Priorconditioned Krylov Subspace Methods for Inverse Problems

反问题在各种应用中变得越来越重要。由于非侵入性诊断技术的发展，它们在医学成像中发挥着重要作用。在某些应用中，例如，在通过测量头骨外部空间中的感应磁场来研究大脑活动时，未知原因与观察到的影响之间的关系可以表示为线性函数。在其他情况下，当关系更复杂时，线性反问题的解决方案可能必须作为更通用的解决方案的一部分来解决。虽然原则上很容易表述，但求解由反问题引起的线性方程组可能极具挑战性，特别是当观测数量和自由度之间存在不匹配以及问题的维度非常大时。当由于相关成本、技术困难或健康风险而导致数据收集出现问题时，所得线性系统中的未知数数量会超过方程的数量。为了为此类系统提供有意义的解决方案，有必要通过有关问题的定性知识来增强标准技术。该项目涉及解决线性不适定问题的计算方法的设计和分析，这些方法自然地将有关数据和解决方案的定性信息或信念转化为定量术语。特别是，通过在贝叶斯推理框架内制定问题，该项目将为大规模问题开发数学上合理且计算高效的方案，其中数据中的干扰可能相当大，并且可能具有与白噪声相当不同的统计数据。贝叶斯框架是表达有关解决方案的先验信念的自然设置。先验信念可能从一个时间实例到另一个时间实例，或者从空间中的一个点到另一个点，变化很大，并且可能需要在分层中表达它们。由于这种方法非常类似于人们表述他们所知道的内容以及知识如何随着新证据的出现而更新的方式，因此预计该方法将被广泛使用。复杂模型在反问题中的日益普及伴随着相关计算成本的增加。作为该项目的一部分开发的方法通过将贝叶斯推理与 Krylov 子空间迭代方法相结合来满足计算效率的需求，这是解决大规模线性系统的自然选择。通过这种方式，贝叶斯框架的哲学吸引力转化为非常强大的具有广泛适用性的贝叶斯遇见克雷洛夫计算方案。该项目提供了数值线性代数和贝叶斯推理之间的重要联系，并将阐明如何将线性算子的谱特性与未知解的统计特征联系起来。用于逆问题和不适定问题的克雷洛夫子空间方法以及逆问题的贝叶斯解决方案是两个非常丰富的研究领域，在过去十年中无论是单独还是共同引起了广泛的兴趣。有实验证据表明，它们的共生合作在各种应用中都非常有利，但对寻求近似解的子空间的变化以及相关 Lanczos 过程中相关特征值的近似仍然缺乏充分的理解。理论和计算工具的结合将填补这一知识空白，并为在顺序蒙特卡罗方法的背景下使用最先进的迭代数值求解器解决非常大的不适定系统开辟了道路。这将缩小统计不确定性量化和数值线性代数之间的差距，为这两个领域带来巨大优势。事实上，克雷洛夫遇上贝叶斯方法的成功在许多不同的环境中得到了证实，特别是在解决不确定问题时，依赖于左和右预处理器用额外的定性信息来增强定量数据。该项目的目标之一是了解离散线性逆问题中由统计启发的预处理器引起的 Krylov 子空间和相关 Lanczos 过程的变化。特别是，数值线性代数的强大工具以及 Krylov 子空间迭代求解器、Lanczos 过程和相关正交多项式之间的联系将被用来阐明与经典方案（包括 Tikhonov 正则化）的联系和差异。在该项目的第一部分中，分析将首先在高斯先验和噪声的情况下进行，然后将扩展到条件高斯先验的情况，其协方差矩阵取决于未知参数，随着我们对主要感兴趣的未知数的了解更多，这些参数通过非线性步骤进行估计。在后一种情况下，随后的先验条件将是参数化矩阵族。了解预处理系统的光谱特性如何随着先验协方差参数的函数而变化将是该项目的一部分；在这里，与高斯型求积规则和矩的联系可能至关重要。

Inverse problems are gaining importance in a wide variety of applications; they play an important role in medical imaging because of the push towards non-invasive diagnostic techniques. In some applications, e.g., in the investigation of the brain activity from the measurement of the induced magnetic field in the space outside the skull, the relation between the unknown causes and the observed effects can be expressed as a linear function. In other cases, when the relationship is more complicated, the solution of linear inverse problems may have to be addressed as part of a more general solution scheme. While in principle easy to state, the solution of a linear system of equations arising from inverse problems can be extremely challenging, in particular when there is a mismatch between the number of observations and the degrees of freedom and when the dimensions of the problems are very large. When data collection is problematic because of the associated costs, technical difficulties, or health risks, the number of unknowns in the resulting linear system exceeds the number of equations. In order to produce a meaningful solution for such systems it is necessary to augment standard techniques with qualitative knowledge about the problem. This project concerns the design and analysis of computational methods for the solution of linear ill-posed problems that naturally translate qualitative information or belief about the data and the solution in quantitative terms. In particular, by formulating the problem within the framework of Bayesian inference, the project will develop mathematically sound and computationally efficient schemes for large scale problems where the disturbance in the data may be rather substantial and may have a statistics rather different from white noise. The Bayesian framework is the natural setting for expressing the a priori beliefs about the solution. The prior beliefs may vary widely from one time instance to another, or