[petsc-users] Why use MATMPIBAIJ?

[Barry Smith]

> On Jan 13, 2016, at 9:57 PM, Justin Chang <jychang48 at gmail.com> wrote:
> 
> Hi all,
> 
> 1) I am guessing MATMPIBAIJ could theoretically have better performance than simply using MATMPIAIJ. Why is that? Is it similar to the reasoning that block (dense) matrix-vector multiply is "faster" than simple matrix-vector?

  See for example table 1 in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.38.7668&rep=rep1&type=pdf

> 
> 2) I am looking through the manual and online documentation and it seems the term "block" used everywhere. In the section on "block matrices" (3.1.3 of the manual), it refers to field splitting, where you could either have a monolithic matrix or a nested matrix. Does that concept have anything to do with MATMPIBAIJ? 

   Unfortunately the numerical analysis literature uses the term block in multiple ways. For small blocks, sometimes called "point-block" with BAIJ and for very large blocks (where the blocks are sparse themselves). I used fieldsplit for big sparse blocks to try to avoid confusion in PETSc. 
> 
> It makes sense to me that one could create a BAIJ where if you have 5 dofs of the same type of physics (e.g., five different primary species of a geochemical reaction) per grid point, you could create a block size of 5. And if you have different physics (e.g., velocity and pressure) you would ideally want to separate them out (i.e., nested matrices) for better preconditioning.

   Sometimes you put them together with BAIJ and sometimes you keep them separate with nested matrices.

> 
> Thanks,
> Justin