Discretization, Solution, and Inversion for Large Systems of PDEs

Series: 
Applied and Computational Mathematics Seminar
Monday, October 29, 2018 - 1:55pm
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
Georgia Tech, School of Computational Science and Engineering
Organizer: 
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.