Hierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs
Hierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs
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偏微分方程反问题中 Hessians 的层次矩阵逼近
DOI：
10.1137/19m1270367
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发表时间：
2020
影响因子：
3.1
通讯作者：
Zampini, Stefano
中科院分区：
文献类型：
--
作者：
Ambartsumyan, Ilona;Boukaram, Wajih;Bui-Thanh, Tan;Ghattas, Omar;Keyes, David;Stadler, Georg;Turkiyyah, George;Zampini, Stefano
Hessian算子在偏微分方程（PDEs）反问题中的应用对于求解确定性反问题的牛顿解以及贝叶斯环境下后验的马尔可夫链蒙特卡罗采样具有重要意义。这些方法需要在Hessian上重复执行操作的能力，例如与任意向量相乘，求解线性系统，求逆和（逆）平方根。不幸的是，海森是一个（正式）密集，隐式定义的运营商，是难以形成明确的实际反问题，需要尽可能多的PDE解决反演参数。当数据包含关于参数的有限信息时，低秩近似是有效的，但是当数据变得更加信息化时，低秩近似变得令人望而却步。然而，在实际应用中出现的许多反问题的海森可以很好地近似矩阵，具有层次低秩结构。层次矩阵表示有望克服密集表示的高复杂性，并提供有效的数据结构和矩阵运算，只有对数线性复杂度。在这项工作中，我们描述的算法，用于构建和更新层次矩阵近似的海森，并说明他们的一些代表性的逆问题，涉及时间相关的扩散，对流为主的运输，频域声波传播，和低频麦克斯韦方程，演示了一个数量级的加速比全球低秩近似。
Hessian operators arising in inverse problems governed by partial differential equations (PDEs) play a critical role in delivering efficient, dimension-independent convergence for Newton solution of deterministic inverse problems, as well as Markov chain Monte Carlo sampling of posteriors in the Bayesian setting. These methods require the ability to repeatedly perform operations on the Hessian such as multiplication with arbitrary vectors, solving linear systems, inversion, and (inverse) square root. Unfortunately, the Hessian is a (formally) dense, implicitly defined operator that is intractable to form explicitly for practical inverse problems, requiring as many PDE solves as inversion parameters. Low rank approximations are effective when the data contain limited information about the parameters but become prohibitive as the data become more informative. However, the Hessians for many inverse problems arising in practical applications can be well approximated by matrices that have hierarchically low rank structure. Hierarchical matrix representations promise to overcome the high complexity of dense representations and provide effective data structures and matrix operations that have only log-linear complexity. In this work, we describe algorithms for constructing and updating hierarchical matrix approximations of Hessians, and illustrate them on a number of representative inverse problems involving time-dependent diffusion, advection-dominated transport, frequency domain acoustic wave propagation, and low frequency Maxwell equations, demonstrating up to an order of magnitude speedup compared to globally low rank approximations.
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