Finite recursion and work accounting for multilevel MGRIT

For one top-level call of Recursive multilevel MGRIT cycles on a finite admissible hierarchy with terminating primitive operations, the recursion terminates. Its visit counts satisfy $v_0=1$ and $v_{\mathrm{\ell }+1}={\nu }_{\mathrm{\ell }}{v}_{\mathrm{\ell }}$ for $0\le \mathrm{\ell }<L$.

Suppose every visit to noncoarsest level $\mathrm{\ell }$ has the same local call counts $n_{\mathrm{\Phi },\mathrm{\ell }},n_{\mathrm{\Psi },\mathrm{\ell }}$, excluding its recursive children. In an additive accounting model with nonnegative call costs $c_{\mathrm{\Phi },\mathrm{\ell }},c_{\mathrm{\Psi },\mathrm{\ell }}$, remaining local work ${\kappa }_{\mathrm{\ell }}$, and work $W_L$ per coarsest solve, its modeled work is $W={\sum }_{\mathrm{\ell }=0}^{L-1}{v}_{\mathrm{\ell }}(n_{\mathrm{\Phi },\mathrm{\ell }}{c}_{\mathrm{\Phi },\mathrm{\ell }}+n_{\mathrm{\Psi },\mathrm{\ell }}{c}_{\mathrm{\Psi },\mathrm{\ell }}+{\kappa }_{\mathrm{\ell }})+v_L{W}_L\mathrm{.}$ Each local F-, C- or FCF-relaxation has the literal fine-call counts $N_{\mathrm{\ell }}-N_{\mathrm{\ell }+1}-1$, $N_{\mathrm{\ell }+1}$ or $2(N_{\mathrm{\ell }}-N_{\mathrm{\ell }+1}-1)+N_{\mathrm{\ell }+1}$ respectively. Residual formation, repeated coarse residuals, restriction, interpolation and correction contribute all their actual calls and overhead; caching incurs no new call. If counts or costs vary by visit, replace each product by the sum of the corresponding visit charges. Outer stopping checks have additional costs.

Parallel execution must respect the order within each F-interval, between the successive recursive visits that update the same correction, and within each exact coarse forward solve, including communication and synchronization. The accounting model implies no measured speedup or multilevel contraction.