[petsc-users] Seeking help for analyzing non-convergence (KSP)

Fri Nov 4 10:44:01 CDT 2016

Hello,

I have a matrix C that is the result of an RBF interpolation.

It is constructured like that:

c_ij = phi( |x_i - x_j| )

x are supporting points. phi is the radial basis functions, here it is a Gaussian: phi(r) = exp( -(1.5*r)^2 ).

The system is augmented by a global polynomial, which results in the first 3 rows being dense.

The matrix is symmetric and of size 2309. It is also sparse since the basis functions are decaying very rapidly. A
picture of the sparsity pattern I have uploaded at [1]

I am having big trouble solving the system.

Using default settings on 10 procs I'm not achieving convergence after 40k iterations:

40000 KSP preconditioned resid norm 2.525546749843e+01 true resid norm 5.470906445602e+01 ||r(i)||/||b|| 3.122197420093e-02

I have uploaded a small python script using petsc4py that simply loads the matrix and the rhs and tries to solve it. It
is bundled with the matrix and rhs and available at [2] (650 KB). Some paths in the Python source need to be adapted.

The condition number using -pc_type svd -pc_svd_monitor did not work, due to incompatible matrix formats.

The condition number estimate using -pc_type none -ksp_type gmres -ksp_monitor_singular_value -ksp_gmres_restart 1000
has been running for over half an hour now, its current line is:

5655 KSP preconditioned resid norm 9.481821216790e-02 true resid norm 9.481821212512e-02 ||r(i)||/||b|| 5.411190635748e-05
5655 KSP Residual norm 9.481821216790e-02 % max 2.853532248325e+02 min 1.073886912921e-06 max/min 2.657199947212e+08

The Eigenvalues, computed with -ksp_compute_eigenvalues -ksp_gmres_restart 1000 -pc_type none did not converge after 40k
iterations, bit they eigenvalues were printed. They are all real, none is zero, but some are pretty close. I've pasted
them at the end of the mail.

I tried a bunch of different PCs, but I'm somehow helpless on how to analyse that problem more systematically.

I know that RBF interpolation tends to produce badly conditioned matrices, but this seems to be (almost) singular. I
have scrutinized my algorithms and in most cases it works just fine, converges and the output results are sane.

I am grateful for any help!

Best,
Florian

[1] http://xgm.de/upload/sparsity.png
[2] http://xgm.de/upload/RBF.tar.bz2

Eigenvalues:

Iteratively computed eigenvalues
-276.515 + 0.i
-276.348 + 0.i
-43.5577 + 0.i
3.27753e-09 + 0.i
5.09352e-06 + 0.i
1.56431e-05 + 0.i
2.79046e-05 + 0.i
0.000145571 + 0.i
0.000184333 + 0.i
0.000283665 + 0.i
0.000395128 + 0.i
0.000488579 + 0.i
0.000568982 + 0.i
0.000704054 + 0.i
0.000864538 + 0.i
0.00101309 + 0.i
0.00119913 + 0.i
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0.00223425 + 0.i
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0.00276749 + 0.i
0.00303141 + 0.i
0.00326647 + 0.i
0.00346721 + 0.i
0.00437335 + 0.i
0.0047525 + 0.i
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