Preconditioning for accurate solutions of ill‐conditioned linear systems

Ye, Qiang

doi:10.1002/nla.2315

Cited by 3 publications

(1 citation statement)

References 26 publications

(45 reference statements)

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“…The fastest vertical mode, although being the most difficult to solve, has a relatively low condition number, C 0 ≃ 500 (corresponding to the current operational configuration) when compared to typical values encountered in recent literature (Ye, 2017, Soleymani, 2013 with typical values of ill-conditioned matrices larger than 10 10 . The remaining N i,i=1...N lev problems are even better conditioned and a very fast convergence using any iterative grid-point solver is expected for the majority of them, confirming the preconditioning effect of this vertical projection.…”

Section: Choice Of Strategymentioning

confidence: 84%

Krylov solvers in a vertical‐slice version of the semi‐implicit semi‐Lagrangian AROME model

Burgot

Auger

Bénard

2021

Quart J Royal Meteoro Soc

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To circumvent the scalability problem due to global communication involved in spectral transforms, a vertical-slice version of the dynamical core where all calculations are performed in grid-point space has been built for AROME, Météo-France's operational limited-area model. It is shown in an idealized but nevertheless physically relevant framework that, despite this major change, it is possible to keep the other main characteristics of the model (constant-coefficient semi-implicit scheme, semi-Lagrangian transport scheme, A-grid, mass-based coordinate, etc.). A Krylov solver is used to solve the implicit problem. Using the solution given by the spectral model as a reference in terms of quality and required accuracy in an operational context, the chosen parameters of the Krylov solver are carefully tuned to maximize its convergence speed. The Krylov solver consists of several applications of a sparse operator, whose stencil is similar to that of the operator applied during the small time step of split-explicit schemes traditionally used in HEVI (Horizontally Explicit/Vertically Implicit) models.HEVI models are considered particularly scalable in the current parallelization paradigm and are used as a reference for scalability in this study. Results show that a fast convergence can be reached, such that only a few applications of the sparse operator are required. This suggests an improvement in scalability compared to the spectral version. Experiments have been formulated regardless of the future strategy considered for temporal and spatial resolutions, i.e., increasing spatial resolution while either maintaining a high Courant number or reducing it.

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Section: Choice Of Strategymentioning

confidence: 84%

Krylov solvers in a vertical‐slice version of the semi‐implicit semi‐Lagrangian AROME model

Burgot

Auger

Bénard

2021

Quart J Royal Meteoro Soc

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Full operator preconditioning and the accuracy of solving linear systems

Mohr,

Nakatsukasa,

Urzúa-Torres

2024

IMA Journal of Numerical Analysis

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Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.

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A Haar wavelets-based direct reconstruction method for the Cauchy problem of the Poisson equation

Rashid¹,

Nachaoui²

2023

DCDS-S

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Preconditioning for accurate solutions of ill‐conditioned linear systems

Cited by 3 publications

References 26 publications

Krylov solvers in a vertical‐slice version of the semi‐implicit semi‐Lagrangian AROME model

Krylov solvers in a vertical‐slice version of the semi‐implicit semi‐Lagrangian AROME model

Full operator preconditioning and the accuracy of solving linear systems

A Haar wavelets-based direct reconstruction method for the Cauchy problem of the Poisson equation

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