2026/07/29 by Shaoliang Yang, Jun Wang, Yunsheng Wang
Computer Science · Mathematics · #cs.CE #cs.NA #math.NA
31 pages; 8 main figures and 4 supplementary figures. Code: https://github.com/nbbllxx0/solver-admissibility-and-guarded-floor-selection
arxiv created 2026/07/29 · arxiv updated 2026/07/30
In a matrix-free geometric-multigrid FGMRES solver for three-dimensional SIMP topology optimization, a reported converged solve is not always a converged solve. On four of 102 held-out states, the projected residual used for stopping falls below 10-6 while a recomputed true residual is 1.35 to 49.5 times the tolerance; in an unguarded optimization trajectory, 22 of 40 state solves reach the iteration cap without raising an error. We formulate floor selection as a verified control problem: probe the frozen state at the original floor, use two residual features to choose the first attempted floor, and accept no solution until ||f-Ku||/||f||≤10-6 is recomputed. The two-feature rule matches 98 of 102 reference classifications; the residual guard detects the four missed escalations, and all 102 selected solves satisfy the tolerance. Relative to always using a 10-3 floor, the policy preserves the original operator on 24 admissible states and avoids mean compliance and gradient changes of 31.0% and 0.340 on those severe random states, and 0.48% and 0.008 on seven optimized designs, at 2.5 times the mean wall time. In a 12-state subset of the held-out states, eight still require escalation at the conventional floor 10-6. In a nine-state control with the preconditioner's adaptive components disabled, every failure is visible and no false acceptance occurs, tying the stopping-estimate drift to the iterate-dependent preconditioner. The recomputed residual is the correctness safeguard; the probe and floor ladder govern an implementation-specific cost-fidelity tradeoff.