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dc.contributor.author
Sanan, Patrick
dc.contributor.author
May, Dave A.
dc.contributor.author
Bollhöfer, Matthias
dc.contributor.author
Schenk, Olaf
dc.date.accessioned
2020-11-20T12:18:57Z
dc.date.available
2020-11-20T05:50:50Z
dc.date.available
2020-11-20T12:18:57Z
dc.date.issued
2020
dc.identifier.issn
1869-9510
dc.identifier.issn
1869-9529
dc.identifier.other
10.5194/se-11-2031-2020
en_US
dc.identifier.uri
http://hdl.handle.net/20.500.11850/452112
dc.identifier.doi
10.3929/ethz-b-000452112
dc.description.abstract
The need to solve large saddle point systems within computational Earth sciences is ubiquitous. Physical processes giving rise to these systems include porous flow (the Darcy equations), poroelasticity, elastostatics, and highly viscous flows (the Stokes equations). The numerical solution of saddle point systems is non-trivial since the operators are indefinite. Primary tools to solve such systems are direct solution methods (exact triangular factorization) or approximate block factorization (ABF) preconditioners. While ABF solvers have emerged as the state-of-the-art scalable option, they are invasive solvers requiring splitting of pressure and velocity degrees of freedom, a multigrid hierarchy with tuned transfer operators and smoothers, machinery to construct complex Schur complement preconditioners, and the expertise to select appropriate parameters for a given coefficient regime – they are far from being “black box” solvers. Modern direct solvers, which robustly produce solutions to almost any system, do so at the cost of rapidly growing time and memory requirements for large problems, especially in 3D. Incomplete LDLT (ILDL) factorizations, with symmetric maximum weighted-matching preprocessing, used as preconditioners for Krylov (iterative) methods, have emerged as an efficient means to solve indefinite systems. These methods have been developed within the numerical linear algebra community but have yet to become widely used in applications, despite their practical potential; they can be used whenever a direct solver can, only requiring an assembled operator, yet can offer comparable or superior performance, with the added benefit of having a much lower memory footprint. In comparison to ABF solvers, they only require the specification of a drop tolerance and thus provide an easy-to-use addition to the solver toolkit for practitioners. Here, we present solver experiments employing incomplete LDLT factorization with symmetric maximum weighted-matching preprocessing to precondition operators and compare these to direct solvers and ABF-preconditioned iterative solves. To ensure the comparison study is meaningful for Earth scientists, we utilize matrices arising from two prototypical problems, namely Stokes flow and quasi-static (linear) elasticity, discretized using standard mixed finite-element spaces. Our test suite targets problems with large coefficient discontinuities across non-grid-aligned interfaces, which represent a common challenging-for-solvers scenario in Earth science applications. Our results show that (i) as the coefficient structure is made increasingly challenging, by introducing high contrast and complex topology with a multiple-inclusion benchmark, the ABF solver can break down, becoming less efficient than the ILDL solver before breaking down entirely; (ii) ILDL is robust, with a time to solution that is largely independent of the coefficient topology and mildly dependent on the coefficient contrast; (iii) the time to solution obtained using ILDL is typically faster than that obtained from a direct solve, beyond 105 unknowns; and (iv) ILDL always uses less memory than a direct solve.
en_US
dc.format
application/pdf
en_US
dc.language.iso
en
en_US
dc.publisher
Copernicus
en_US
dc.rights.uri
http://creativecommons.org/licenses/by/4.0/
dc.title
Pragmatic solvers for 3D Stokes and elasticity problems with heterogeneous coefficients: evaluating modern incomplete LDLT preconditioners
en_US
dc.type
Journal Article
dc.rights.license
Creative Commons Attribution 4.0 International
dc.date.published
2020-11-10
ethz.journal.title
Solid Earth
ethz.journal.volume
11
en_US
ethz.journal.issue
6
en_US
ethz.journal.abbreviated
Solid earth
ethz.pages.start
2031
en_US
ethz.pages.end
2045
en_US
ethz.version.deposit
publishedVersion
en_US
ethz.identifier.wos
ethz.identifier.scopus
ethz.publication.place
Göttingen
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02330 - Dep. Erdwissenschaften / Dep. of Earth Sciences::02506 - Institut für Geophysik / Institute of Geophysics::03698 - Tackley, Paul / Tackley, Paul
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02330 - Dep. Erdwissenschaften / Dep. of Earth Sciences::02506 - Institut für Geophysik / Institute of Geophysics::03698 - Tackley, Paul / Tackley, Paul
ethz.date.deposited
2020-11-20T05:50:59Z
ethz.source
SCOPUS
ethz.eth
yes
en_US
ethz.availability
Open access
en_US
ethz.rosetta.installDate
2020-11-20T12:19:08Z
ethz.rosetta.lastUpdated
2021-02-15T20:51:50Z
ethz.rosetta.versionExported
true
ethz.COinS
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