Adaptive Techniques for Improving the Performance of Incomplete Factorization Preconditioning

Gupta, Anshul; George, Thomas

doi:10.1137/080727695

Cited by 33 publications

(33 citation statements)

References 44 publications

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“…That is, we either keep or drop an entire row of a supernode when it is formed at the crrent step. This is similar to what was first proposed by Gupta and George for incomplete Cholesky factorization [11]. Our dropping criterion is the second rule shown in Figure 1.…”

Section: Threshold-based Dropping Criteriasupporting

confidence: 54%

“…Hénon et al developed a general scheme for identifying supernodes in ILU(k) [12], but it is not directly applicable to threshold-based dropping. Our algorithm is most similiar to the method proposed by Gupta and George [11], and we extended it to the case of unsymmetric factorization with partial pivoting.…”

Section: Introductionmentioning

confidence: 99%

“…In Saad's ILU(τ,p) approach [19], p is the largest number of nonzeros (not the level-of-fill) allowed in echo row of F (in a row-wise algorithm). Gupta and George suggested using p( j) = γ · nnz(A(:, j)) for the j-th column instead of a constant, where γ is an upper bound of the fill ratio defined by a user [11]. They also proposed a method of computing a secondary dropping tolerance by an interpolation formula rather than sorting the largest p entries, which is cheaper than the original ILU(τ,p).…”

Section: Secondary Dropping To Control Fill-in Adaptivelymentioning

confidence: 99%

“…This is an acceptable solution when not many zero pivots occur, otherwise, the preconditioner can be quite ill-conditioned even though the factorization completes, making this ineffective. Some other methods were proposed to handle the breakdown, such as the delayed pivoting [11] and the multilevel method [1]. We plan to investigate them in the future.…”

Section: Handling Breakdown Due To Zero Pivotsmentioning

confidence: 99%

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A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting

Shao

2011

ACM Trans. Math. Softw.

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We present a new supernode-based incomplete LU factorization method to construct a preconditioner for solving sparse linear systems with iterative methods. The new algorithm is primarily based on the ILUTP approach by Saad, and we incorporate a number of techniques to improve the robustness and performance of the traditional ILUTP method. These include the new dropping strategies that accommodate the use of supernodal structures in the factored matrix. We present numerical experiments to demonstrate that our new method is competitive with the other ILU approaches and is well suited for today's high performance architectures.

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Section: Threshold-based Dropping Criteriasupporting

confidence: 54%

Section: Introductionmentioning

confidence: 99%

Section: Secondary Dropping To Control Fill-in Adaptivelymentioning

confidence: 99%

Section: Handling Breakdown Due To Zero Pivotsmentioning

confidence: 99%

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A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting

Shao

2011

ACM Trans. Math. Softw.

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“…The results of this chapter show that, by taking advantage of this block structure, the solver can be more robust and efficient. Other recent studies on block ILU preconditioners have drawn similar conclusions on the importance of exposing dense blocks during the construction of the incomplete LU factorization for better performance, in the design of incomplete multifrontal LUfactorization preconditioners [61] and adaptive blocking approaches for blocked incomplete Cholesky factorization [62]. We believe that the proposed VBARMS method can be useful for solving linear systems also in other areas, such as in Electromagnetics applications [63][64][65].…”

Section: Discussionmentioning

confidence: 69%

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

Carpentieri¹,

Bonfiglioli²

2018

Computational Fluid Dynamics - Basic Instruments and Applications in Science

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Implicit methods based on the Newton's rootfinding algorithm are receiving an increasing attention for the solution of complex Computational Fluid Dynamics (CFD) applications due to their potential to converge in a very small number of iterations. This approach requires fast convergence acceleration techniques in order to compete with other conventional solvers, such as those based on artificial dissipation or upwind schemes, in terms of CPU time. In this chapter, we describe a multilevel variable-block Schur-complement-based preconditioning for the implicit solution of the Reynoldsaveraged Navier-Stokes equations using unstructured grids on distributed-memory parallel computers. The proposed solver detects automatically exact or approximate dense structures in the linear system arising from the discretization, and exploits this information to enhance the robustness and improve the scalability of the block factorization. A complete study of the numerical and parallel performance of the solver is presented for the analysis of turbulent Navier-Stokes equations on a suite of threedimensional test cases.

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HILUCSI: Simple, robust, and fast multilevel ILU for large‐scale saddle‐point problems from PDEs

Chen

Ghai

Jiao

2021

Numerical Linear Algebra App

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Incomplete factorization is a widely used preconditioning technique for Krylov subspace methods for solving large‐scale sparse linear systems. Its multilevel variants, such as ILUPACK, are more robust for many symmetric or unsymmetric linear systems than the traditional, single‐level incomplete LU (or ILU) techniques. However, the previous multilevel ILU techniques still lacked robustness and efficiency for some large‐scale saddle‐point problems, which often arise from systems of PDEs. We introduce HILUCSI, or Hierarchical Incomplete LU‐Crout with Scalability‐oriented and Inverse‐based dropping. As a multilevel preconditioner, HILUCSI statically and dynamically permutes individual rows and columns to the next level for deferred factorization. Unlike ILUPACK, HILUCSI applies symmetric preprocessing techniques at the top levels but always uses unsymmetric preprocessing and unsymmetric factorization at the coarser levels. The deferring combined with mixed preprocessing enabled a unified treatment for nearly or partially symmetric systems and simplified the implementation by avoiding mixed 1×1 and 2×2 pivots for symmetric indefinite systems. We show that this combination improves robustness for indefinite systems without compromising efficiency. Furthermore, to enable superior efficiency for large‐scale systems with millions or more unknowns, HILUCSI introduces a scalability‐oriented dropping in conjunction with a variant of inverse‐based dropping. We demonstrate the effectiveness of HILUCSI for dozens of benchmark problems, including those from the mixed formulation of the Poisson equation, Stokes equations, and Navier–Stokes equations. We also compare its performance with ILUPACK and the supernodal ILUTP in SuperLU.

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Adaptive Techniques for Improving the Performance of Incomplete Factorization Preconditioning

Cited by 33 publications

References 44 publications

A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting

A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

HILUCSI: Simple, robust, and fast multilevel ILU for large‐scale saddle‐point problems from PDEs

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