RILU conditioning for the Neumann problem on smooth planar domains
Let $\Omega \subset \mathbb{R}^2$ be a bounded connected domain with smooth boundary, and let $h > 0$. Number $1, \dots, N_h$, in lexicographic order, the points $p \in h\mathbb{Z}^2$ whose control cells $C_p = p + [-h/2, h/2]^2$ intersect $\Omega$. For two such points whose cells share an edge put $w_{pq} = \operatorname{length}(C_p \cap C_q \cap \Omega)/h$, and $w_{pq} = 0$ otherwise. The Purvis–Burkhalter matrix $A$ of the Neumann problem has diagonal entries $a_p = \sum_q w_{pq}$ and off-diagonal entries $A_{pq} = -w_{pq}$; let $L$ be its strictly lower triangular part. For $0 < r < 1$ the relaxed incomplete LU (RILU) preconditioner is $M = (L + E)E^{-1}(E + L^T)$, where $E = \operatorname{diag}(e_1, \dots, e_{N_h})$ is computed for $p = 1, 2, \dots, N_h$ in turn by $$e_p = a_p - \sum_{q < p,\ w_{pq} > 0} \frac{w_{pq}}{e_q} \Bigl(w_{pq} + (1 - r) \sum_{s > q,\ s \ne p} w_{qs}\Bigr),$$ so that $e_1 = a_1$. When the pivots $e_p$ are positive, let $\kappa_+(M^{-1}A)$ be the largest divided by the smallest positive eigenvalue of $M^{-1/2} A M^{-1/2}$, which excludes the zero eigenvalue of the Neumann problem. Lee and Min proved $\kappa_+(M^{-1}A) = O(h^{-1})$ for $r = ch^2$, with $c > 0$ a constant, when $\Omega$ is a rectangle tiled by the control cells, which does not have a smooth boundary. Their experiments in three dimensions do not support the analogous statement there.
Problem. For every such $\Omega$, are there constants $c_\Omega, K_\Omega, h_\Omega > 0$ such that, for every $0 < h < h_\Omega$, the ratio $r = c_\Omega h^2$ is less than $1$, the pivots $e_p$ are positive, and $\kappa_+(M^{-1}A) \le K_\Omega h^{-1}$?
Source: Park, Kim, Jung, Lee and Min (2018), Section 3, for their MILU–ILU mixture; restated as the RILU conjecture by Lee and Min (2021), Section 3. The explicit constants and the positivity of the pivots are OpenProblemsInNLA's.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Keywords
- condition numbers, incomplete factorizations, OpenProblemsInNLA, preconditioners
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{park2018,
author = {Park, Yesom and Kim, Jeongho and Jung, Jinwook and Lee, Euntaek and Min, Chohong},
title = {An efficient {MILU} preconditioning for solving the {2D} {P}oisson equation with {N}eumann boundary condition},
journal = {J. Comput. Phys.},
volume = {356},
year = {2018},
pages = {115--126},
doi = {10.1016/j.jcp.2017.11.028},
}
@article{lee2021,
author = {Lee, Byungjoon and Min, Chohong},
title = {Optimal preconditioners on solving the {P}oisson equation with {N}eumann boundary conditions},
journal = {J. Comput. Phys.},
volume = {433},
year = {2021},
pages = {110189},
doi = {10.1016/j.jcp.2021.110189},
}
@misc{openproblemsinnla-ie24,
author = {Townsend, Alex},
title = {{IE-24}: RILU conditioning for the Neumann problem on smooth planar domains},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/linear-systems-and-elimination/IE-24/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00191, "RILU conditioning for the Neumann problem on smooth planar domains", clopen.solutions.