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dc.contributor.authorAmbartsumyan, Ilona
dc.contributor.authorBoukaram, Wagih Halim
dc.contributor.authorBui-Thanh, Tan
dc.contributor.authorGhattas, Omar
dc.contributor.authorKeyes, David E.
dc.contributor.authorStadler, Georg
dc.contributor.authorTurkiyyah, George
dc.contributor.authorZampini, Stefano
dc.identifier.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
dc.description.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.
dc.description.sponsorshipThis 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.
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)
dc.rightsArchived with thanks to SIAM Journal on Scientific Computing
dc.titleHierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs
dc.contributor.departmentComputer Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentExtreme Computing Research Center
dc.contributor.departmentOffice of the President
dc.identifier.journalSIAM Journal on Scientific Computing
dc.contributor.institutionOden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712 USA.
dc.contributor.institutionCourant Institute of Mathematical Sciences, New York University, New York, NY 10012 USA.
dc.contributor.institutionDepartment of Computer Science, American University of Beirut, Beirut 1107-2020, Lebanon.
kaust.personBoukaram, Wagih Halim
kaust.personKeyes, David E.
kaust.personZampini, Stefano
kaust.acknowledged.supportUnitOffice of Sponsored Research (OSR)

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