<div dir="ltr"><div class="gmail_extra"><div class="gmail_quote">On Fri, Nov 4, 2016 at 10:44 AM, Florian Lindner <span dir="ltr"><<a href="mailto:mailinglists@xgm.de" target="_blank">mailinglists@xgm.de</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Hello,<br>
<br>
I have a matrix C that is the result of an RBF interpolation.<br>
<br>
It is constructured like that:<br>
<br>
c_ij = phi( |x_i - x_j| )<br>
<br>
x are supporting points. phi is the radial basis functions, here it is a Gaussian: phi(r) = exp( -(1.5*r)^2 ).<br>
<br>
The system is augmented by a global polynomial, which results in the first 3 rows being dense.<br>
<br>
The matrix is symmetric and of size 2309. It is also sparse since the basis functions are decaying very rapidly. A<br>
picture of the sparsity pattern I have uploaded at [1]<br>
<br>
I am having big trouble solving the system.<br>
<br>
Using default settings on 10 procs I'm not achieving convergence after 40k iterations:<br>
<br>
40000 KSP preconditioned resid norm 2.525546749843e+01 true resid norm 5.470906445602e+01 ||r(i)||/||b|| 3.122197420093e-02<br>
<br>
I have uploaded a small python script using petsc4py that simply loads the matrix and the rhs and tries to solve it. It<br>
is bundled with the matrix and rhs and available at [2] (650 KB). Some paths in the Python source need to be adapted.<br>
<br>
The condition number using -pc_type svd -pc_svd_monitor did not work, due to incompatible matrix formats.<br>
<br>
The condition number estimate using -pc_type none -ksp_type gmres -ksp_monitor_singular_value -ksp_gmres_restart 1000<br>
has been running for over half an hour now, its current line is:<br>
<br>
5655 KSP preconditioned resid norm 9.481821216790e-02 true resid norm 9.481821212512e-02 ||r(i)||/||b|| 5.411190635748e-05<br>
5655 KSP Residual norm 9.481821216790e-02 % max 2.853532248325e+02 min 1.073886912921e-06 max/min 2.657199947212e+08<br>
<br>
The Eigenvalues, computed with -ksp_compute_eigenvalues -ksp_gmres_restart 1000 -pc_type none did not converge after 40k<br>
iterations, bit they eigenvalues were printed. They are all real, none is zero, but some are pretty close. I've pasted<br>
them at the end of the mail.<br>
<br>
I tried a bunch of different PCs, but I'm somehow helpless on how to analyse that problem more systematically.<br></blockquote><div><br></div><div>Small block RASM is a good preconditioner. Here is a lengthy explanation</div><div><br></div><div>  <a href="https://arxiv.org/abs/0909.5413">https://arxiv.org/abs/0909.5413</a></div><div><br></div><div>  Thanks,</div><div><br></div><div>     Matt</div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
I know that RBF interpolation tends to produce badly conditioned matrices, but this seems to be (almost) singular. I<br>
