Collaborative Research: Efficient Coupling of Multilevel Partial Differential Equation Solvers and Advanced Sampling Methods

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We know today how to simulate many processes on computers, such as air flow around airfoils or how objects deform when a force is applied. However, from a practical perspective, it is often desirable to determine to "quantify the uncertainty" of our predictions, i.e., to be able to say how accurate or inaccurate a computer simulation is when the inputs are not exactly known. An example is how an airfoil behaves when the air ahead is turbulent. In these cases, one needs to simulate many scenarios, an exceedingly expensive proposition. This project aims to develop mathematical tools to make this process more feasible and computationally affordable by taking advantage of multilevel hierarchies in the sampling and solution approach.Many processes in the sciences and engineering are described by parameter-dependent partial differential equations (PDEs) whose numerical solution is expensive. While their forward solution may be feasible, the computational cost makes it difficult or impossible to statistically evaluate questions such as (i) to quantify the uncertainty in output variables given a known or assumed distribution of parameter values, or (ii) to solve the statistical inverse problem of inferring a probability distribution among parameters from noisy measurements. Both of these questions are typically answered through high-dimensional and consequently very expensive sampling methods. This project will use hierarchical and multilevel decompositions of PDE solvers, coupled with error estimates, to derive vastly faster methods to obtain a sufficient number of samples to achieve statistical sampling with prescribed accuracy. The approach implies that whenever possible, samples and PDE solutions can happen on coarse, cheap levels of their respective hierarchies, and expensive solves are performed only when necessary. The combined use of these hierarchies therefore promises to deliver more, and more informative, samples at a cheaper cost - thereby enabling applications in the sciences and engineering that were not possible before.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.