Hierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs
Boukaram, Wagih Halim
Keyes, David E.
KAUST DepartmentComputer Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program
Extreme Computing Research Center
Office of the President
KAUST Grant NumberOSR-2018-CARF-3666
Preprint Posting Date2020-03-23
Online Publication Date2020-10-22
Print Publication Date2020-01
Permanent link to this recordhttp://hdl.handle.net/10754/662368
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AbstractHessian 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.
CitationAmbartsumyan, I., Boukaram, W., Bui-Thanh, T., Ghattas, O., Keyes, D., Stadler, G., … Zampini, S. (2020). Hierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs. SIAM Journal on Scientific Computing, 42(5), A3397–A3426. doi:10.1137/19m1270367
SponsorsThis work was supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No: OSR-2018-CARF-3666.