具有Whittle-Matérn先验的贝叶斯逆问题的有效算法
Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors
摘要：
本文研究了基于Whittle-Matérn高斯随机场的先验贝叶斯反问题的有效方法。Whittle-Matérn先验的特征在于一个平均函数和协方差算子，该算子被视为椭圆微分算子的负幂。这种方法是灵活的，因为它可以包含广泛的先验信息，包括非平稳效应，但它目前是计算上的优势，只有整数值的指数。在本文中，我们得到了一个有效的方法来处理所有允许的非整数值的指数。该方法首先离散的协方差算子，使用有限元和正交，并使用预处理Krylov子空间求解器移位线性系，有效地将所得的协方差阵的向量。这种方法可以通过两种不同的方式从分布中生成样本：通过求解随机偏微分方程，以及通过使用截断的Karhunen-Loève展开。我们将展示如何将这种先验表示到无限维贝叶斯公式，并展示如何有效地计算最大后验估计，并近似后验方差。虽然本文的重点是贝叶斯逆问题，这里开发的技术适用于解决系统的分数拉普拉斯和高斯随机场。数值实验表明，性能和可扩展性的求解器和它们的适用性模型和实时数据的反问题，在层析成像和时间相关的热方程。
英文摘要：
This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle--Matérn Gaussian random fields. The Whittle--Matérn prior is characterized by a mean function and a covariance operator that is taken as a negative power of an elliptic differential operator. This approach is flexible in that it can incorporate a wide range of prior information including non-stationary effects, but it is currently computationally advantageous only for integer values of the exponent. In this paper, we derive an efficient method for handling all admissible noninteger values of the exponent. The method first discretizes the covariance operator using finite elements and quadrature, and uses preconditioned Krylov subspace solvers for shifted linear systems to efficiently apply the resulting covariance matrix to a vector. This approach can be used for generating samples from the distribution in two different ways: by solving a stochastic partial differential equation, and by using a truncated Karhunen-Loève expansion. We show how to incorporate this prior representation into the infinite-dimensional Bayesian formulation, and show how to efficiently compute the maximum a posteriori estimate, and approximate the posterior variance. Although the focus of this paper is on Bayesian inverse problems, the techniques developed here are applicable to solving systems with fractional Laplacians and Gaussian random fields. Numerical experiments demonstrate the performance and scalability of the solvers and their applicability to model and real-data inverse problems in tomography and a time-dependent heat equation.
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