SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3397-A3426, January 2020.
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.

中文翻译：

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偏微分方程控制的逆问题中引起的黑森州人的层次矩阵逼近

SIAM科学计算杂志，第42卷，第5期，第A3397-A3426页，2020年1月。
由偏微分方程（PDE）控制的反问题中出现的黑森算子在确定性反问题的牛顿解以及贝叶斯环境中后验的马尔可夫链蒙特卡罗采样的有效，维数无关收敛中起着关键作用。这些方法要求能够在Hessian上重复执行操作，例如与任意向量相乘，求解线性系统，求反和（求逆）平方根。不幸的是，Hessian是（形式上）密集的，隐式定义的运算符，对于实际的反问题而言，它难以显式地形成，因此需要与反演参数一样多的PDE求解。当数据包含有关参数的有限信息时，低秩近似是有效的，但随着数据变得更具有信息性，低秩近似将变得令人望而却步。但是，对于具有实际应用中出现的许多反问题的Hessian，可以通过具有低等级结构的矩阵很好地近似。分层矩阵表示法有望克服密集表示法的高复杂度，并提供仅具有对数线性复杂度的有效数据结构和矩阵运算。在这项工作中，我们描述了用于构造和更新Hessians的层次矩阵近似的算法，并针对涉及时间依赖性扩散，对流占主导的传输，频域声波传播和低频Maxwell方程的许多代表性逆问题对它们进行了说明，