论文标题
弹性半空间中断层逆问题的随机算法，并证明了融合的证明
A stochastic algorithm for fault inverse problems in elastic half space with proof of convergence
论文作者
论文摘要
[11]中引入了用于解决混合线性和非线性问题的一般随机算法。我们在本文中表明，如何使用它来解决故障逆问题，其中弹性半空间中的平面断层必须从嘈杂的表面位移测量值中重建该故障。随着给出包含由M表示的故障的平面的参数和C的线性部分的正则化参数，由C表示为c的逆问题，均以随机变量为模型，我们得出了m的后边缘的公式。建模C作为一个随机变量允许扫描广泛的可能值，这被证明优于选择固定值[11]。我们证明，随着测量点的数量和离散滑移空间的尺寸增加，M的后边缘是收敛的。简而言之，我们的证明仅假定处理测量的正规离散误差功能与1个正交规则有关，并且有限维空间的离散滑移的联合是密集的。我们的证明依靠痕量类操作者理论表明，正确的决定因素是统一的界限。我们还解释了只要满足某些基本要求，我们的证明如何扩展到整个逆问题。最后，我们显示了数字模拟，以说明算法的数值收敛性。
A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [11]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic half-space and a slip on that fault have to be reconstructed from noisy surface displacement measurements. With the parameter giving the plane containing the fault denoted by m and the regularization parameter for the linear part of the inverse problem denoted by C, both modeled as random variables, we derive a formula for the posterior marginal of m. Modeling C as a random variable allows to sweep through a wide range of possible values which was shown to be superior to selecting a fixed value [11]. We prove that this posterior marginal of m is convergent as the number of measurement points and the dimension of the space for discretizing slips increase. Simply put, our proof only assumes that the regularized discrete error functional for processing measurements relates to an order 1 quadrature rule and that the union of the finite-dimensional spaces for discretizing slips is dense. Our proof relies on trace class operator theory to show that an adequate sequence of determinants is uniformly bounded. We also explain how our proof can be extended to a whole class of inverse problems, as long as some basic requirements are met. Finally, we show numerical simulations that illustrate the numerical convergence of our algorithm.
