Finite-budget recursive nonlinear FAS V/W cycle

Fix a finite hierarchy of levels $j=0,1,\dots ,J$ with $J\ge 1$. At level $j$, let $F_j:U_j\to {\mathbb{R}}^{n_j}$ be defined on an open $U_j\subseteq {\mathbb{R}}^{n_j}$, let $f_j\in {\mathbb{R}}^{n_j}$, and provide the linear transfers $I_j$, $R_j$, and $P_j$ with the dimensions in Finite-budget nonlinear FAS two-grid cycle. For each $j>0$, provide the two diagonal smoother matrices, damping factors, and finite pre/post sweep counts from that algorithm. Choose a finite base solver at level $0$ with a declared total operation budget $B_0$ and statuses exact-root, inexact-root, branch-failure, domain-failure, and budget-exhausted. Its exact-root and inexact-root statuses must report $\mathrm{\parallel }{F}_0(v)-f_0{\mathrm{\parallel }}_2$ as $0$ and a positive number, respectively. At each noncoarsest level choose ${\gamma }_j\in \{1,2\}$, representing V and W recursion, respectively. For every $j>0$, the recursive level-$(j-1)$ solver call in the two-grid procedure is metered against its declared finite coarse-call budget; if it exceeds that budget, it returns budget-exhausted with its last valid iterate.

The base-solver contract also requires it to return its initial value unchanged with status exact-root whenever that value already satisfies $F_0(z)=f_0$.

Define $\mathrm{Cycle}(0,u,f_0)$ to call the base solver. For $j>0$, provide the level-$(j-1)$ cycle as the coarse-solver interface to Finite-budget nonlinear FAS two-grid cycle: its reported residual is measured as $\mathrm{\parallel }{F}_{j-1}(v)-g{\mathrm{\parallel }}_2$, where $v$ is the returned value, with exact-root only at residual zero and inexact-root only at positive residual. The two-grid procedure makes ${\gamma }_j$ calls with the same coarse right side $g$ and the current coarse iterate $z$. Return the two-grid output, its status, its level residual, and the finite diagnostic list of coarse-call residuals. Any failure status propagates immediately with the last valid fine-level iterate. Each recursive call lowers $j$, and all within-level loops have the finite budgets required by Finite-budget nonlinear FAS two-grid cycle.

The V-cycle is the choice ${\gamma }_j=1$ on each noncoarsest level; the W-cycle is ${\gamma }_j=2$. This definition does not assert that the returned iterate solves the nonlinear equations, that the cycles converge, or that their work is linear in the number of fine unknowns.