Date and time: Jan 19 at 2:00 pm (Parker 328)

Title: On energy laws and stability of Runge-Kutta methods for linear seminegative problems

Abstract: In scientific and engineering applications, people usually encounter time-dependent systems with a decaying total energy. A natural question to ask is that when using a time integration method to approximate such systems, whether and how the energy-decay law will be preserved at the discrete level. In this talk, we will focus on linear autonomous systems and present a general framework on deriving discrete energy identities for both explicit and implicit Runge-Kutta (RK) method. The established identities provide a precise characterization on energy generation or dissipation in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we also present a general energy identity for all the diagonal Padé approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices.

Date and time: Feb 2 at 2:00 pm (Parker 328)

Title: Global existence and asymptotic behavior of classical solutions of chemotaxis systems with logistic source and consumption on \(\mathbb{R}^{n}\)

Abstract: The current talk is concerned with the global existence and asymptotic behavior of classical solutions of chemotaxis systems with logistic source and consumption on the whole space \(\mathbb{R}^n\). We will first consider the global existence of positive classical solutions. We then discuss the asymptotic behavior of solutions with strictly positive initials,  and the effect of chemotaxis on the rate of spread of the mobile species when the initial have compact support. 

Date and time: Feb 9 at 2:00 pm (Parker 328)

Title: Movement versus no movement: When is it good to move?

Abstract: In a famous paper published in 1998  on the evolution of slow dispersal, Dockery and his collaborators showed that,  when two identical diffusive competing species live together in the same ecological environment and are different in their dispersal rates,  the slower moving species will eventually drive the faster species to extinction.  In a subsequent work in 2008, Cantrell and his collaborators then asked: would this competitive advantage still be preserved if one of these two species doesn't move?  This talk will focus on providing some satisfactory answers to this question.  In fact, our results go beyond the simple setting of identical species. Furthermore, we will also examine the dynamics of the classical solutions of a  consumer-resource system comprising a  non-dispersing single resource species whose density function evolves and is consumed by three competitors adopting three different dispersal mechanisms.   Our results reveal that the existence of a non-sink area is beneficial to the non-dispersing species.

Date and time: Feb 23 at 2:00 pm (Parker 328)

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, the model 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. 

Date and time: March 1 at 2:00 pm (Parker 328)

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. 

Date and time: March 15 at 11:00 am (Parker 328)

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.  

Date and time: March 15 at 2:00 pm (Parker 328)

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.

Date and time: March 29 at 2:00 pm (Zoom)

Title:  Coupling Dynamics in Moist Geophysical Flows: The Impact of Phase Transitions on Energy Transfer and Nonlinear Interaction

Abstract: The interaction between slow and fast components in geophysical fluid dynamics, especially under the influence of phase changes, poses significant analytical challenges. Our study develops a fast-wave averaging framework for the moist Boussinesq system, expanding past dry dynamics to include phase changes between water vapor and liquid water. We examine whether these phase transitions induce coupling between slow and fast waves or if the slow component evolves independently. Numerical simulations with a range of Froude and Rossby numbers reveal that phase changes may disrupt the proportionality of wave influence on the slow component, evidenced by a nonzero time-averaged wave component due to phase transitions. Furthermore, inverse energy transfer to larger scales is investigated in rotating and stratified flows, including water effects and rapid cloud microphysics. The findings suggest that potential vorticity, phase boundaries, and vertical velocity contribute to the formation of coherent structures in strongly rotated and stratified flows, altering the traditional view of energy cascades in geophysical fluids.

Date and time: April 5 at 2:00 pm (Parker 328)

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.

Date and time: April 12 at 2:00 pm (Zoom)

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 show 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.  

Date and time: April 19 at 2:00 pm (Parker 328)

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.

Date and time: April 26 at 2:00 pm (Parker 328)

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.