have scrutinized my algorithms and in most cases it works just fine, converges and the output results are sane.<br>
<br>
I am grateful for any help!<br>
<br>
Best,<br>
Florian<br>
<br>
<br>
[1] <a href="http://xgm.de/upload/sparsity.png" rel="noreferrer" target="_blank">http://xgm.de/upload/sparsity.<wbr>png</a><br>
[2] <a href="http://xgm.de/upload/RBF.tar.bz2" rel="noreferrer" target="_blank">http://xgm.de/upload/RBF.tar.<wbr>bz2</a><br>
<br>
Eigenvalues:<br>
<br>
Iteratively computed eigenvalues<br>
-276.515 + 0.i<br>
-276.348 + 0.i<br>
-43.5577 + 0.i<br>
3.27753e-09 + 0.i<br>
5.09352e-06 + 0.i<br>
1.56431e-05 + 0.i<br>
2.79046e-05 + 0.i<br>
0.000145571 + 0.i<br>
0.000184333 + 0.i<br>
0.000283665 + 0.i<br>
0.000395128 + 0.i<br>
0.000488579 + 0.i<br>
0.000568982 + 0.i<br>
0.000704054 + 0.i<br>
0.000864538 + 0.i<br>
0.00101309 + 0.i<br>
0.00119913 + 0.i<br>
0.00142954 + 0.i<br>
0.00156907 + 0.i<br>
0.00178931 + 0.i<br>
0.00200295 + 0.i<br>
0.00223425 + 0.i<br>
0.00246876 + 0.i<br>
0.00276749 + 0.i<br>
0.00303141 + 0.i<br>
0.00326647 + 0.i<br>
0.00346721 + 0.i<br>
0.00437335 + 0.i<br>
0.0047525 + 0.i<br>
0.0049999 + 0.i<br>
0.0051701 + 0.i<br>
0.00556462 + 0.i<br>
0.00605938 + 0.i<br>
0.00628264 + 0.i<br>
0.00697463 + 0.i<br>
0.00756472 + 0.i<br>
0.00783031 + 0.i<br>
0.00822392 + 0.i<br>
0.00868428 + 0.i<br>
0.00928107 + 0.i<br>
0.00954981 + 0.i<br>
0.0104767 + 0.i<br>
0.0108791 + 0.i<br>
0.0111159 + 0.i<br>
0.0119535 + 0.i<br>
0.0125402 + 0.i<br>
0.0131083 + 0.i<br>
0.0135975 + 0.i<br>
0.0142472 + 0.i<br>
0.0148179 + 0.i<br>
0.0156007 + 0.i<br>
0.0160567 + 0.i<br>
0.0169899 + 0.i<br>
0.0174895 + 0.i<br>
0.0182683 + 0.i<br>
0.0189543 + 0.i<br>
0.0196781 + 0.i<br>
0.0206413 + 0.i<br>
0.0212257 + 0.i<br>
0.0220118 + 0.i<br>
0.0229829 + 0.i<br>
0.023606 + 0.i<br>
0.0244077 + 0.i<br>
0.0253428 + 0.i<br>
0.0263089 + 0.i<br>
0.0269905 + 0.i<br>
0.0278572 + 0.i<br>
0.028664 + 0.i<br>
0.0293435 + 0.i<br>
0.0306571 + 0.i<br>
0.0315797 + 0.i<br>
0.0320843 + 0.i<br>
0.0334732 + 0.i<br>
0.0344048 + 0.i<br>
0.0349148 + 0.i<br>
0.0361261 + 0.i<br>
0.037176 + 0.i<br>
0.0382401 + 0.i<br>
0.0390156 + 0.i<br>
0.0404156 + 0.i<br>
0.0413084 + 0.i<br>
0.0421763 + 0.i<br>
0.0434521 + 0.i<br>
0.0452247 + 0.i<br>
0.0460417 + 0.i<br>
0.0467954 + 0.i<br>
0.0479258 + 0.i<br>
0.0489899 + 0.i<br>
0.0506182 + 0.i<br>
0.0512171 + 0.i<br>
0.0529938 + 0.i<br>
0.0537732 + 0.i<br>
0.0549977 + 0.i<br>
0.056272 + 0.i<br>
0.0574546 + 0.i<br>
0.0585867 + 0.i<br>
0.0595438 + 0.i<br>
0.0614507 + 0.i<br>
0.0625339 + 0.i<br>
0.0634786 + 0.i<br>
0.0651556 + 0.i<br>
0.066393 + 0.i<br>
0.067585 + 0.i<br>
0.0701521 + 0.i<br>
0.0709748 + 0.i<br>
0.0725257 + 0.i<br>
0.073676 + 0.i<br>
0.0739372 + 0.i<br>
0.0758159 + 0.i<br>
