Numerical Methods for Multiscale Inverse Problems and Applications to Sonar Imaging

首席研究人员的目标是通过使用理论上合理的预测模型和捕捉复杂物理过程内在特征的自适应数值方法，为数据驱动的研究开发数学和计算工具。这一研究项目旨在为一类在数学、商业工业和国防作战中重要的反问题的可解性提供新的保证。该项目将涉及本科生和研究生，他们将接受数值方法、分析和科学应用方面的培训。首席研究员将使用一类新的有效方法从大规模数据集中提取细节，这些方法利用在多个尺度上发生的过程的问题相关特征。数值格式的设计适应于定性的科学知识，并有可能显著提高当前反演方法的可及性和性能。该方法的基础是选择描述关键微观细节的低维参数，并开发数值方法，在求解大规模模型时保留参数值的内在知识，从而大大降低计算成本。该项目的成果将是新的方法，以及声纳成像中大规模逆问题的科学应用，其中主要挑战是捕捉适当的物理，同时保持计算时间和内存需求对当前计算机体系结构的可接受。

The principal investigator aims to develop mathematical and computational tools for data-driven research by using theoretically-sound prediction models and adaptive numerical methods that capture intrinsic features of complex physical processes. This research project is intended to provide new guarantees for the solvability of a class of inverse problems important in mathematics, commercial industries, and defense operations. The project will involve undergraduate and graduate students, who will receive training in numerical methods, analysis, and scientific applications. The principal investigator will pursue an original strategy for extracting details from large scale datasets using a new class of efficient methods that exploit problem-dependent features of processes occurring on multiple scales. The design of numerical schemes is adaptable to qualitative scientific knowledge and has the potential to significantly enhance the accessibility and performance of current inversion methods. The basis of the approach is the selection of a low-dimensional parameter that describes key microscopic details and the development of numerical methods that retain an intrinsic knowledge of parameter values while solving large-scale models, substantially reducing computational costs. Deliverables of the project will be the new methodology, as well as scientific applications to large scale inverse problems in sonar imaging, where the main challenge is to capture the appropriate physics while maintaining computational time and memory demands acceptable for current computer architectures.