Sub-Linear Complexity Methods for Multiscale Problems Without Scale Separation

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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.