The all-at-once bidiagonal time operator and C/F points

Let $d,N\ge 1$, let ${\mathrm{\Phi }}_i\in {\mathbb{R}}^{d\times d}$ for $1\le i\le N$, and let $u_0\in {\mathbb{R}}^d$ be fixed. The linear part of a fine propagator in Fine and coarse propagators on a common macro grid is represented by ${\mathrm{\Phi }}_i$. For forcing vectors $g_i\in {\mathbb{R}}^d$, write the fine recurrence as $u_i={\mathrm{\Phi }}_i{u}_{i-1}+g_i$. On the unknown vector $v=(v_1,\dots ,v_N)\in ({\mathbb{R}}^d{)}^N$, define $A_t$ by $(A_{t}v{)}_1=v_1$ and $(A_{t}v{)}_i=v_i-{\mathrm{\Phi }}_i{v}_{i-1}$ for $2\le i\le N$. With $b_1={\mathrm{\Phi }}_1{u}_0+g_1$ and $b_i=g_i$ for $i\ge 2$, the recurrence is exactly $A_{t}v=b$.

For a coarsening factor $m\ge 2$ dividing $N$, the coarse indices are $0,m,2m,\dots ,N$. Among unknowns, let $C=\{m,2m,\dots ,N\}$ and $F=\{1,\dots ,N\}\setminus C$. Reordering the unknown vector as $(v_F,v_C)$ gives $A_t=\left(\begin{array}{cc}{A}_{FF}& A_{FC}\\ A_{CF}& A_{CC}\end{array}\right)\mathrm{.}$ A practical coarse propagator ${\mathrm{\Psi }}_j$ on each interval is the linear subcase of the coarse maps $G_n$ in Fine and coarse propagators on a common macro grid; the exact coarse operator is determined by eliminating F-points and is not generally the same as this practical choice. This definition asserts no stability or convergence property.