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

科学和工程的许多领域都需要大规模的计算，例如气候建模、天气预报和可持续结构的设计。相应的数学模型往往涉及广泛的时间和空间尺度，仿真需要解决这些问题。有效地解决这些多尺度结构一直是科学计算中的一个长期挑战。该项目提出了形状变形模式作为一种新的计算方法，它将极大地减少模拟多尺度系统的计算时间和内存需求。形状变形模式是自适应地改变其形状和位置以有效捕捉各种时间和空间尺度的计算元素。由此产生的计算加速将使我们能够执行以前方法无法访问的实时预测、优化和控制任务。时空系统的动力学通常用依赖于时间的偏微分方程(PDE)来描述。这些偏微分方程组的解通常表现出时变的局部化结构，具有尖锐的梯度，周围环绕着大尺度运动的区域。这种多尺度偏微分方程组出现在许多应用中，如飞机设计、天气预报、海洋和气候建模，其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.自适应方法，它动态地进化空间离散化，使得计算网格围绕局部结构进行细化，而在静止区域则较少。2.多分辨率方法，如小波，在基中对各种尺度进行编码，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依赖于时间的函数基础，这些函数会随着时间和空间自动变形形状，以便有效地解析所有比例。与现有的自适应方法相比，该方法无网格，大大降低了计算成本。此外，由于模式自适应于偏微分方程组的解，因此需要更少的模式来解析所有尺度。这大大减少了对内存的要求，从而超过了现有的多分辨率方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.