4.3.1.2 Change of vector types

Obviously, we can apply addition/substraction and similar operations without communication on vectors of the same type.
The conversion of type II (distributed) vector into a type I (accumulated) vector requires communication :

$$\boxed{ \underline{{\ensuremath{\color{red}\mathfrak{w}}}}_i \;=\... ..._{i=1}^p A_i^T \underline{{\ensuremath{\color{green}{\sf r}}}}_i \enspace . }$$ (4.4)

The other direction, conversion of a type-I vector into a type-II vector is not unique. One possibility consists in dividing locally each vector component by its priority (number of subdomains a node belongs to), i.e.,

$$\boxed{ \underline{{\ensuremath{\color{green}{\sf r}}}}_i \;=\; R^{-1} \underline{{\ensuremath{\color{red}\mathfrak{w}}}}_i }$$ (4.5)

with $R$ defined in (4.4).