Algorithms for Inverse Problems that Exploit Kronecker Product and Tensor Structures

在许多科学和工程应用中，有必要解决一个反问题;也就是说，通过间接测量来确定定义对象或系统的量。用于在X射线计算机断层扫描（CT）和磁共振成像（MRI）等设备中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非常复杂，以至于不可能找到精确的解析解，并且必须使用基于一组更简单的方程的近似。然而，可能需要许多方程;例如，在医学成像中，通过求解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穷多个解，或者如果间接测量中存在小误差，则解可能会急剧变化，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）来稳定数值方法。正则化的类型和数量取决于问题，要求算法能够容易地适应用户和/或问题规范。该项目的目的是开发数学工具和算法，利用与构成逆问题的方程相关的矩阵中的特定数学结构（称为克罗内克积和张量）。利用这些结构将允许更好的近似，更有效的算法，并更好地促进正则化方法的结合。一个有针对性的应用是在乳腺癌检测的图像重建中出现的逆问题;在这一领域的进步将有一个明显的好处society.Efficient奇异值分解（SVD）的近似方法产生的离散不适定逆问题的大规模矩阵将被开发。该方法将利用固有的克罗内克积和张量结构，并将是一个计算平台的基础上，大规模不适定问题的有效解决方案。有效的方法来解决克罗内克产品和张量结构SVD更新问题将开发。迭代方法，可以结合正规化，稀疏和低秩约束的解决方案也将被考虑。在这个项目中开发的SVD近似和更新方法可以作为工具来获得不适定的反问题的近似解，作为预条件，以加速迭代求解器，或作为工具来建立非线性问题的解决方法。基于SVD近似的计算平台将对需要计算大规模不适定逆问题的解决方案的应用产生广泛的科学影响，包括天文学，宇宙学，天体物理学，显微镜和医学成像。

In many scientific and engineering applications, it is necessary to solve an inverse problem; that is, to determine quantities defining an object or a system through indirect measurements. The mathematics used to produce images in devices such as X-Ray Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are excellent examples. The mathematical models describing an inverse problem are often so complicated that it is not possible to find an exact analytical solution, and it is necessary to use an approximation based on a set of simpler equations. However, many equations may be needed; in medical imaging, for example, it is not unusual to compute approximations by solving millions of equations. Additional complications arise because an inverse problem may have infinitely many solutions, or the solutions may change dramatically if there are small errors in the indirect measurements. Thus, additional constraints and mathematical tools (often referred to as regularization) are needed to stabilize the numerical methods. The type and amount of regularization is problem dependent, requiring the algorithms to be able to easily adapt to user and/or problem specifications. The aim of this project is to develop mathematical tools and algorithms that exploit particular mathematical structures (called Kronecker product and tensors) in the matrices associated with equations that make up the inverse problem. Exploiting these structures will allow for better approximations, more efficient algorithms, and better facilitate the incorporation of regularization methods. One targeted application is for inverse problems that arise in image reconstruction for breast cancer detection; advancements in this area would have a clear benefit to society.Efficient singular value decomposition (SVD) approximation methods for large-scale matrices that arise in discrete ill-posed inverse problems will be developed. The approach will exploit inherent Kronecker product and tensor structures, and will be the basis for a computational platform for the efficient solution of large-scale ill-posed problems. Efficient approaches to solve Kronecker product and tensor structured SVD updating problems will be developed. Iterative methods that can incorporate regularization, sparse and low-rank constraints on the solution will also be considered. The SVD approximations and updating methods developed in this project can be used as tools to obtain approximate solutions of ill-posed inverse problems, as preconditioners to accelerate iterative solvers, or as tools to build solution methods for nonlinear problems. The computational platform, based on SVD approximations, developed will have a broad scientific impact for applications where it is necessary to compute solutions of large-scale ill-posed inverse problems, including astronomy, cosmology, geophysics, microscopy, and medical imaging.