Algorithms for Inverse Problems that Exploit Kronecker Product and Tensor Structures

在许多科学和工程应用中，需要求解逆问题；也就是说，通过间接测量来确定定义一个物体或系统的量。在x射线计算机断层扫描（CT）和磁共振成像（MRI）等设备中用于产生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，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，因此会产生额外的复杂性。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，要求算法能够轻松地适应用户和/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。一个目标应用是在乳腺癌检测的图像重建中出现的逆问题；这一领域的进步将对社会有明显的好处。对于离散病态反问题中出现的大规模矩阵，将发展有效的奇异值分解（SVD）逼近方法。该方法将利用固有的克罗内克积和张量结构，并将成为有效解决大规模病态问题的计算平台的基础。将开发解决Kronecker积和张量结构化SVD更新问题的有效方法。我们还将考虑将正则化、稀疏和低秩约束结合在解上的迭代方法。本课题开发的SVD逼近和更新方法可以作为获得病态逆问题近似解的工具，可以作为加速迭代求解的前提条件，也可以作为构建非线性问题解方法的工具。基于SVD近似的计算平台的开发将对天文学、宇宙学、地球物理学、显微镜学和医学成像等需要计算大规模病态逆问题解的应用产生广泛的科学影响。

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.