Uncertainty Quantification in PDE-Based Modelling and Optimization

在许多当代工程挑战中，发生在不同长度尺度上的复杂多物理场级联决定了宏观现象。本研究计划将集中于两个问题，其中简单的宏观现象是在较低尺度上复杂相互作用的结果。具体来说，我们将问1)超导体内部的交流损耗是如何由表面附近的涡旋柱的微米级相互作用引起的，以及2)当小振动导致孔隙压力增加时，饱和土壤是如何突然失去阻力的。在这两种情况下，我们将从数学角度研究微米过程，以测量这些固有不稳定的过程如何导致定义良好且稳定的宏观量。我们的重点将放在系统的稳定性特性上，以及低尺度的不确定性如何相互抵消或相互合作，从而对宏观量做出贡献。

In many of the contemporary engineering challenges, a cascade of complex multiphysics occurring at different length scales determines macroscopic phenomena. This research program will focus on two problems where simple macroscopic phenomena are the result of complex interactions at the lower scales. Specifically we will ask 1) how AC losses inside superconductors are caused by the micrometer interaction of vortex columns near the surface, and 2) how saturated soils may suddenly lose their resistance when small vibrations lead to an increase in pore pressure. In both cases, we will be examining the micrometer processes for a mathematical perspective in order to measure how those inherently unstable processes lead to well-defined and stable macroscale quantities. Our focus will be on the stability properties of the systems and how uncertainties at the lower scales either negate each other, or cooperate with each other, to contribute to macroscale quantities. High-temperature superconductors are a new generation of superconductors that can be cooled with liquid nitrogen, rather than expensive and difficult to manage liquid helium. Their production costs have been decreasing dramatically, but their promise of revolutionizing applications such as MRI machines, MagLev trains, or even high-power devices such as transformers and fault-current limiters, has yet to come to fruition because the numerical models are too computationally costly to enable iterative and adaptive design. It is important to estimate the AC losses generated during AC loading because these will often be the main design parameter that must be minimized. Unfortunately, AC losses are related to subtle interactions between vortex columns, the external electric field, and heterogeneities within the superconductor. Although we walk and build on compacted soils formed of millimeter grains, overall they act as solids and can sustain incredible amounts of stress. Yet, when certain soils become water-saturated, small vibrations may lead to increases in the pressure of the fluid between the pores, hence pushing the grains apart. This phenomena is called liquefaction and has been at the origin of many catastrophic events in the St-Lawrence Valley, in part because the region contains large deposits of Leda clay which are susceptible to such phenomena. We intend to develop a novel stochastic model for assemblies of grains that will help to isolate the properties of grain geometries that are responsible for the macroscale properties of soils. This new model will also introduce new methods to measure the stability of these soils to perturbations, and in particular to the uncertainties inherent in such heterogeneous assemblies.