and martingale methods and obtained an error bound for parameters of the neural network. Finally, we present two numerical examples to show the efficiency and robustness of our algorithm.
This is based on joint work with Prof. George Yin and Prof. Qing Zhang.
DMS Applied and Computational Mathematics Seminar
Mar 15, 2024 02:00 PM
328 Parker Hall
Speaker: Ke Chen (University of Maryland)
Title: Towards efficient deep operator learning for forward and inverse PDEs: Theory and algorithms
Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in scientific computing. This talk delves into efficient training for PDE operator learning in both the forward and inverse PDE settings. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), and compare its numerical performance with baseline models on several inverse problems, including optical tomography and inverse scattering.
DMS Applied and Computational Mathematics Seminar
Mar 15, 2024 11:00 AM
328 Parker Hall
Title: Population dynamics subject to nitrogen-to-carbon stoichiometric constraints
Abstract: Ecological processes depend on the flow and balance of essential elements, such as carbon, nitrogen, and phosphorus. The balance of these essential elements in ecological interactions is studied in the theory of Ecological Stoichiometry (ES). The integration of ES into population dynamics has provided insights into understanding how stoichiometric constraints affect food webs and nutrient cycles. I will present stoichiometric ordinary differential equation mathematical models that focus on the essential elements of nitrogen and carbon to investigate topics such as disease-ecosystem feedback loops and nitrogen fixation.
DMS Applied and Computational Mathematics Seminar
Mar 01, 2024 02:00 PM
328 Parker Hall
Speaker: Thi Thao Phuong Hoang (Auburn University)
Title: Space-time mixed finite element method for reduced fracture flow models on nonmatching grids
Abstract: We propose a space-time mixed variational formulation of the flow problem in a fractured porous medium, where the fracture is treated as a lower dimensional object embedded in the rock matrix. Different spatial and temporal grids can be used in the subdomains and in the fracture to adapt to the heterogeneity of the problem. To efficiently solve the coupled system between the subdomains and the fracture, substructuring techniques are used to derive a space-time interface problem which is solved iteratively by GMRES. Analysis and numerical results for the proposed methods will be presented.
DMS Applied and Computational Mathematics Seminar
Feb 23, 2024 02:00 PM
328 Parker Hall
Speaker: Pengjun Wang (Auburn)
Title: A pseudo-reversible normalizing flow for stochastic differential equation and uncertainty quantification
Abstract: We present a pseudo-reversible normalizing flow method for efficiently generating samples of the state of a stochastic differential equation (SDE) with various initial distributions. Once trained,the normalizing flow model can directly generate samples of the SDE's final state without simulating trajectories. Notably, themodel requires only a single training session, after which it can accommodate a range of initial distributions. This feature can provide a significant computational saving in studies of how the final state varies with the initial distribution. Additionally, we introduce a conditional pseudo-reversible normalizing flow for quantifying forward and inverse uncertainty propagation. The convergence analysis in the Kullback–Leibler divergence and numerical experiments will be provided.