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.

在各种应用中，反问题变得重要。由于推进非侵入性诊断技术，它们在医学成像中起着重要作用。在某些应用中，例如，在颅骨外部空间中诱导的磁场的测量中研究大脑活性时，未知原因与观察到的效果之间的关系可以表示为线性函数。在其他情况下，当关系更加复杂时，线性反问题的解决方案可能必须作为更通用的解决方案方案的一部分解决。虽然原则上易于说明，但逆问题引起的线性方程系统的解决方案可能非常具有挑战性，特别是当观察数和自由度之间以及问题的尺寸很大时，观察次数和自由度之间存在不匹配时。当数据收集由于相关成本，技术困难或健康风险而存在问题时，所得线性系统中未知数的数量超过方程数。为了为此类系统产生有意义的解决方案，有必要增强有关该问题的定性知识的标准技术。该项目涉及用于解决线性不足问题的计算方法的设计和分析，这些问题自然地以定量术语转化了有关数据和解决方案的定性信息或信念。特别是，通过在贝叶斯推论的框架内提出问题，该项目将开发出数学上的声音和计算有效方案，以解决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环境。先前的信念可能会因一次性实例而异，或者从一个空间中的一个点到另一个点，并且可能有必要以层次结构层表达它们。由于这种方法非常类似于人们制定自己所知道的知识以及随着新证据的到来的更新方式的方式，因此预计该方法学将被广泛使用。在反问题中，复杂模型的普及越来越多，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断与Krylov子空间迭代方法相结合，这是解决计算效率的需求，这是大规模线性系统解决方案的自然选择。通过这种方式，贝叶斯框架的哲学吸引力在非常强大的贝叶斯 - 凯斯 - 克里洛夫计算方案中转变为广泛适用性。该项目在数值线性代数和贝叶斯推理之间提供了重要的联系，并将阐明如何将线性运算符的光谱特性与未知解决方案的统计特征联系起来。 Krylov子空间方法用于逆问题和不良问题，贝叶斯逆问题解决方案是两个非常丰富的研究领域，在过去的十年中，它们单独和共同引起了很多兴趣。有实验证据表明，他们的共生合作在各种应用中可能非常有利，但是对寻求近似解决方案的子空间的变化以及相关兰开斯过程中相关特征值的近似值的近似值仍然很大。理论和计算工具的组合将填补这一智力差距，并为在顺序的蒙特卡洛方法的背景下，为非常大的不良系统使用最新的迭代数值求解器开辟了道路。这将减少统计不确定性定量和数值线性代数之间的差距，从而对两个字段都有很大的优势。实际上，Krylov-Meets-Bayes方法的成功，在许多不同的环境中，尤其是在解决不确定的问题的解决方案中确认，依赖于左右预处理程序，以增强定量数据，并提供其他定性信息。了解Krylov子空间的变化以及在离散线性反问题中受统计启发的预处理引起的相关兰开斯过程中的变化是该项目的目的之一。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，兰开斯过程和相关的正交多项式之间的连接来启发与包括Tikhonov正则化在内的经典方案的连接和差异。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，随后将扩展到有条件的高斯先验的情况下，其协方差矩阵取决于未知参数，这些参数是通过非线性步骤估算的，因为我们了解了有关主要利益未知的主要利益的更多信息。在后一种情况下，随后的先前护发素将是一个参数化的矩阵家族。了解预处理系统的光谱特性如何随着先前协方差参数的功能而变化，将成为项目的一部分；在这里，与高斯型正交规则和时刻的连接可能至关重要。