TY - JOUR

T1 - Hierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs

AU - Ambartsumyan, Ilona

AU - Boukaram, Wagih Halim

AU - Bui-Thanh, Tan

AU - Ghattas, Omar

AU - Keyes, David E.

AU - Stadler, Georg

AU - Turkiyyah, George

AU - Zampini, Stefano

N1 - KAUST Repository Item: Exported on 2020-10-30
Acknowledged KAUST grant number(s): OSR-2018-CARF-3666
Acknowledgements: This 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.

PY - 2020/10/23

Y1 - 2020/10/23

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/662368

UR - https://epubs.siam.org/doi/10.1137/19M1270367

U2 - 10.1137/19m1270367

DO - 10.1137/19m1270367

M3 - Article

SN - 1064-8275

VL - 42

SP - A3397-A3426

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 5

ER -