Kernel Methods for Bayesian Elliptic Inverse Problems on Manifolds
Kernel Methods for Bayesian Elliptic Inverse Problems on Manifolds
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DOI：
10.1137/19m1295222
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
J. Harlim;D. Sanz-Alonso;Ruiyi Yang
中科院分区：
文献类型：
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作者：
J. Harlim;D. Sanz-Alonso;Ruiyi Yang
本文研究了贝叶斯反问题的制定和实施，以学习定义在流形上的偏微分方程（PDE）的输入参数。具体来说，我们研究的逆问题，确定一个封闭的流形上的二阶椭圆型偏微分方程的扩散系数的噪声测量的解决方案。受流形学习技术的启发，我们用一个基于核的积分算子来近似椭圆微分算子，该积分算子可以通过蒙特-卡罗离散化，而不需要参考黎曼度量。由此产生的计算方法是无网格的，易于实现，并且可以在不完全了解底层流形的情况下应用，前提是流形样本的点云可用。我们采用贝叶斯的角度来看，逆问题，并建立一个上限的总变化之间的距离的真实后验和近似后定义的核前向映射。支持的数值结果表明所提出的方法的有效性。
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the diffusion coefficient of a second-order elliptic PDE on a closed manifold from noisy measurements of the solution. Inspired by manifold learning techniques, we approximate the elliptic differential operator with a kernel-based integral operator that can be discretized via Monte-Carlo without reference to the Riemannian metric. The resulting computational method is mesh-free and easy to implement, and can be applied without full knowledge of the underlying manifold, provided that a point cloud of manifold samples is available. We adopt a Bayesian perspective to the inverse problem, and establish an upper-bound on the total variation distance between the true posterior and an approximate posterior defined with the kernel forward map. Supporting numerical results show the effectiveness of the proposed methodology.