Iteratively Regularized Broyden-Type Algorithms for Nonlinear Inverse Problems

该项目的目标是解决科学家和工程师在寻求提高求解大规模逆问题的数值算法的准确性和效率时面临的主要计算挑战。在这种情况下，对未知量的直接测量是不可行的，需要通过使用（通常是非线性的）数学和统计模型来确定“因果关系”。由此产生的问题是众所周知的不适定（或不稳定），从某种意义上说，即使输入数据中的小测量误差也可能在恢复的解决方案中引起大量的噪声传播，直至该解决方案完全被破坏。因此，必须将称为“正则化”的特殊技术与高速优化过程相结合，以便从可用数据中获得关于未知效应的可靠信息。关键应用领域包括成像和传感技术、机器学习、重力探测、海洋声学和数据科学。本项目旨在发展迭代正则化broyden型数值算法，用于解决有限维或无限维空间中的非线性病态反问题。一组新的正则化方法将被设计用于解决大规模的不稳定最小二乘问题，其中离散非线性算子的雅可比矩阵很难甚至不可能计算。为了克服这一障碍，pi考虑了一类高斯-牛顿和Levenberg-Marquardt算法，其中Frechet导数算子通过使用broyden型单秩更新递归地重新计算。为了平衡精度和稳定性，在迭代过程的每一步都以特定问题的方式正则化无导数雅可比矩阵的伪逆。将研究各种各样的过滤器，以便更灵活地利用针对每一具体应用问题的定性和定量先验资料。本文提出的迭代正则化方法将在确定性和随机环境下进行研究。对于随机过程，最小化函数的评估受到随机误差的影响，由于不精确的计算，以降低每次迭代成本，和/或不可避免的环境噪声和波动。在提出的研究框架内，pi将对新算法进行全面的收敛性分析，包括收敛率和正则化参数和步长选择的最优策略。除了理论研究之外，本项目的一个重要组成部分是通过对现实世界非线性逆问题的大量数值实验来评估所提出的算法。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。

The goal of this project is to tackle major computational challenges faced by scientists and engineers in their quest to improve the accuracy and efficiency of numerical algorithms for solving large-scale inverse problems. This is a scenario where direct measurements of the unknown quantities are not feasible, and one needs to identify "cause from effect" by using (generally nonlinear) mathematical and statistical models. The resulting problems are notoriously ill-posed (or unstable), in a sense that even small measurement errors in the input data may give rise to a substantial noise propagation in the recovered solution, to the extent that this solution gets entirely destroyed. For this reason, special techniques called "regularization" must be combined with high-speed optimization procedures, so that reliable information on the unknown effect could be obtained from the available data. The key areas of application include imaging and sensing technology, machine learning, gravitational sounding, ocean acoustics, and data sciences.This project aims at the development of iteratively regularized Broyden-type numerical algorithms for solving nonlinear ill-posed inverse problems in either finite or infinite dimensional spaces. A family of new regularization methods will be designed to solve large-scale unstable least squares problems, where the Jacobian of a discretized nonlinear operator is difficult or even impossible to compute. To overcome this obstacle, PIs consider a family of Gauss-Newton and Levenberg-Marquardt algorithms with the Frechet derivative operator recalculated recursively by using Broyden-type single rank updates. To balance accuracy and stability, the pseudo-inverse for the derivative-free Jacobian is regularized in a problem-specific manner at every step of the iteration process. A variety of filters will be investigated, yielding greater flexibility in the use of qualitative and quantitative a priori information available for each particular applied problem. The proposed iteratively regularized methods will be studied in both deterministic and stochastic settings. For stochastic processes, the minimization functionals are evaluated subject to stochastic errors due to inexact computations to lower per-iteration cost, and/or unavoidable environmental noise and fluctuations. In the framework of the proposed research, PIs will conduct comprehensive convergence analysis of the new algorithms, including convergence rates and optimal policies for the selection of regularization parameters and step sizes. In addition to the theoretical investigation, a significant component of this project is to evaluate the proposed algorithms using extensive numerical experiments on real-world nonlinear inverse problems.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.