AI & ML
Compensated Splitting For Generalized Lyapunov Equations
arXiv cs.NA0 views
arXiv:2606.22245v2 Announce Type: replace Abstract: The generalized Lyapunov equation has a natural splitting that leads to the standard fixed-point iteration (sFPI) for its numerical solution. sFPI is convergent, for an arbitrarily given initial guess, if and only if the spectral radius of the associated linear operator is smaller than $1$. This means that sFPI may diverge if the spectral radius is $1$ or bigger. In this paper, we propose a compensated splitting scheme that aims to reduce the spectral radius so that the resulting compensated fixed-point iteration (cFPI) has a better convergence property, namely, cFPI may still converge even if sFPI does not or cFPI converges faster than sFPI does when the latter is also convergent. Numerical results are presented to demonstrate the superiority of cFPI to sFPI.
Source：https://arxiv.org/abs/2606.22245PreviousClassification of Double Saddle-Point SystemsNextSharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data
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