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I am using the METIS to partition a matrix and then using domain decomposition to solve the subdomains in parallel using the Restricted Additive Schwarz method.

I am currently trying to solve some SPD matrices. Using METIS' METIS_PartGraphKway(...), I partition the matrix. I think this function minimizes edge cuts. I am sorry, my knowledge on graph partitioning is limited. So my question is, after the partition, is the SPD-ness of each of the sub-domain matrix preserved ?

For simple cases of Laplacian 2d or other matrices such as bcsstk10 , bcsstk16 from the Matrix market which have more or less quite regular structure and non-zero entries near the diagonals, this property seems to hold, but for slightly more irregular matrices such as bcsstk38 (which is still SPD) it seems that the local solver does not converge.

Thank you.

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  • $\begingroup$ Sorry, I just realized that this question is stupid. The property is preserved because any partition has to exchange both column and corresponding row and it does not affect the eigenvalue as well. Should I delete this question ? What are the rules ? $\endgroup$ – Mathnoob May 17 '18 at 23:14
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The block SPD-ness is preserved because any partitioning does not change the SPD property of the matrix as it would not alter the eigenvalues as partitioning would involve row and column swaps which would only re-arrange the matrix but not change the fact that the eigenvalues of its block be greater than 0.

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