Porous media and conduit coupled systems are heavily used in a variety of areas such as groundwater system, petroleum extraction, and biochemical transport. A coupled dual porosity Stokes model has been proposed to simulate the fluid flow in a dual-porosity media and conduits coupled system. Data assimilation is the discipline that studies the combination of mathematical models and observations. It…

We consider a so-called fractional gradient flow: an evolution equation aimed at the minimization of a convex and l.s.c. energy, but where the evolution has memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so- called Caputo derivative of the state. We introduce a…

We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account…

 Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretized, they lead to ill-conditioned linear systems, often of huge dimensions: regularization consists in replacing the original system by a nearby problem with better numerical properties, in order to find a meaningful approximation of its solution. After briefly surveying some standard regularization…

The resurging interest in deep learning is commonly attributed to advances in hardware and growing data sizes and less so to new algorithmic improvements. However, cutting-edge numerical methods are needed to tackle ever larger and more complex learning problems. In this talk, I will illustrate numerical analysis tools for improving the effectiveness of deep learning…

Two-dimensional crystals have been intensely investigated both experimentally and theoretically since graphene was exfoliated from graphite. Physicists have recently developed the ability to stack one layer on another with a twist angle controlled to the scale of .1 degree with the goal of creating two dimensional materials with desired electronic, optical, and mechanical properties. Unusual…

Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering).…

The magnetohydrodynamics (MHD) equations are continuum models used in the study of a wide range of plasma physics systems. However, efficient numerical solution methods for MHD are extremely challenging due to disparate time and length scales, strong hyperbolic phenomena, and nonlinearity. Therefore the development of scalable, implicit MHD algorithms and high-resolution adaptive mesh refinement strategies…

 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…