Shape-Morphing Modes for Efficient Computation of Multiscale Evolution Partial Differential Equations with Conserved Quantities

在科学和工程的许多领域都需要大规模的计算，如气候建模、天气预报和可持续结构的设计。相应的数学模型往往涉及大范围的时间和空间尺度，这是模拟需要解决的问题。有效地求解这些多尺度结构一直是科学计算中的一个长期挑战。该项目提出形状变形模式作为一种新的计算方法，将大大减少模拟多尺度系统的计算时间和内存需求。形状变形模式是一种自适应改变其形状和位置以有效捕获不同时空尺度的计算元素。由此产生的计算加速将使我们能够执行实时预测、优化和控制任务，这些任务是以前的方法无法实现的。时空系统的动力学通常用时变偏微分方程（PDEs）来描述。这些偏微分方程的解通常表现出时变的局部结构，具有尖锐的梯度，周围是大规模运动的区域。这种多尺度偏微分方程出现在许多应用中，如飞机设计、天气预报、海洋和气候建模，在这些应用中，解决小尺度结构仍然是一个主要挑战。目前，有两大类方法来解决这一挑战：1。动态演化空间离散化的自适应方法，使计算网格在局部结构周围细化，而在静态区域细化。2. 多分辨率方法，如小波，在基中编码各种尺度，而不是离散化。该项目将开发一种新的计算效率高的方法，称为形状变形模式。这种方法背后的主要思想是使用与时间相关的函数，这些函数会随着时间和空间的变化自动改变它们的形状，以便有效地解决所有的尺度。该方法不需要网格，与现有的自适应方法相比，大大降低了计算成本。此外，由于模态自适应于PDE的解，因此需要更少的模态来解决所有尺度。这大大降低了内存需求，从而优于现有的多分辨率方法。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。

Large-scale computations are needed in many areas of science and engineering, such as climate modeling, weather forecast, and design of sustainable structures. The corresponding mathematical models often involve a wide range of time and spatial scales which the simulations need to resolve. Efficiently resolving these multiscale structures has been a long-standing challenge in scientific computing. This project proposes shape-morphing modes as a new computational method that will drastically reduce the computational time and memory requirements of simulating multiscale systems. Shape-morphing modes are computational elements that adaptively change their shape and location to efficiently capture various temporal and spatial scales. The resulting computational speedup will enable us to perform real-time prediction, optimization, and control tasks that had been inaccessible to previous methods. The dynamics of spatiotemporal systems are routinely described by time-dependent partial differential equations (PDEs). The solutions of these PDEs often exhibit time-varying localized structures, with sharp gradients, surrounded by regions of large-scale motion. Such multiscale PDEs arise in numerous applications, such as aircraft design, weather prediction, ocean and climate modeling, where resolving small scale structures remains a major challenge. Currently, there are two broad classes of methods for addressing this challenge: 1. Adaptive methods which dynamically evolve the spatial discretization so that the computational grid is refined around the localized structure and less so in the quiescent regions. 2. Multiresolution methods, such as wavelets, which encode various scales in the basis instead of the discretization. This project will develop a new and computationally efficient method called shape-morphing modes. The main idea behind this method is to use a time-dependent basis of functions that automatically morph their shapes over time and space in order to efficiently resolve all scales. Being mesh-free, the proposed method substantially reduces the computational cost as compared to existing adaptive methods. Furthermore, since the modes adapt themselves to the solution of the PDE, far fewer modes are needed to resolve all scales. This significantly reduces the memory requirements, thus outperforming the existing multiresolution methods.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.