Sub-Linear Complexity Methods for Multiscale Problems Without Scale Separation

科学和工程中的许多问题涉及空间和时间尺度之间的复杂相互作用，由于巨大的模拟成本，这些问题在计算上具有挑战性。提出的工作旨在以显著降低模拟成本的方式提取多尺度问题的显著特征。该研究将使大规模计算问题的快速模拟方法成为可能，包括材料性能的优化设计，如电导率、弹性和电池的长寿命周期等。此外，地震学和声学科学界也将从地下和水下物理的调查和研究中受益，如目标探测、定位、材料分类等。该项目还考虑在数值天气预报方法中的应用，这些方法使用大量样本来量化天气预报模式中的不确定性，从而显著提高预报精度。该项目将资助一名研究生在项目的第二年。该项目的总体目标是应用于不可分多尺度问题的新颖亚线性复杂性方法。亚线性复杂度提供了显著提高的效率，可以在不计算解决所有活动尺度的情况下提取问题的本质和显著特征。该项目的基础是通过无缝应用分离尺度问题的标准方法提取有效行为。提出的研究提供了一种独特的方法来解决不可分尺度问题，而不需要特别的参数调整，同时保持低模拟成本。待开发的数学方法允许对双尺度问题的均匀化理论进行明智的应用。因此，该项目具有很大的潜力，可以提高为双尺度问题开发的标准计算方法对广泛问题的适用性。此外，数值天气预报的应用和验证将有助于连接确定性和随机多尺度模式框架。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many problems in science and engineering involve complicated interactions between a wide range of scales in space and time, which are computationally challenging to solve for all relevant scales due to a huge simulation cost. The proposed work aims to extract salient features of multiscale problems with a significantly reduced simulation cost. This research will enable fast simulation methods for large-scale computational problems, including optimal design of material properties such as conductivity, elasticity, and long-life cycle of batteries, etc. Also, seismology and acoustic scientific communities will benefit from the proposed work in investigating and studying underground and underwater physics such as object detection, localization, material classification, etc. The project also considers applications in numerical weather forecast methods that significantly improve the prediction accuracy using a large number of samples to quantify uncertainties in the weather forecast models. This project will fund one graduate student in year 2 of the project.The overarching goal of the project is novel sub-linear complexity methods that apply to non-separable multiscale problems. The sub-linear complexity provides a significantly improved efficiency that extracts essential and salient features of the problems without computationally resolving all active scales. The basis of the project is the extraction of effective behaviors through a seamless application of the standard method for separated scale problems. The proposed research offers a unique way to tackle non-separable scale problems without ad-hoc parameter tuning while maintaining a low simulation cost. The mathematical methods to be developed allow judicious applications of the homogenization theory for two-scale problems. Thus, the project has a significant potential to enhance the applicability of the standard computational methods developed for two-scale problems to a wide range of problems. Also, the application and validation in the context of the numerical weather forecast will contribute to connecting deterministic and stochastic multiscale modeling frameworks.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.