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.

在包括纳米尺度在内的许多尺度上操纵物质的能力方面的最新进展，为开发计算机和传感器做出了重大贡献，这些计算机和传感器的能力在十年前是无法想象的。这反过来又拉近了计算实验作为物理实验的真正替代品的前景。这种开发的好处非常明显：在昂贵的生产开始之前，产品可以被设计成最小的细节；飞机和化工厂等复杂系统可以被忠实地设计和认证，同时绕过非常昂贵的“物理测试”阶段；假设假设的情景，如恐怖袭击或流星相遇，是可以先发制人的，或者至少是做好了准备的。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：每次在岸上观测到波浪，每次测量到地震，从地球上挖出一个土样，分别观察到波浪、地震或土壤的不同和独特的特征。然后，“计算现实”面临的一个挑战是再现这种真实世界的分散的能力。拟议的研究通过使用数学概率理论对这种变化的未知根本原因进行建模来解决这个问题。因此，导致观察到的自然界变异性的不确定性成为预测模型的内在组成部分，使其具有更现实地再现现实的能力。作为对这一总体问题的重大贡献，本研究具体讨论了两个问题。首先，至关重要的是，赋予模型的特定形式的不确定性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测相一致的随机过程模型的理论和算法。这些模型将被开发成可以有效地嵌入到目前由最先进的预测工具使用的计算算法中。为达到这一目的，将使用由PI开创的随机表示法。其次，必须指出的是，任何捕捉自然界可变性的努力都充满了复杂性，其中尤其是列举在某种意义上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来解决这一复杂性，这些算法可以有效和忠实地再现自然的可变性。这将通过利用基础物理中的特定结构以及假定控制感兴趣的物理行为的数学形式来实现。为了解决这一问题，将开发新的模型和新的算法。这项研究为实现复杂系统的合理风险管理和资源分配做出了重要贡献，因为复杂系统的准确行为需要大规模计算复杂数学方程的解。