0.078728 + 0.i<br>
0.0803127 + 0.i<br>
0.0813565 + 0.i<br>
0.0819371 + 0.i<br>
0.0824692 + 0.i<br>
0.0847166 + 0.i<br>
0.0878818 + 0.i<br>
0.0901822 + 0.i<br>
0.0906192 + 0.i<br>
0.0912192 + 0.i<br>
0.0921386 + 0.i<br>
0.0964252 + 0.i<br>
0.0978271 + 0.i<br>
0.0991256 + 0.i<br>
0.100197 + 0.i<br>
0.101757 + 0.i<br>
0.103163 + 0.i<br>
0.104427 + 0.i<br>
0.105316 + 0.i<br>
0.108255 + 0.i<br>
0.110149 + 0.i<br>
0.111963 + 0.i<br>
0.113533 + 0.i<br>
0.114857 + 0.i<br>
0.11807 + 0.i<br>
0.119241 + 0.i<br>
0.121196 + 0.i<br>
0.124063 + 0.i<br>
0.124824 + 0.i<br>
0.125867 + 0.i<br>
0.129153 + 0.i<br>
0.129692 + 0.i<br>
0.130804 + 0.i<br>
0.13186 + 0.i<br>
0.136671 + 0.i<br>
0.137159 + 0.i<br>
0.139948 + 0.i<br>
0.140338 + 0.i<br>
0.143836 + 0.i<br>
0.145398 + 0.i<br>
0.146868 + 0.i<br>
0.14813 + 0.i<br>
0.149113 + 0.i<br>
0.152292 + 0.i<br>
0.154569 + 0.i<br>
0.1592 + 0.i<br>
0.159868 + 0.i<br>
0.16113 + 0.i<br>
0.163854 + 0.i<br>
0.16612 + 0.i<br>
0.167395 + 0.i<br>
0.169089 + 0.i<br>
0.171137 + 0.i<br>
0.173693 + 0.i<br>
0.176847 + 0.i<br>
0.177794 + 0.i<br>
0.178951 + 0.i<br>
0.179342 + 0.i<br>
0.185936 + 0.i<br>
0.18772 + 0.i<br>
0.189272 + 0.i<br>
0.191293 + 0.i<br>
0.193085 + 0.i<br>
0.195212 + 0.i<br>
0.197571 + 0.i<br>
0.199457 + 0.i<br>
0.201105 + 0.i<br>
0.204097 + 0.i<br>
0.207915 + 0.i<br>
0.208748 + 0.i<br>
0.212964 + 0.i<br>
0.214298 + 0.i<br>
0.217664 + 0.i<br>
0.218679 + 0.i<br>
0.220846 + 0.i<br>
0.222277 + 0.i<br>
0.226707 + 0.i<br>
0.228239 + 0.i<br>
0.229806 + 0.i<br>
0.231871 + 0.i<br>
0.235329 + 0.i<br>
0.237857 + 0.i<br>
0.241059 + 0.i<br>
0.242546 + 0.i<br>
0.244337 + 0.i<br>
0.246314 + 0.i<br>
0.248005 + 0.i<br>
0.250223 + 0.i<br>
0.252628 + 0.i<br>
0.255812 + 0.i<br>
0.256945 + 0.i<br>
0.258131 + 0.i<br>
0.266436 + 0.i<br>
0.269103 + 0.i<br>
0.270135 + 0.i<br>
0.271536 + 0.i<br>
0.273719 + 0.i<br>
0.279941 + 0.i<br>
0.28122 + 0.i<br>
0.281853 + 0.i<br>
0.285513 + 0.i<br>
0.28672 + 0.i<br>
0.28773 + 0.i<br>
0.291966 + 0.i<br>
0.296344 + 0.i<br>
0.299661 + 0.i<br>
0.303254 + 0.i<br>
0.304844 + 0.i<br>
0.306891 + 0.i<br>
0.309339 + 0.i<br>
0.311826 + 0.i<br>
0.313826 + 0.i<br>
0.315959 + 0.i<br>
0.31859 + 0.i<br>
0.319327 + 0.i<br>
0.322979 + 0.i<br>
0.329764 + 0.i<br>
0.330846 + 0.i<br>
0.335162 + 0.i<br>
0.336212 + 0.i<br>
0.338197 + 0.i<br>
0.34279 + 0.i<br>
0.345071 + 0.i<br>
0.34884 + 0.i<br>
0.34954 + 0.i<br>
0.354234 + 0.i<br>
0.355576 + 0.i<br>
0.359178 + 0.i<br>
0.361957 + 0.i<br>
0.364873 + 0.i<br>
0.367973 + 0.i<br>
0.370043 + 0.i<br>
0.3732 + 0.i<br>
0.375047 + 0.i<br>
0.376219 + 0.i<br>
0.38884 + 0.i<br>
0.390239 + 0.i<br>
0.392451 + 0.i<br>
0.394641 + 0.i<br>
0.39603 + 0.i<br>
0.40011 + 0.i<br>
0.402863 + 0.i<br>
0.40477 + 0.i<br>
0.407166 + 0.i<br>
0.409253 + 0.i<br>
0.41114 + 0.i<br>
