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

这项研究受到了在研究传染病传播动力学方面的众多挑战的启发。研究人员将专注于估计数据启用的传染病数学模型的参数，旨在产生对未来发病病例的预测。该项目将为这种数据估计开发正则化的计算算法，这带来了许多挑战，并包括不确定性。要使用的模型和优化方法在预测疾病管理所需的资源方面很重要。研究人员将研究过去和现在的传染病数据，调查人员将与该大学的公共卫生学院合作。除了在流行病学中的应用，这个项目还将对科学学科产生广泛的影响，包括信号和图像处理、生物医学成像、重力测深、混沌理论、海洋声学等。该项目包括通过参与研究对研究生进行培训。该项目旨在开发使用最优化估计参数的计算算法。从最优化的观点来看，从数据中估计和预测参数归结为求解一个由常微分方程组或偏微分方程组约束的不适定极小化问题。对于不确定性量化，必须执行多次反转算法，最好是实时的。为了应对这一挑战，该项目将构建一系列信赖域优化算法，对雅可比算子进行低阶更新，从而减少拟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。在具有非零残差的非线性最小二乘的情况下，将研究用于稳定Hessian估计的低阶更新。新方法的理论和数值分析将首先对正常可解的不适定算子方程进行，然后扩展到本质不适定问题。这个项目的成功完成将促进对不适定逆问题的理解，并促进更稳定和有效的模拟。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.