[petsc-users] Customizeing MatSetValuesBlocked(...)

[Jed Brown]

On Thu, Aug 9, 2012 at 11:28 AM, Jinquan Zhong <jzhong at scsolutions.com>wrote:

>    Hope this is not too technical. J****
>
> ** **
>
> **Jed, A’=[A^-1 U B], not the transpose of A.****
>
> ** **
>
> I understand that it's not the transpose. I still don't have a clue what
> the notation [A^{-1} U B] means. I would think it means some block
> decomposed thing, but I don't think you mean just adding columns or taking
> a product of matrices. Using some standard mathematical notation would help.
> ****
>
> ** **
>
> >> A’=[A^-1 U B] means matrix A’ consists all elements from A^-1, the
> inverted A, and all elements from B.  One simple equation is ****
>
> ** **
>
>            K=[Kst] =A’ = ASSEMBLE_ij (A^-1_ij) + ASSEMBLE_i’j’(B_i’j’),
>  here (s,t), (i,j) and (i’,j’) = different index sets****
>
> ** **
>
> i.e., entry K_st= A^-1_st+ B_st, ****
>
>  = 0 when A^-1_st=0 and B_st=0****
>
> =  A^-1_st when B_st =0****
>
> =  B_st when A^-1_st =0
>

Oh, so U meant "union" (in some funny sense). We would often write things
like

K = \sum_i R_i^T A_i^{-1} R_i + B

where R_i is a restriction to "part" i. The great thing about mathematical
notation is that we can communicate precisely what we mean. In this case,
we would have been able to ask how the restriction operators R_i are
connected (e.g. how much they overlap).


> ****
>
>
> ****
>
>    We need to solve for A’*x=b. A is dense matrix.****
>
>  ** **
>
> Where does A come from? Why is it dense?****
>
> ** **
>
> >> A comes from wave propagation.  It is dense since it denotes a function
> of Green function.
>
What is the kernel? Is it a pure integral operator (a discretized first
kind Fredholm integral equation), identity plus a compact integral operator
(a second kind Fredholm integral operator), or something else? This matters
a great deal.


>  Is A^{-1} an operator on some subdomain? Are you trying to implement a
> substructuring algorithm? What is B physically?****
>
> ** **
>
> >> NO.  A^{-1} denotes the inverted A.  B is a sparse matrix of much
> larger order.
>

What is B physically? In what way is it coupled to A?

What does QA stand for? Can you explain why B needs the entries of A^{-1}?**
> **
>
> ** **
>
> >> QA means quality assurance.  It is a procedure to ensure product
> quality.  In Eq. ****
>
> ** **
>
> K=A’=ASSEMBLE_ij (A^-1_ij)+ ASSEMBLE_i’j’(B_i’j’)****
>
>                 ****
>
> Entry B_i’j’ and A^-1_ij may or may not locate at the same row and col.
> That why we need explicitly each entry in B_i’j’ and A^-1_ij to assemble
> K.  The big picture is that K is the final sparse matrix we need to solve
> K*x=A’*x=b.  However, K indexed by (s,t) needs to be constructed in terms
> of dense matrix A and sparse matrix B using index sets (i,j) and (i’,j’).
> ****
>

It is very likely that you can (and should) solve K x = b without computing
explicit entries for K. You would apply its action by some other procedure,
likely not involving explicit construction of the A_i^{-1} matrices either.
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