Georgia Institute of Technology College of Sciences

In this talk, we will discuss some computational issues when applying the large deviation theory to study small-noise-induced rare events in differential equations. We focus on two specific problems: the most probable transition path for an ordinary differential equation and the asymptotically efficient simulation of rare events for an elliptic problem. Both problems are related to the large deviation theory. From a computational point of view, the former one is a variational problem while the latter one is a sampling problem. For the first problem, we have developed an hp adaptive minimum action method, and for the second problem, we will present an importance sampling estimator subject to a sufficient and necessary condition for its asymptotic efficiency.

We are often forced to make important decisions with imperfect and incomplete data. In model-based inference, our efforts to extract useful information from data are aided by models of what occurs where we have no observations: examples range from climate prediction to patient-specific medicine. In many cases, these models can take the form of systems of PDEs with critical-yet-unknown parameter fields, such as initial conditions or material coefficients of heterogeneous media. A concrete example that I will present is to make predictions about the Antarctic ice sheet from satellite observations, when we model the ice sheet using a system of nonlinear Stokes equations with a Robin-type boundary condition, governed by a critical, spatially varying coefficient. This talk will present three aspects of the computational stack used to efficiently estimate statistics for this kind of inference problem. At the top is an posterior-distribution approximation for Bayesian inference, that combines Laplace's method with randomized calculations to compute an optimal low-rank representation. Below that, the performance of this approach to inference is highly dependent on the efficient and scalable solution of the underlying model equation, and its first- and second- adjoint equations. A high-level description of a problem (in this case, a nonlinear Stokes boundary value problem) may suggest an approach to designing an optimal solver, but this is just the jumping-off point: differences in geometry, boundary conditions, and otherconsiderations will significantly affect performance. I will discuss how the peculiarities of the ice sheet dynamics problem lead to the development of an anisotropic multigrid method (available as a plugin to the PETSc library for scientific computing) that improves on standard approaches.At the bottom, to increase the accuracy per degree of freedom of discretized PDEs, I develop adaptive mesh refinement (AMR) techniques for large-scale problems. I will present my algorithmic contributions to the p4est library for parallel AMR that enable it to scale to concurrencies of O(10^6), as well as recent work commoditizing AMR techniques in PETSc.

In recent years, metamaterials have drawn a great deal of attention in the scientific community due to their unusual properties and useful applications. Metamaterials are artificial materials made of subwavelength microstructures. They are well known to exhibit exotic properties and could manipulate wave propagation in a way that is impossible by using nature materials.In this talk, I will present our recent works on membrane-type acoustic metamaterials (AMMs). First, I will talk about how to achieve near-zero density/index AMMs using membranes. We numerically show that such an AMM can be utilized to achieve angular filtering and manipulate wave-fronts. Next, I will talk about the design of an acoustic complimentary metamaterial (CMM). Such a CMM can be used to acoustically cancel out aberrating layers so that sound transmission can be greatly enhanced. This material could find usage in transcranial ultrasound beam focusing and non-destructive testing through metal layers. I will then talk about our recent work on using membrane-type AMMs for low frequency noise reduction. We integrated membranes with honeycomb structures to design simultaneously lightweight, strong, and sound-proof AMMs. Experimental results will be shown to demonstrate the effectiveness of such an AMM. Finally, I will talk about how to achieve a broad-band hyperbolic AMM using membranes.

Following the seminal work of Nesterov, accelerated optimization methods (sometimes referred to as momentum methods) have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios were second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with an attraction basin large enough to accommodate the initial overshoot. This behavior has made accelerated search methods particularly popular within the machine learning community where stochastic variants have been proposed as well. So far, however, accelerated optimization methods have been applied to searches over finite parameter spaces. We show how a variational setting for these finite dimensional methods (recently formulated by Wibisono, Wilson, and Jordan) can be extended to the infinite dimensional setting, both in linear functional spaces as well as to the more complicated manifold of 2D curves and 3D surfaces.