Development of Geometrically-Flexible Physics-Based Convolution Kernels

数据压缩在许多技术领域都是必不可少的，例如卫星成像、语音识别、数据库设计等等。然而，这种理解大多是基于具有良好属性的数据。 越来越多的技术，可以应用于复杂或不完整的数据，所谓的灵活数据的必要性。该项目有助于开发可应用于灵活数据的更先进的压缩技术。 该项目的主要目的是开发基于物理的计算技术，通过使数据压缩算法更加准确和高效来增强数据压缩算法。 在这个项目中开发的想法适用于更传统的计算流体动力学应用，如高超音速和大气建模以及机器学习等领域。 除了科学影响外，该项目还扩大了妇女对计算科学的参与。 它包括对学生指导，实习和保留的支持。 学生将获得的计算技能是广泛适用的，并允许他们获得各种职业选择，包括在国家需求很大的领域。PI希望在此提案中开发的工具也将包括在未来面向公众的推广会谈中。这项研究的总体目标是开发创新的、数学上严格的、几何上灵活的、基于物理的多维卷积核，这些卷积核在数据压缩、冲击滤波、后处理和机器学习领域都很有用。GEOCONKER（基于几何灵活物理学的卷积核）项目不仅致力于建立一个强大的分析框架，而且致力于这些内核的有效实现。这将允许增强包括来自传感器数据的信息的多尺度物理的准确捕获。这些技术将能够以不同的方式应用于不同类型的数据。 这些卷积核将通过在应用中建立数学、物理、数值和几何之间的相互作用来帮助建立可证明的高阶分辨率滤波器。 这些新技术将使用（灵活的）样条函数，可以适应给定数据的几何形状。这将允许高效的计算代码，这将增强多尺度物理的准确捕获和过滤。 这将包括在冲击捕获、精度增强和噪声数据过滤中有用的内核。该奖项反映了NSF的法定使命，并被认为是值得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的支持。

Data compression is essential in many areas of technology such as satellite imaging, speech recognition, database design, and much more. However, most of this understanding is based on data with good properties.  Increasingly, there is a necessity for techniques that can be applied to complex or incomplete data, so-called flexible data. This project contributes to the development of more advanced compression techniques that can be applied to flexible data.  The main purpose of the project is the development of physics-based computational techniques that enhance data compression algorithms by making them more accurate and efficient.  The ideas developed in this project are applicable to more traditional computational fluid dynamics applications such as hypersonics and atmospheric modeling as well as areas such as machine learning.  In addition to the scientific impact, this project broadens the participation of women in the computational sciences.  It includes support for student mentorship, traineeship, and retention.   The computational skills that the students will obtain are broadly applicable and allows them access to a variety of career options, including in areas of great national need. The PI expects that the tools developed in this proposal will also be included in future outreach talks to the general public.The overall goal of this research is to develop innovative, mathematically rigorous, geometrically flexible, physics-based, multi-dimensional convolution kernels that are useful in areas of data compression, shock filtering, post-processing, and machine learning. The GEOCONKER (GEOmetrically-flexible physics-based CONvolution KERnels) project will not only concentrate on establishing a robust analytical framework, but also on the efficient implementation of these kernels. This will allow for enhancing accurate capturing of multi-scale physics that includes information from sensor data. These techniques will be able to be applied to different types of data and in different manners.   These convolution kernels will aid in establishing provable high-order resolution filters by establishing the interaction between the mathematics, physics, numerics, and geometry in applications.  These novel techniques will use (flexible) spline functions that can adapt to the geometry of the given data on the fly. This will allow for efficient computational codes that will enhance the accurate capturing and filtering of multi-scale physics.  This will include kernels that are useful in shock capturing, accuracy enhancement, and filtering of noisy data. 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.