Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization
Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization
复制标题
反演问题中 Hessians 的分层非对角低秩逼近，及其在冰盖模型初始化中的应用
DOI：
10.1088/1361-6420/acd719
复制
发表时间：
2023
期刊：
影响因子：
2.1
通讯作者：
Petra, Noémi
中科院分区：
文献类型：
--
作者：
Hartland, Tucker;Stadler, Georg;Perego, Mauro;Liegeois, Kim;Petra, Noémi
在偏微分方程（PDE）反问题中，获得轻量级和精确的离散化目标泛函Hessian近似是使确定性和贝叶斯统计大规模反问题计算上易于处理的关键。稠密线性代数任务的立方计算复杂性，如Cholesky因式分解，提供了一种方法来采样高斯分布和确定牛顿线性系统的解决方案，是一个大规模的计算瓶颈。这些任务可以减少到对数线性的复杂性，利用分层非对角低秩（HODLR）矩阵近似。在这项工作中，我们表明，一类海森所产生的偏微分方程的逆问题，以及近似的HODLR矩阵格式。特别是，我们研究由偏微分方程模型的瞬时粘性流冰盖的反问题。在这些问题中，我们寻求一个空间分布的基底滑动参数场，使冰盖模型预测的流量与冰盖表面速度观测结果一致。我们证明了使用HODLR Hessian近似有效地采样后验分布的拉普拉斯近似，协方差进一步近似HODLR矩阵压缩。进行计算研究，说明冰盖问题的制度，高斯-牛顿数据失配海森更有效地近似HODLR矩阵格式比低秩（LR）格式。然后，我们证明，HODLR近似可以是有利的，相比全球LR近似，大规模的问题，通过研究数据失配海森与反问题的一阶斯托克斯流动模型的洪堡冰川和格陵兰冰盖。
Obtaining lightweight and accurate approximations of discretized objective functional Hessians in inverse problems governed by partial differential equations (PDEs) is essential to make both deterministic and Bayesian statistical large-scale inverse problems computationally tractable. The cubic computational complexity of dense linear algebraic tasks, such as Cholesky factorization, that provide a means to sample Gaussian distributions and determine solutions of Newton linear systems is a computational bottleneck at large-scale. These tasks can be reduced to log-linear complexity by utilizing hierarchical off-diagonal low-rank (HODLR) matrix approximations. In this work, we show that a class of Hessians that arise from inverse problems governed by PDEs are well approximated by the HODLR matrix format. In particular, we study inverse problems governed by PDEs that model the instantaneous viscous flow of ice sheets. In these problems, we seek a spatially distributed basal sliding parameter field such that the flow predicted by the ice sheet model is consistent with ice sheet surface velocity observations. We demonstrate the use of HODLR Hessian approximation to efficiently sample the Laplace approximation of the posterior distribution with covariance further approximated by HODLR matrix compression. Computational studies are performed which illustrate ice sheet problem regimes for which the Gauss–Newton data-misfit Hessian is more efficiently approximated by the HODLR matrix format than the low-rank (LR) format. We then demonstrate that HODLR approximations can be favorable, when compared to global LR approximations, for large-scale problems by studying the data-misfit Hessian associated with inverse problems governed by the first-order Stokes flow model on the Humboldt glacier and Greenland ice sheet.
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