Preconditioning Parametrized Linear Systems

Arielle, Carr,; Sturler, Eric de; Gugercin, Serkan

doi:10.1137/20m1331123

Cited by 6 publications

(5 citation statements)

References 68 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Figure 1 displays the sparsity pattern of

L+U

on the first four levels of AMG using the AMGToolbox 29 for matrix dimension

N=14186

. Here, the drop tolerance is set to 1.e

-15

and fill limit to 200 per row using the ILUTP implementation in 30 with pivoting turned off 6 . A very small drop tolerance and large fill‐in are unreasonable choices for an ILU factorization because they can substantially increase the cost associated with computing, applying and, storing the factors.…”

Section: Iterative Triangular Solvesmentioning

confidence: 99%

Scaled ILU smoothers for Navier–Stokes pressure projection

Thomas,

Carr,

Mullowney

et al. 2023

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

Incomplete LU (ILU) smoothers are effective in the algebraic multigrid (AMG) ‐cycle for reducing high‐frequency components of the error. However, the requisite direct triangular solves are comparatively slow on GPUs. Previous work has demonstrated the advantages of Jacobi iteration as an alternative to direct solution of these systems. Depending on the threshold and fill‐level parameters chosen, the factors can be highly nonnormal and Jacobi is unlikely to converge in a low number of iterations. We demonstrate that row scaling can reduce the departure from normality, allowing us to replace the inherently sequential solve with a rapidly converging Richardson iteration. There are several advantages beyond the lower compute time. Scaling is performed locally for a diagonal block of the global matrix because it is applied directly to the factor. Further, an ILUT Schur complement smoother maintains a constant GMRES iteration count as the number of MPI ranks increases, and thus parallel strong‐scaling is improved. Our algorithms have been incorporated into hypre, and we demonstrate improved time to solution for linear systems arising in the Nalu‐Wind and PeleLM pressure solvers. For large problem sizes, GMRESAMG executes at least five times faster when using iterative triangular solves compared with direct solves on massively parallel GPUs.

show abstract

“…Figure 1 displays the sparsity pattern of

L+U

on the first four levels of AMG using the AMGToolbox 29 for matrix dimension

N=14186

. Here, the drop tolerance is set to 1.e

-15

Section: Iterative Triangular Solvesmentioning

confidence: 99%

Scaled ILU smoothers for Navier–Stokes pressure projection

Thomas,

Carr,

Mullowney

et al. 2023

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, these preconditioners have a matrix-free implementation if the initial preconditioner can also be applied in a matrixfree regime, as for instance the polynomial preconditioner recently resumed in [14], reducing the CPU resources of the matrix allocation considerably. Another strategy to update preconditioners for sequences of linear systems, based on the sparse minimization of the Frobenius norm of a suitable error function, has been recently studied in [15].…”

Section: Spatial and Time Discretizationsmentioning

confidence: 99%

Strategies of Preconditioner Updates for Sequences of Linear Systems Associated with the Neutron Diffusion

Carreño

Bergamaschi

Martínez

et al. 2022

Computational and Mathematical Methods

View full text Add to dashboard Cite

The time-dependent neutron diffusion equation approximates the neutronic power evolution inside a nuclear reactor core. Applying a Galerkin finite element method for the spatial discretization of these equations leads to a stiff semi-discrete system of ordinary differential equations. For time discretization, an implicit scheme is used, which implies solving a large and sparse linear system of equations for each time step. The GMRES method is used to solve these systems because of its fast convergence when a suitable preconditioner is provided. This work explores several matrix-free strategies based on different updated preconditioners, which are constructed by low-rank updates of a given initial preconditioner. They are two tuned preconditioners based on the bad and good Broyden’s methods, initially developed for nonlinear equations and optimization problems, and spectral preconditioners. The efficiency of the resulting preconditioners under study is closely related to the selection of the subspace used to construct the update. Our numerical results show the effectiveness of these methodologies in terms of CPU time and storage for different nuclear benchmark transients, even if the initial preconditioner is not good enough.

show abstract

“…[20][21][22] Nevertheless, optimizing the aforementioned solvers so as to attain a uniformly fast convergence for multiple parameter instances, as required in multi-query problems, remains a challenging task to this day. To tackle this problem, several works suggest the use of interpolation methods tasked with constructing approximations of the system's inverse operator for different parameter values, [23][24][25] which can then be used as preconditioners. Another approach can be found in Reference 26, where primal and dual FETI decomposition methods with customized preconditioners are developed in order to accelerate the solution of stochastic problems in the context of Monte Carlo simulation, as well as intrusive Galerkin methods.…”

Section: Introductionmentioning

confidence: 99%

AI‐enhanced iterative solvers for accelerating the solution of large‐scale parametrized systems

Nikolopoulos,

Kalogeris,

Stavroulakis

et al. 2023

Numerical Meth Engineering

View full text Add to dashboard Cite

SummaryRecent advances in the field of machine learning open a new era in high performance computing for challenging computational science and engineering applications. In this framework, the use of advanced machine learning algorithms for the development of accurate and cost‐efficient surrogate models of complex physical processes has already attracted major attention from scientists. However, despite their powerful approximation capabilities, surrogate model predictions are still far from being near to the ‘exact’ solution of the problem. To address this issue, the present work proposes the use of up‐to‐date machine learning tools in order to equip a new generation of iterative solvers of linear equation systems, capable of very efficiently solving large‐scale parametrized problems at any desired level of accuracy. The proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed using a standard finite element methodology and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using a combination of deep feedforward neural networks and convolutional autoencoders. This mapping serves as a means of obtaining very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the Algebraic Multigrid method in combination with Proper Orthogonal Decomposition, termed POD‐2G, is developed that successively refines the initial predictions of the surrogate model towards the exact solution. The application of POD‐2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its strong superiority over conventional iterative solution schemes.

show abstract

Preconditioning Parametrized Linear Systems

Cited by 6 publications

References 68 publications

Scaled ILU smoothers for Navier–Stokes pressure projection

Scaled ILU smoothers for Navier–Stokes pressure projection

Strategies of Preconditioner Updates for Sequences of Linear Systems Associated with the Neutron Diffusion

AI‐enhanced iterative solvers for accelerating the solution of large‐scale parametrized systems

Contact Info

Product

Resources

About