Priorconditioned Krylov Subspace Methods for Inverse Problems

反问题的先验 Krylov 子空间方法

• 批准号: 1522334
• 负责人: Daniela Calvetti
• 金额: $ 22万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522334&HistoricalAwards=false
• 关键词: Priorconditioned; Krylov; Subspace; Methods; Inverse

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.

逆问题在各种各样的应用中越来越重要;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，在通过测量颅骨外空间中的感应磁场来研究脑活动时，未知原因和观察到的效应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，线性逆问题的解决方案可能必须作为更一般的解决方案的一部分来解决。虽然在原则上很容易陈述，但由逆问题产生的线性方程组的求解可能极具挑战性，特别是当观测数量和自由度之间不匹配时，以及当问题的维度非常大时。当数据收集由于相关成本、技术困难或健康风险而出现问题时，所得线性系统中的未知数数量超过方程的数量。为了产生一个有意义的解决方案，这样的系统，有必要增加标准技术与定性知识的问题。该项目涉及设计和分析用于解决线性不适定问题的计算方法，这些问题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，通过制定贝叶斯推理的框架内的问题，该项目将开发数学健全和计算效率高的计划，大规模的问题，在数据中的干扰可能是相当大的，可能有一个统计，而不同于白色噪音。贝叶斯框架是表达关于解决方案的先验信念的自然设置。先验信念可能会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，并且可能有必要在分层中表达它们。由于这种方法非常类似于人们表达他们所知道的东西以及随着新证据的到来如何更新知识的方式，因此预计这种方法将得到广泛使用。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解决了计算效率的需要，通过结合贝叶斯推理与Krylov子空间迭代方法，自然选择的解决方案的大规模线性系统。以这种方式，贝叶斯框架的哲学吸引力被转化为一个非常强大的贝叶斯满足克雷洛夫计算方案的广泛适用性。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。Krylov子空间方法的反问题和不适定问题和贝叶斯反问题的解决方案是两个非常丰富的研究领域，已经收到了很大的兴趣，单独和联合，在过去的十年。有实验证据表明，它们的共生合作可以是非常有利的，在各种应用中，但一个坚实的理解，在子空间的变化，其中的近似解决方案，并在相关的Lanczos过程中的相关特征值的近似仍然在很大程度上失踪。理论和计算工具的结合将填补这一智力空白，并开辟了使用国家的最先进的迭代数值求解器非常大的不适定系统的顺序蒙特卡罗方法的背景下。这将减少统计不确定性量化和数值线性代数之间的差距，对这两个领域都有很大的好处。事实上，Krylov-meets-Bayes方法的成功，在许多不同的设置，特别是在欠定问题的解决方案中得到了证实，依赖于左和右预条件子，以增加定量数据与额外的定性信息。了解Krylov子空间的变化，并在相关的Lanczos过程中引起的统计启发的预处理离散线性逆问题是这个项目的目标之一。特别是，数值线性代数的强大工具和Krylov子空间迭代求解器，Lanczos过程和相关的正交多项式之间的连接将被用来启发与经典计划，包括吉洪诺夫正则化的连接和差异。在项目的第一部分中，分析将首先在高斯先验和噪声的情况下进行，随后将扩展到条件高斯先验的情况下，其协方差矩阵取决于未知参数，这些参数通过非线性步骤进行估计，因为我们了解更多关于主要感兴趣的未知数。在后一种情况下，随后的先验条件将是一个参数化的矩阵族。了解预处理系统的谱特性如何作为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是至关重要的。