Flexible Krylov Subspace Projection Methods for Inverse Problems

在许多科学和工程应用中，求解逆问题是必要的；也就是说，通过间接测量来确定定义一个物体或一个系统的量。在x射线计算机断层扫描（CT）和磁共振成像（MRI）等设备中用于生成图像的数学就是很好的例子。描述反问题的数学模型通常是如此复杂，以至于不可能找到一个精确的解析解，并且有必要使用基于一组更简单方程的近似。然而，可能需要许多方程；例如，在医学成像中，通过解决数百万个方程来计算近似值并不罕见。由于反问题可能有无限多个解，或者如果间接测量中存在小误差，解可能会发生巨大变化，因此会出现额外的复杂性。因此，需要额外的约束和数学工具（通常称为正则化）来将搜索范围缩小到一组适当的可行解，并稳定计算。正则化的类型和数量取决于问题，要求算法能够轻松地适应用户和/或问题规范。该项目的目的是开发一个灵活和适应性强的计算平台，可用于这类重要的问题，这些问题出现在许多应用领域，包括空间物体识别、地球物理学、显微镜和医学成像。因此，该项目取得的进展有可能有益于国防、能源勘探和人类健康。将发展计算大规模离散病态反问题正则解的有效迭代方法。该方法将使用灵活的Krylov子空间迭代方法，并将利用数据、解和算子的低秩结构。该方法将允许合并各种正则化技术，包括解决方案上的稀疏和低秩约束。该项目中开发的方法可以用作获得病态逆问题近似解的工具，优化实现将包含在一个名为IRTools（迭代正则化工具）的新MATLAB包中。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。

In many scientific and engineering applications, it is necessary tosolve an inverse problem; that is, to determine quantities defining anobject or a system through indirect measurements. The mathematicsused to produce images in devices such as X-Ray Computed Tomography(CT) and Magnetic Resonance Imaging (MRI) are excellent examples. Themathematical models describing an inverse problem are often socomplicated that it is not possible to find an exact analyticalsolution, and it is necessary to use an approximation based on a setof simpler equations. However, many equations may be needed; inmedical imaging, for example, it is not unusual to computeapproximations by solving millions of equations. Additionalcomplications arise because an inverse problem may have infinitelymany solutions, or the solutions may change dramatically if there aresmall errors in the indirect measurements. Thus, additionalconstraints and mathematical tools (often referred to asregularization) are needed to narrow the search to a set ofappropriate feasible solutions, and to stabilize the computations.The type and amount of regularization is problem dependent, requiringthe algorithms to be able to easily adapt to user and/or problemspecifications. The aim of this project is develop a flexible andadaptable computational platform that can be used for this importantclass of problems, which arise in many application areas, includingspace object identification, geophysics, microscopy and medicalimaging. Thus advancements made from this project have the potential tobenefit national defense, energy exploration, and human health. Efficient iterative methods to compute regularized solutions oflarge-scale discrete ill-posed inverse problems will be developed.The approach will use flexible Krylov subspace iterative methods, andwill exploit low-rank structure of data, solutions and operators. Theapproach will allow to incorporate a variety of regularizationtechniques, including sparse and low-rank constraints on the solution.The methods developed in this project can be used as tools to obtainapproximate solutions of ill-posed inverse problems, and optimizedimplementations will be included in a new MATLAB package called IRTools (Iterative Regularization Tools).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.