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

今天，我们知道如何在计算机上模拟许多过程，例如机翼周围的空气流动或物体在施加力时如何变形。然而，从实际的角度来看，决定“量化预测的不确定性”通常是可取的，也就是说，当输入不完全已知时，能够说出计算机模拟的准确性或不准确性。一个例子是机翼如何表现时，前方的空气是紊流。在这些情况下，需要模拟许多场景，这是一个非常昂贵的提议。该项目旨在开发数学工具，通过利用采样和解决方法中的多层次层次结构，使这一过程更加可行和计算负担得起。科学和工程中的许多过程都是用参数相关偏微分方程（PDEs）来描述的，其数值解是昂贵的。虽然它们的正解可能是可行的，但计算成本使得统计评估问题变得困难或不可能，例如(i)在给定已知或假设的参数值分布的情况下量化输出变量的不确定性，或（ii）解决从噪声测量中推断参数之间概率分布的统计逆问题。这两个问题通常通过高维采样方法来回答，因此非常昂贵。该项目将使用PDE求解器的分层和多层分解，加上误差估计，以获得更快的方法来获得足够数量的样本，以达到规定精度的统计抽样。这种方法意味着，只要可能，样本和PDE解决方案可以在它们各自层次结构的粗糙、廉价的层次上发生，而昂贵的解决方案只在必要时执行。因此，这些层次结构的结合使用有望以更低的成本提供更多、更有信息量的样本——从而使科学和工程领域的应用成为可能，这在以前是不可能的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.