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

Abstract

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.

Robust and efficient multilevel‐ILU preconditioning of hybrid Newton–GMRES for incompressible Navier–Stokes equations

“…Like single-level ILU, the permutation matrices P and Q can be statically constructed. One can also apply pivoting 53 or deferring 30,54 in MLILU. For this two-level ILU, PMQ T provides a preconditioner of A.…”

“…The computational kernel of HILUNG is a robust and efficient multilevel ILU preconditioner, called HILUCSI (or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping), which the authors developed recently. 30 HILUCSI shares some similarities with other MLILU (such as ILUPACK 33 ) in its use of the Crout version of ILU factorization, 61 its dynamic deferring of rows and columns to ensure the well-conditioning of B in (17) at each level, 54 and its inverse-based dropping for robustness. 54 Different from ILUPACK, however, HILUCSI improved the robustness for saddle-point problems from PDEs by using static deferring of small diagonals and by utilizing a combination of symmetric and unsymmetric permutations at the top and lower levels, respectively.…”

“…We have previously conducted a thorough analysis of HILUCSI in terms of its accuracy 63 and its efficiency. 30 In particular, HILUCSI was shown to satisfy an 𝜖-accuracy criterion (Definition 3.8 of Jiao and Chen 63 ). As a result, with sufficiently small dropping thresholds, HILUCSI guarantees the convergence of right-preconditioned GMRES, and it converges to an optimal preconditioner as the thresholds decrease, enabling the right-preconditioned GMRES to converge in one iteration at the limit (see Lemma 4.3 and Theorem 3.6 of Jiao and Chen 63 ).…”

“…In addition, we proved that HILUCSI has (near) linear space and time complexities in its factorization cost and its solve cost per GMRES iteration. 30 In contrast, neither the block triangular approximate Schur complement preconditioners 2 nor the augmented Lagrangian preconditioners 23,24 satisfy the 𝜖-accuracy criterion, and the complete factorizations typically used for their leading blocks 26,31,32 have superlinear space and time complexity. For these reasons, we expect HILUCSI to perform well as the preconditioner for INS, as we will confirm empirically in Section 4.…”

“…We build our preconditioner based on HILUCSI (or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping), which the authors and co-workers introduced recently for indefinite linear systems from partial differential equations (PDEs), such as saddle-point problems. 30 In this work, we incorporate HILUCSI into Newton-GMRES to develop HILUNG, for nonlinear saddle-point problems from Navier-Stokes equations. To this end, we introduce sparsifying operators based on (4) and ( 5), develop adaptive refactorization and thresholding to avoid potential "over-factorization" (i.e., too dense incomplete factorization or too frequent refactorization), and introduce iterative refinement during preconditioning to reduce memory requirement.…”