Recursive multilevel MGRIT cycles

Input: levels $\mathrm{\ell }=0,\dots ,L$ with $L\ge 1$; fine-grid systems $A_{\mathrm{\ell }}{u}_{\mathrm{\ell }}=b_{\mathrm{\ell }}$ of the form in The all-at-once bidiagonal time operator and C/F points; coarsening factors $m_{\mathrm{\ell }}\ge 2$ dividing $N_{\mathrm{\ell }}$ with $N_{\mathrm{\ell }+1}=N_{\mathrm{\ell }}\mathrm{/}{m}_{\mathrm{\ell }}$; transfers $P_{\mathrm{\ell }},R_{\mathrm{\ell }}$ from the F/C construction in A declared two-level MGRIT coarse-correction cycle; and pre/post-relaxations selected from its F- and FCF-relaxations. At each level $\mathrm{\ell }<L$, form the practical coarse matrix $A_{\mathrm{\ell }+1}$ with identity diagonal blocks and subdiagonal blocks $-{\mathrm{\Psi }}_{\mathrm{\ell },j}$ as in A declared two-level MGRIT coarse-correction cycle. Each noncoarsest level has a positive integer recursive visit count ${\nu }_{\mathrm{\ell }}$. Also supply an initial fine-level iterate, a chosen norm on the fine-level residual space as in A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, outer cycle budget $J_{\mathrm{o}\mathrm{u}\mathrm{t}}\ge 0$, and residual tolerance $\epsilon \ge 0$. The hierarchy must provide a nonsingular coarse system at every level and an exact solve on level $L$. V-cycles use ${\nu }_{\mathrm{\ell }}=1$ and W-cycles use ${\nu }_{\mathrm{\ell }}=2$. This algorithm gives no existence or convergence claim for a hierarchy that fails these input conditions.

For a recursive call at level $\mathrm{\ell }$, perform the following operations on an initial guess $u_{\mathrm{\ell }}$:

1. If $\mathrm{\ell }=L$, solve $A_L{u}_L=b_L$ exactly by forward substitution and return.
2. Apply the selected pre-relaxation to $A_{\mathrm{\ell }}{u}_{\mathrm{\ell }}=b_{\mathrm{\ell }}$; compute $r_{\mathrm{\ell }}=b_{\mathrm{\ell }}-A_{\mathrm{\ell }}{u}_{\mathrm{\ell }}$ and restrict it to $r_{\mathrm{\ell }+1}=R_{\mathrm{\ell }}{r}_{\mathrm{\ell }}$.
3. Set the coarse correction $z_{\mathrm{\ell }+1}=0$. Repeat ${\nu }_{\mathrm{\ell }}$ times: form ${\rho }_{\mathrm{\ell }+1}=r_{\mathrm{\ell }+1}-A_{\mathrm{\ell }+1}{z}_{\mathrm{\ell }+1}$; recursively apply the level $\mathrm{\ell }+1$ cycle to $A_{\mathrm{\ell }+1}{\delta }_{\mathrm{\ell }+1}={\rho }_{\mathrm{\ell }+1}$ from zero initial correction; then set $z_{\mathrm{\ell }+1}\leftarrow z_{\mathrm{\ell }+1}+{\delta }_{\mathrm{\ell }+1}$.
4. Correct $u_{\mathrm{\ell }}\leftarrow u_{\mathrm{\ell }}+P_{\mathrm{\ell }}{z}_{\mathrm{\ell }+1}$ and apply the selected post-relaxation.

Return the updated fine-level vector when the call completes. At level $0$, check the residual before the first outer cycle and after every completed cycle; return status converged only when the residual tolerance is met. After $J_{\mathrm{o}\mathrm{u}\mathrm{t}}$ cycles without a verified residual certificate, return the last iterate with status budget-exhausted. Inexact level solves contribute explicit update defects; no such defect is hidden in the exact-arithmetic cycle.

No two-grid contraction is transferred to this recursive cycle; convergence requires a bound for its actual full recursive error operator.