Efficient Algorithms for Uncertainty Quantification in High Dimensions

不确定性量化（UQ）已经成为当今科学计算的一个组成部分，因为它对于理解各种不确定输入（边界和初始数据，参数值，几何形状等）的影响至关重要。数字预测。因此，UQ是至关重要的许多重要的实际问题，如气候建模，天气预报，海洋动力学，生物化学反应等，在UQ计算的最大挑战之一是模拟成本，因为UQ使传统的计算在更高维的参数空间。对于大型和复杂的系统，标准基线确定性仿真可能非常耗时，并且进行UQ仿真将进一步增加仿真成本，并且可能非常昂贵。这正是本项目要解决和研究的核心问题。将开发一套新的高效UQ算法，使UQ模拟适合大型和复杂的系统。新的算法将显着推进目前的国家的最先进的UQ方法。新算法的一个突出特点是，它们的目的是产生数学上最优的UQ仿真结果的基础上，给定的负担得起的仿真能力。虽然传统的UQ方法寻求在固定精度下提供最小的成本，但新方法将以固定的可负担成本提供最佳结果。这一新特性使得新算法非常适合于大型复杂系统的实际UQ仿真，并将对UQ至关重要的各个多学科领域产生深远的影响。新算法的核心组将基于随机排序（SC）。为了在给定和有限的模拟能力下提供最佳预测，本项目将开发一套新颖而有效的随机配置方法，重点是使用很少的样本数来处理高维问题。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，并由从业人员。 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种主要方法：（1）任意插值型SC;（2）稀疏回归型SC;和（3）多保真度SC。在所有情况下，高保真模拟的数量被假定为有限的，并由可用的模拟能力，并构造方法来产生最好的UQ模拟。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近似理论;并且也很容易实现，因为它们是非侵入式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.