Efficient Algorithms for Uncertainty Quantification in High Dimensions

不确定性量化(UQ)已成为当今科学计算中不可或缺的一部分，因为它对于理解各种不确定输入(边界和初始数据、参数值、几何等)的影响至关重要。到数字预测。因此，UQ对于气候建模、天气预报、海洋动力学、生物化学反应等许多重要的实际问题都是至关重要的。UQ计算的最大挑战之一是模拟成本，因为UQ是在高维参数空间进行传统计算的。对于大型和复杂的系统，标准的基线确定性仿真可能非常耗时，而进行UQ仿真将进一步增加仿真成本，并且可能昂贵得令人望而却步。这正是本项目打算解决和研究的核心问题。将开发一套新的高效UQ算法，使UQ模拟适用于大型和复杂系统。新的算法将极大地推进当前UQ方法的最先进水平。新算法的一个突出特点是，它们被设计为基于给定的负担得起的模拟能力来产生数学上最优的UQ模拟结果。传统的UQ方法寻求在固定精度下提供最小的成本，而新方法将在固定的负担得起的成本下提供最佳结果。这一新特征使得新算法非常适合于大型复杂系统的实际UQ仿真，并将在UQ至关重要的各个多学科领域产生深远的影响。新算法的核心将基于随机排序(SC)。为了在给定和有限的模拟能力下提供最优预测，本项目旨在开发一套新颖而有效的随机配置方法，专注于使用极少的样本来解决高维问题。一个重要的和基本的假设是：样本运行的数量和样本的位置是任意的，由从业者给出。然后，目标是基于给定的样本，在潜在的非常高维的参数空间中寻求最佳近似。所提出的研究包括三个主要方法：(1)任意内插型SC；(2)稀疏回归型SC；(3)多保真SC。在所有情况下，假设高保真模拟的数量是有限的，并由可用的模拟能力给出，并且构建了产生最佳UQ模拟的方法。所有的方法在数学上都是严格的，因为它们的构造在很大程度上依赖于高维的近似理论；而且也很容易实现，因为它们是非侵入性的SC方法。

Uncertainty quantification (UQ) has become an integral part of today's scientific computing, as it is essential to the understanding of the impacts of various uncertain inputs (boundary and initial data, parameter values, geometry, etc.) to numerical predictions. UQ is thus critical to many important practical problems such as climate modeling, weather prediction, ocean dynamics, bio-chemical reactions, etc. One of the biggest challenges in UQ computations is the simulation cost, as UQ makes the traditional computations in much higher dimensional parameter spaces. For large and complex systems, the standard baseline deterministic simulations can be very time consuming, and conducing UQ simulations will further increase the simulation cost and can be prohibitively expensive. This is precisely the core issue this project intends to address and study. A novel set of highly efficient UQ algorithms will be developed to make UQ simulations amenable for large and complex systems. The new algorithms will significantly advance the current state-of-the-art of UQ methods. One prominent feature of the new algorithms is that they are designed to produce mathematically optimal UQ simulation results based on given affordable simulation capacity. While the traditional UQ methods seek to provide the smallest cost at fixed accuracy, the new methods will provide the best results at fixed affordable cost. This new feature thus makes the new algorithms ideally suited for practical UQ simulations of large and complex systems, and will have a profound impacts in various multidisciplinary fields where UQ is critical.The core group of the new algorithms will be based on stochastic collation (SC). In order to provide optimal prediction under given and limited simulation capacity, this project is to develop a set of novel and efficient stochastic collocation methods, with a focus on high dimensional problems using very few number of samples. An important and fundamental assumption is made: the number of the sample runs and the location of the samples are arbitrary and given by practitioners. The goal is then to seek the best approximation in the potentially very high dimensional parameter space, based on the given samples. The proposed research consists of three major approaches: (1) arbitrary interpolation type SC; (2) sparse regression type SC; and (3) multi-fidelity SC. In all cases, the number of high-fidelity simulations is assumed to be limited and given by the available simulation capacity, and methods are constructed to produce the best possible UQ simulations. All methods will be mathematically rigorous, as their constructions rely heavily on approximation theories in high dimensions; and also easy to implement, as they are the non-intrusive SC methods.