Nonlinear preconditioning refers to transforming a nonlinear algebraic system to a form for which Newton-type algorithms have improved success through quicker advance to the domain of quadratic convergence. We place these methods, which go back at least as far as the Additive Schwarz Preconditioned Inexact Newton (ASPIN, 2002) in the context of a proliferation distinguished…

A rank-revealing QR factorization (RRQR) of an mxn matrix A can be an efficient alternative to the singular value decomposition.  Given 1≤k<n,  the problem of computing an RRQR is selecting k linearly independent columns of A. In this talk, we discuss the RRQR and present an efficient two-staged randomized algorithm to compute one. We analyze…

We propose novel numerical methods for solving a class of high-order hyperbolic PDEs on general geometries, which involve 2nd-order derivatives in time and up-to 4th-order derivatives in space. These PDEs are widely used in modeling thin-walled elastic structures such as beams, plates and shells, etc. High-order spatial derivatives together with general geometries bring a number…

In this talk, I will present our recent numerical methodology studies for unsteady moving interface problems and applications to dynamic fluid-structure interaction (FSI) problems. Our numerical methodologies include the body-fitted mesh method (arbitrary Lagrangian−Eulerian (ALE) method), the body-unfitted mesh method (fictitious domain (FD) method), combining with the mixed finite element approximation, as well as the…

Time-delayed feedback control, attributed to Pyragas (1992 Phys. Lett. 170), is a method known to stabilise periodic orbits in low dimensional chaotic dynamical systems. A system of the form dx/dt=f(x) has an additional term G(x(t)-x(t-T)) introduced where G is some 'gain matrix' and T a time delay. This form of the delay term is such…

Incommensurate stacked 2D materials have gained significant attention after the recent discovery of a new mechanism of superconductivity in systems with small twist angles. Theoretically, the electronics of such systems are studied through tight-binding models. These models can be studied in several different spaces, though momentum space is often the leading favorite for physicists because…

In this talk we will introduce a new stability theory for implicit-explicit (IMEX) time integration schemes—which treat some terms in a differential equation implicitly (for stability) and others explicitly (for efficiency).  Our focus will be on devising new efficient stable schemes for several fluid equations ranging from the incompressible Navier-Stokes equations, nonlinear diffusion equations, and…

Monsoons are significant atmospheric phenomena that manifest over various regions in the tropics and have massive social and economic consequences. We will present hybrid methods that combine ideas from dynamical systems-based nonlinear time series analysis and machine learning and analyze the dynamics of the South Asian monsoon. Specifically, we will show two sets of results,…

Constructing efficient numerical solution methods for the equation of radiative transfer (ERT)remains as a challenging task in scientific computing despite of the tremendous development on the subject in recent years. We present in this work a simple fast computational algorithm for solving the ERT in isotropic media in steady state setting and time-dependent setting. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly-averaged ERT solution using iterative schemes such as the GMRES method.The computation in this step is accelerated with a fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the ERT solution. The algorithm does not require the underlying medium be homogeneous. We present numerical  simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous media. Then we will…

Catalysts are usually made of a dense but porous material such as activated carbon, zeolites, etc. that provide a large surface area. Liquids that are produced as a by-product of a gas reaction at the catalyst site transport to the surface of the porous material, slowing down transport of the gaseous reactants to the catalyst…

Traditionally, linear programming (LP) is solved using Simplex or Interior Point Method whose core computational operation is factorization. Recently, there has been a push in the optimization community towards developing methods whose core computational operation is instead matrix-vector multiplications. Compared with factorization, matrix-vector multiplications are less likely to run out of memory on large-scale problems…

Developing suitable approximate models for analyzing and simulating complex nonlinear systems is practically important. This paper aims at exploring the skill of a rich class of nonlinear stochastic models, known as the conditional Gaussian nonlinear system (CGNS), as both a cheap surrogate model and a fast preconditioner for facilitating many computationally challenging tasks. The CGNS…

In this talk, I will present recent research that builds fast and accurate data-driven surrogate models (or emulators) for various complex and high-dimensional systems. Furthermore we will use scientific machine learning techniques in lieu of black-box data-driven methods. In other words, not only will our models be informed by data, but they will also be…

Dry eye disease is caused by a breakdown of a uniform tear film, which occurs when the layer of tears experiences breakup. To better understand this ocular condition, the dynamics of the tear film can be studied using fluorescence imaging.   Many parameters affect tear film thickness and fluorescent intensity distributions over time; exact values or…

The capability of using imperfect statistical reduced-order models to capture crucial statistics in complex turbulent systems is investigated. Much simpler and more tractable block-diagonal models are proposed to approximate the complex and high-dimensional turbulent dynamical equations using both parameterization and machine learning strategies. A systematic framework of correcting model errors with empirical information theory is…

Gaussian processes (GPs) are popular models for random functions in computational and applied mathematics, statistics, machine learning and data science. However, GP methodology scales poorly to large data-sets due to the need to factorize a dense covariance matrix. In spatial statistics, a standard approach to surmount this challenge is to represent Matérn GPs using finite…

The objective of this talk is to introduce a fairness interpretability framework for measuring and explaining the bias in classification and regression models at the level of a regressor distribution. In our work, we measure the model bias across sub-population distributions in the model output using the Wasserstein metric. To properly quantify the contributions of…

Stochastic optimization problems with deterministic constraints commonly appear in machine learning, finance, and engineering applications. This talk presents a new adaptive solution strategy for this important class of problems. The aim is to decrease the computational cost while maintaining an optimal convergence rate. The guiding principle is to adjust the batch size (or sample size)…

There is a well-established linear algebraic lens for studying consensus dynamics on networks, which has yielded significant theoretical results in areas like distributed computing, modeling of opinion dynamics, and ranking methods.  Recently, strong connections have been made between problems of consensus dynamics on networks and classical iterative methods in numerical linear algebra.  This talk will…

Analysis of physical and engineering systems is characterized by unique computational challenges associated with high dimensionality of parameter spaces, large cost of simulations or experiments, as well as existence of uncertainty. For a wide range of these problems the goal is to either quantify uncertainty and compute risk for critical events, optimize parameters or control…