On Low-Rank Regularization for Ill-Posed Nonlinear Parameter Estimation

这项研究受到了研究传染病传播动力学的众多挑战的启发。研究者将专注于估计传染病数据数学模型的参数，旨在产生未来发病率的预测。该项目将为该数据估计开发正则化计算算法，该数据估计提出了许多挑战，包括不确定性。所使用的模型和优化方法对于预测疾病管理所需的资源非常重要。研究人员将研究过去和现在的传染病数据，并与该大学的公共卫生学院合作。除了在流行病学方面的应用外，该项目还将对信号和图像处理、生物医学成像、重力探测、混沌理论、海洋声学等科学学科产生广泛影响。该项目包括通过参与研究来培养研究生。该项目旨在开发使用优化估计参数的计算算法。从优化的角度来看，数据的参数估计和预测归结为解决由常微分方程或偏微分方程系统约束的不适定最小化问题。对于不确定度的量化，必须多次运行反演算法，最好是实时运行。为了应对这一挑战，该项目将构建一系列具有低秩更新雅可比算子的信任区域优化算法，这将减少准牛顿步骤的计算成本，同时在迭代过程中加入额外的稳定性层。对于残差非零的非线性最小二乘，我们将研究稳定Hessian评价的低秩更新。新方法的理论和数值分析将首先对通常可解的病态算子方程进行，然后将其推广到本质病态问题。该项目的成功完成将促进对病态逆问题的理解，并促进更稳定和有效的模拟。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。

This research has been inspired by numerous challenges in studying the transmission dynamics of infectious diseases. The investigator will focus on estimating parameters of data-enabled mathematical models of infectious disease aiming to generate forecasts of future incidence cases. The project will develop regularized computational algorithms for this data estimation which presents many challenges and includes uncertainties. The models and optimization methods to be used are important in anticipating the resources needed for disease management. Data on past and present infectious diseases will be studied, and the investigator will collaborate with the university's School of Public Health. Apart from applications in epidemiology, this project will have a broad impact on scientific disciplines including signal and image processing, biomedical imaging, gravitational sounding, chaos theory, ocean acoustics, and others. The project includes graduate student training through involvement in the research.The project aims to develop computational algorithms for estimating parameters using optimization. From an optimization standpoint, parameter estimation and forecasting from data comes down to solving an ill-posed minimization problem constrained by a system of ordinary or partial differential equations. For uncertainty quantification, multiple runs of the inversion algorithm must be carried out, preferably in real time. To address this challenge, the project will construct a family of trust-region optimization algorithms with low-rank updates for the Jacobian operator that will reduce the computational cost of a quasi-Newton step and, at the same time, incorporate an extra layer of stability in the iterative process. In case of nonlinear least squares with non-zero residuals, low-rank updates for stable Hessian evaluation will be investigated. Theoretical and numerical analysis of the new methods will be first carried out for normally solvable ill-posed operator equations and then extended to essentially ill-posed problems. The successful completion of this project will advance the understanding of ill-posed inverse problems and facilitate more stable and efficient simulations.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.