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

AMC-SS：随机偏微分方程的计算算法和简化模型

• 批准号: 0512231
• 负责人: Roger Ghanem
• 金额: $ 25万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0512231&HistoricalAwards=false
• 关键词: AMC; SS; Computational; Algorithms; Reduced

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.

在包括纳米级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仅在十年前就无法想象的计算机和传感器做出了重大贡献。 反过来，这更接近了计算实验作为物理实验的真实替代物的承诺。 这一开发的好处非常明确：在昂贵的生产开始之前，产品可以设计为最小的细节；可以忠实地设计和认证的复杂系统，例如飞机和化学工厂，同时绕过非常昂贵的“物理测试”阶段；如果情况涉及涉及恐怖袭击或流星遭遇等假设灾难的情况，可以被抢占或至少要准备好。要实现这一“计算现实”的诺言的重要组成部分是认识到现实是可变的：每次在海岸上观察到浪潮，每次都会观察到地震的范围。土壤分别。 然后，对“计算现实”的挑战是重现这种现实世界散射的能力。拟议的研究通过使用概率的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一问题。 因此，有助于自然界观察到的可变性的不确定性成为预测模型的内在部分，使其具有更现实地再现现实的能力。 作为对这个总体问题的重大贡献，本研究在本研究中得到了两个问题。 首先，至关重要的是，赋予模型的特定不确定性形式确实与现实中观察到的相对应。 因此，在本研究的第一个组成部分中，将开发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，以便可以将它们有效地嵌入到当前由最先进的预测工具使用的计算算法中。 由PI开创的随机表示将用于该目的。 其次，必须指出的是，捕获自然界变异性的任何努力都充满了复杂性，其中最重要的是，从某种意义上说，枚举自然的所有可能状态。 本研究的第二个组成部分通过创新的计算算法来解决这种复杂性，这些算法可以有效而忠实地再现自然变异性。 这将通过在基础物理学以及假定控制感兴趣的身体行为的数学形式中利用一定的结构来完成。 将开发新的模型和新算法来解决这个问题。这项研究为复杂系统的准确行为需要大规模计算解决复杂的数学方程式的大规模计算解决方案提供了重要的贡献。