Uniform or mildly regularization-dependent convergence constants

Prove convergence estimates for the alternating score--system iteration whose constants remain uniform in, or depend only mildly on, the Tikhonov regularization parameter lambda as lambda tends to zero.

Background

The factorial convergence theorem is valid for every fixed positive lambda, but its constants deteriorate rapidly as lambda decreases: the score stability constant is of order lambda{-2}, while the force and Volterra constants also diverge. Numerical experiments converge far faster than these worst-case bounds predict.

Closing this gap would be needed to make the estimates useful for selecting iteration counts and solver tolerances in the small-regularization regime.

References

Thus the theorem guarantees convergence, the experiments exhibit its speed, and closing the gap between the two, that is, proving convergence with constants uniform or mildly dependent on $\lambda$, is an open problem.

— Convergence of alternating score-system iterations for regularized Landau particle dynamics  (2610.25780 - Kim et al., 19 Oct 2026) in Section 3, Remark 3.7 ("lambda-scaling")

whether an evolving family $g\lambda_t$ solving eq:density-level-regularized-Coulomb satisfies it uniformly in $\lambda$ and $t$ is not known to us, so the rate eq:density-level-zero-lambda-rate is at present a criterion without a verified instance.

eq:density-level-regularized-Coulomb:

$\partial_tg^\lambda+\nabla\cdot \left(g^\lambda U[g^\lambda,S_\lambda(g^\lambdad v)]\right)=0 $

eq:density-level-zero-lambda-rate:

sup⁡0≤t≤TKL(gtλ∥ft)=OT(λ),sup⁡0≤t≤Tgtλ−ftL1=OT(λ).\sup_{0\le t\le T}KL(g_t^\lambda\Vert f_t)=O_T(\lambda), \qquad \sup_{0\le t\le T}{g_t^\lambda-f_t}_{L^1}=O_T(\sqrt\lambda).

— Convergence of alternating score-system iterations for regularized Landau particle dynamics  (2610.25780 - Kim et al., 19 Oct 2026) in Section 9.2, Remark 9.8 ("Conditional hypotheses")