AMC-SS: Computational Algorithms and Reduced Models for Stochastic PDEs

在包括纳米尺度在内的许多尺度上操纵物质的能力的最新进展，对开发计算机和传感器的能力做出了重大贡献，这些能力在十年前是无法想象的。这反过来又使计算实验成为物理实验的真正替代品的希望更加接近。这种发展的好处非常明显：产品可以在昂贵的生产开始之前设计到最小的细节；复杂的系统，如飞机和化工厂，可以忠实地设计和认证，而绕过非常昂贵的“物理测试”阶段；如果涉及假想灾难（如恐怖袭击或流星遭遇）的场景可以先发制人，或者至少可以做好准备。实现这一“计算现实”承诺的一个重要组成部分是认识到现实是可变的：每次在海岸上观察波浪，每次测量地震，从地下挖出土壤样本，分别观察到波浪，地震或土壤的不同和独特的特征。然后，对“计算现实”的挑战是重现这种真实世界的分散的能力。提出的研究通过使用概率数学理论对这种可变性的未知根本原因进行建模来解决这个问题。因此，导致自然界中观测到的变率的不确定性成为预测模式的固有部分，使其具有更真实地再现现实的能力。作为对这一总体问题的重大贡献，本研究特别解决了两个问题。首先，至关重要的是，赋予模型的不确定性的特定形式确实与现实中观察到的不确定性相对应。因此，在本研究的第一个组成部分中，将开发用于构建与实验观察一致的随机过程模型的理论和算法。这些模型将被开发，以便它们可以有效地嵌入到当前使用的最先进的预测工具的计算算法中。由PI首创的随机表示将用于此目的。其次，必须指出的是，捕捉自然界的可变性的任何努力都充满了复杂性，其中最重要的是在某种意义上列举所有可能的自然状态的负担。本研究的第二个组成部分通过创新的计算算法解决了这种复杂性，该算法可以有效和忠实地再现自然变异性。这将通过利用基础物理和数学形式中的某种结构来实现，这些结构被假定为控制感兴趣的物理行为。新模型和新算法都将被开发来解决这个问题。该研究为实现复杂系统的合理风险管理和资源分配提供了重要贡献，这些系统的精确行为需要复杂数学方程的大规模计算解。

Recent advances in the ability to manipulate matter across many scales, including the nanoscale, have contributed significantly towards developing computers and sensors whose capabilities could not be imagined only a decade ago. This has, in turn, brought closer the promise of computational experiments as true surrogates for physical experiments. The benefits of this development are very clear: products can be designed to their smallest detail before expensive production begins; complex systems such as airplanes and chemical plants can be faithfully designed and certified while bypassing the very expensive "physical testing" phase; what-if scenarios involving hypothetical disasters such as terrorist attacks or meteor encounters can be preempted or at least be readied for.An essential component towards fulfilling this promise of "computational reality" is the realization that reality is variable: every time a wave is observed on the shore, every time an earthquake is measured, a soil specimen dug out from the earth, different and unique features are observed of the wave, the quake, or the soil, respectively. Then, a challenge to "computational reality" is the ability to reproduce this real-world scatter.The proposed research addresses this very issue by modeling the unknown root cause of this variability using the mathematical theory of probabilities. Thus, uncertainties that contribute to the observed variability in nature become an intrinsic part of the predictive model, endowing it with the ability to more realistically reproduce reality. As significant contributions to this overarching problem, two issues are specifically addressed in the present research. First, it is vital that the particular form of uncertainty with which the model is endowed does indeed correspond to that observed in reality. Thus, in the first component of the present research, theory and algorithms for constructing models of stochastic processes that are consistent with experimental observations will be developed. These models will be developed such that they can be efficiently embedded into computational algorithms currently in use by state-of-the-art predictive tools. Stochastic representations pioneered by the PI will be used to that end. Secondly, it must be noted that any effort at capturing the variability in nature is fraught with complexity, not the least of which is the burden of enumerating, in some sense, all possible states of nature. The second component of the present research addresses this complexity through innovative computational algorithms that can efficiently and faithfully reproduce natural variability. This will be done by capitalizing on a certain structure both in the underlying physics as well as in the mathematical form assumed to govern the physical behavior of interest. Both new models and new algorithms will be developed to tackle this problem.This research provides a significant contribution towards enabling rational risk management and resource allocation for complex systems whose accurate behavior requires the large-scale computational solution of complex mathematical equations.