Collaborative Research: Innovative Integrated Strategies for Nonlinear Parametric Inversion

研究人员的目标是通过将新的参数水平集方法和非线性最小二乘方法与参数系统的线性求解器、预处理和模型简化方面的创新相结合，大幅降低数值反演的成本(例如，在医学成像中)。将四种策略结合在一起，在不降低解的准确性的情况下减少了必要的计算量。首先，通过发展低阶参数反演方法取代通常的基于体素的反演方法，大大降低了反问题的维度。其次，参数反演背后的优化结合了专门为处理病态雅可比而设计的新型非线性最小二乘求解器。第三，通过特别适合于所考虑的反问题的结构的新的模型简化技术，降低了解决许多大型正问题的高成本。第四，通过Krylov子空间循环的创新和对参数化线性系统预条件的有效重用，降低了计算简化模型和求解正问题的高成本。这里研究的反问题涉及恢复描述未知量的诊断兴趣(如电导率)如何分布在给定介质(如人体组织或土壤)中的图像。这些图像可以揭示异常的存在或不存在，例如人体组织中的肿瘤或土壤中的污染羽毛。在合理的时间内从含噪声的表面测量中提取高质量的图像是一项非常困难的任务。随着技术的快速进步，进行更多测量成为可能，计算瓶颈变得越来越严重，阻碍了医学和其他成像领域的创新。该项目旨在结合计算线性代数、系统论和最优化领域中不同领域的创新，创建显著改进的图像提取策略。

The investigators aim to reduce drastically the costs of numerical inversion (as occurs, for example, in medical imaging) by blending new parametric level-set approaches and nonlinear least squares methods together with innovations in linear solvers, preconditioning, and model reduction of parameterized systems. Four strategies are combined to reduce the computation necessary while not degrading accuracy of the solution. First, the dimension of the inverse problem is drastically reduced by developing low-order parametric inversion methods replacing the usual voxel-based inversion. Second, the optimization underlying parametric inversion incorporates novel nonlinear least-squares solvers specifically designed to deal with ill-conditioned Jacobians. Third, the high cost of solving many large forward problems is reduced through new model reduction techniques that are particularly well-suited to the structure of the inverse problems under consideration. Fourth, the high costs of computing reduced models and solving forward problems is reduced by innovations in Krylov subspace recycling and efficient reuse of preconditioners for parameterized linear systems. The inverse problems studied here involve recovery of images describing how unknown quantities of diagnostic interest (such as electrical conductivity) are distributed throughout a given medium (such as human tissue or soil). These images can reveal the presence or absence of anomalies, such as tumors in human tissue or contaminant plumes in soil. The computational extraction of high quality images from noisy surface measurements in reasonable time is a very difficult task. As rapid advances in technology make it possible to take vastly more measurements, computational bottlenecks become ever more acute, impeding innovation in medical and other areas of imaging. This project aims to combine innovations in diverse fields within computational linear algebra, systems theory, and optimization to create dramatically improved strategies for image extraction.