from one point in space to another, and it may be necessary to express them in hierarchical layers. Since this approach very closely resembles the way in which people formulate what they know and how knowledge is updated as new evidence arrives, it is expected that the methodology will be widely utilized. The increasing popularity of complex models in inverse problems comes with an increase in associated computational costs. The methodology developed as part of this project addresses the need for computational efficiency by combining Bayesian inference with the Krylov subspace iterative methods, the natural choice for the solution of large scale linear systems. In this manner the philosophical appeal of the Bayesian framework is transformed in a very powerful Bayes-meets-Krylov computational scheme of wide applicability. The project provides an important connection between numerical linear algebra and Bayesian inference and will shed some light on how to link spectral properties of linear operators with statistical features of the unknown solution. Krylov subspace methods for inverse and ill-posed problems and the Bayesian solution of inverse problems are two very rich research areas which have received much interest, individually and jointly, in the last decade. There is experimental evidence that their symbiotic cooperation can be very advantageous in a variety of applications, but a solid understanding of the changes in the subspaces where the approximate solutions are sought and in approximation of the relevant eigenvalues in the associated Lanczos processes is still largely missing. The combination of theoretical and computational tools will fill this intellectual gap and open the way for the use of state-of-the-art iterative numerical solvers for very large ill-posed systems in the context of sequential Monte Carlo methods. This will reduce the gap between statistical uncertainty quantification and numerical linear algebra, to great advantage for both fields. In fact, the success of the Krylov-meets-Bayes approach, confirmed in a number of different settings and particularly in the solution of underdetermined problems, relies on left and right preconditioners to augment the quantitative data with additional qualitative information. Understanding the changes in the Krylov subspaces and in the associated Lanczos process induced by the statistically inspired preconditioners in discrete linear inverse problems is one of the aims of this project. In particular, the powerful tools of numerical linear algebra and the connection between Krylov subspace iterative solvers, the Lanczos process, and the associated orthogonal polynomials will be utilized to enlighten the connections and differences with classical schemes, including Tikhonov regularization. In the first part of the project, the analysis will be first carried out in the case of Gaussian prior and noise, and will be subsequently extended to the case of conditionally Gaussian prior, whose covariance matrix depends on unknown parameters, which are estimated via a nonlinear step as we learn more about the unknown of primary interest. In the latter case, the ensuing prior conditioners will be a parametrized family of matrices. Understanding how the spectral properties of the preconditioned systems change as functions of the parameters of the prior covariance will be part of the project; here, the connections with Gauss-type quadrature rules and moments may turn out be crucial.