Collaborative Research: Reduced Order Modeling of Realistic Noisy Flows

工程、地球物理和医学中的许多流动对计算提出了两个重大挑战。首先，可用于数值模拟的计算资源只能适应低空间和时间分辨率。因此，标准数值方法通常得出的结果极不准确。为了缓解这种情况，最先进的数值方法通常使用空间滤波来消除噪声（即数值伪影）。这些现实流动带来的第二个挑战是，它们需要大量的重复运行（例如，在汽车设计或心血管流动模拟中确定最佳参数，或在天气预报和气候建模中找到适当的初始条件）。这些重复运行会极大地增加数值模拟的计算成本。因此，通常使用仅针对主要流结构的低成本代理模型（称为降阶模型）。结合最先进的数据生成方法和降阶建模，需要一个准确和有效的数值模拟现实的流动。然而，将这两种方法结合起来的简单尝试注定要失败，因为数值不稳定、有噪声的数据和建模不一致。该项目旨在开发一个框架，将降阶建模转化为一个强大的工具，可以解决工程、地球物理和医学中现实噪声流带来的挑战。许多实际流动的数值模拟充满了困难（数值分辨率不足；数值不稳定；需要重复运行）。为了应对这些挑战，需要最先进的数值方法：大涡模拟（LES）和正则化模型解决了缺乏数值分辨率和不稳定性的问题，而基于适当正交分解（POD）的降阶模型（ROMs）在需要重复运行时平衡了计算成本和准确性。然而，将LES和正则化模型与标准rom相结合的简单尝试注定要失败，原因如下：(i)标准rom受到数值不稳定性的困扰；（ii）虽然LES和正则化模型稳定了数值模拟，但它们为rom生成的数据本质上是有噪声的；数据生成（即正则化模型和LES模型）和rom之间的建模不一致可能产生不准确的结果。该项目将开发一个建模、理论和计算框架，将降阶建模转化为一个强大的工具，可以解决现实噪声流带来的挑战。主要的创新是显式POD空间过滤器，它弥合了数据生成（即正则化模型和LES模型）与rom之间的不一致性差距。这一突破为新型正则化ROM的发展铺平了道路，并在ROM设置中引入了使用近似反卷积来恢复子滤波器尺度信息的真正LES模型。在过去的几十年里，大量的正则化和LES模型在工程和地球物理界得到了高度发展。显式POD空间过滤器表示最终允许在降阶建模中利用这些成功方法的缺失环节。

Many flows in engineering, geophysics, and medicine pose two significant challenges for computations. First, the computational resources that are available for the numerical simulations can accommodate only low spatial and temporal resolutions. Therefore, standard numerical methods usually yield extremely inaccurate results. To alleviate this, state-of-the-art numerical methods generally use spatial filtering to eliminate the noise (i.e., numerical artifacts). The second challenge posed by these realistic flows is that they require numerous repeated runs (e.g., to determine optimal parameters in automobile design or cardiovascular flow simulation, or to find appropriate initial conditions in weather forecasting and climate modeling). These repeated runs can tremendously increase the computational cost of the numerical simulations. Thus, low cost surrogate models (called reduced-order models) that target only the dominant flow structures are generally used. Combining state-of-the-art data generation methods and reduced-order modeling is required for an accurate and efficient numerical simulation of realistic flows. A simplistic attempt to combine these two approaches is, however, doomed to fail due to numerical instability, noisy data, and modeling inconsistency. This project aims to develop a framework that will transform reduced-order modeling into a robust tool that can tackle the challenges raised by realistic noisy flows in engineering, geophysics, and medicine. The numerical simulation of many realistic flows is fraught with difficulties (insufficient numerical resolution; numerical instability; need for repeated runs). To address these challenges, state-of-the-art numerical approaches are needed: large eddy simulation (LES) and regularized models tackle the lack of numerical resolution and the instability, whereas reduced-order models (ROMs) based on proper orthogonal decomposition (POD) balance the computational cost and accuracy when repeated runs are needed. A simplistic attempt to combine LES and regularized models with standard ROMs is, however, doomed to fail due to the following reasons: (i) standard ROMs are plagued by numerical instability; (ii) although LES and regularized models stabilize the numerical simulations, the data that they generate for ROMs is inherently noisy; and (iii) the modeling inconsistency between data generation (i.e., regularized and LES models) and ROMs can yield inaccurate results. This project will develop a modeling, theoretical, and computational framework that will transform reduced-order modeling into a robust tool that can tackle the challenges raised by realistic noisy flows. The main innovation is the explicit POD spatial filter, which bridges the inconsistency gap between the data generation (i.e., regularized and LES models) and ROMs. This breakthrough paves the way for the development of novel regularized ROMs and the introduction in a ROM setting of genuine LES models that use approximate deconvolution to recover subfilter-scale information. Over the last decades, a wealth of regularized and LES models have been highly developed in the engineering and geophysics communities. The explicit POD spatial filter represents the missing link that finally allows the leverage of these successful approaches in reduced-order modeling.