0.417392 + 0.i<br>
0.419475 + 0.i<br>
0.421597 + 0.i<br>
0.42226 + 0.i<br>
0.425458 + 0.i<br>
0.432817 + 0.i<br>
0.434498 + 0.i<br>
0.437448 + 0.i<br>
0.440073 + 0.i<br>
0.441784 + 0.i<br>
0.44471 + 0.i<br>
0.449445 + 0.i<br>
0.450416 + 0.i<br>
0.454865 + 0.i<br>
0.456899 + 0.i<br>
0.459119 + 0.i<br>
0.464975 + 0.i<br>
0.466739 + 0.i<br>
0.470461 + 0.i<br>
0.473138 + 0.i<br>
0.473366 + 0.i<br>
0.482849 + 0.i<br>
0.487335 + 0.i<br>
0.487996 + 0.i<br>
0.490971 + 0.i<br>
0.492149 + 0.i<br>
0.498317 + 0.i<br>
0.49924 + 0.i<br>
0.501193 + 0.i<br>
0.502928 + 0.i<br>
0.505282 + 0.i<br>
0.507326 + 0.i<br>
0.5094 + 0.i<br>
0.510887 + 0.i<br>
0.52977 + 0.i<br>
0.529973 + 0.i<br>
0.534443 + 0.i<br>
0.536762 + 0.i<br>
0.539341 + 0.i<br>
0.542418 + 0.i<br>
0.543103 + 0.i<br>
0.545815 + 0.i<br>
0.547044 + 0.i<br>
0.548941 + 0.i<br>
0.553764 + 0.i<br>
0.558755 + 0.i<br>
0.55932 + 0.i<br>
0.566552 + 0.i<br>
0.566763 + 0.i<br>
0.571683 + 0.i<br>
0.571809 + 0.i<br>
0.587266 + 0.i<br>
0.58805 + 0.i<br>
0.588659 + 0.i<br>
0.58981 + 0.i<br>
0.593885 + 0.i<br>
0.598992 + 0.i<br>
0.60122 + 0.i<br>
0.602449 + 0.i<br>
0.604899 + 0.i<br>
0.607548 + 0.i<br>
0.609311 + 0.i<br>
0.611774 + 0.i<br>
0.621952 + 0.i<br>
0.623849 + 0.i<br>
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0.624863 + 0.i<br>
0.635872 + 0.i<br>
0.63847 + 0.i<br>
0.639046 + 0.i<br>
0.646496 + 0.i<br>
0.650915 + 0.i<br>
0.652575 + 0.i<br>
0.654514 + 0.i<br>
0.656047 + 0.i<br>
0.658775 + 0.i<br>
0.66205 + 0.i<br>
0.664198 + 0.i<br>
0.666337 + 0.i<br>
0.670879 + 0.i<br>
0.675989 + 0.i<br>
0.677792 + 0.i<br>
0.678938 + 0.i<br>
0.6845 + 0.i<br>
0.684931 + 0.i<br>
0.687205 + 0.i<br>
0.692041 + 0.i<br>
0.692177 + 0.i<br>
0.693601 + 0.i<br>
0.711189 + 0.i<br>
0.716514 + 0.i<br>
0.71922 + 0.i<br>
0.72388 + 0.i<br>
0.72405 + 0.i<br>
0.730072 + 0.i<br>
0.730319 + 0.i<br>
0.732633 + 0.i<br>
0.73299 + 0.i<br>
0.737779 + 0.i<br>
0.74204 + 0.i<br>
0.744661 + 0.i<br>
0.746251 + 0.i<br>
0.746947 + 0.i<br>
0.751408 + 0.i<br>
0.751786 + 0.i<br>
0.753859 + 0.i<br>
0.761784 + 0.i<br>
0.762905 + 0.i<br>
0.765961 + 0.i<br>
0.767479 + 0.i<br>
0.769471 + 0.i<br>
0.769529 + 0.i<br>
0.774108 + 0.i<br>
0.794173 + 0.i<br>
0.799031 + 0.i<br>
0.799111 + 0.i<br>
0.811928 + 0.i<br>
0.812351 + 0.i<br>
0.813483 + 0.i<br>
0.815524 + 0.i<br>
0.815749 + 0.i<br>
0.826079 + 0.i<br>
0.826832 + 0.i<br>
0.831323 + 0.i<br>
0.832443 + 0.i<br>
0.8415 + 0.i<br>
0.842963 + 0.i<br>
0.843567 + 0.i<br>
0.84377 + 0.i<br>
0.847571 + 0.i<br>
0.8493 + 0.i<br>
0.849347 + 0.i<br>
0.85882 + 0.i<br>
0.859196 + 0.i<br>
0.862321 + 0.i<br>
0.863661 + 0.i<br>
0.867476 + 0.i<br>
0.8677 + 0.i<br>
0.884245 + 0.i<br>
0.884266 + 0.i<br>
0.893719 + 0.i<br>
0.893886 + 0.i<br>
0.907204 + 0.i<br>
0.907405 + 0.i<br>
0.908615 + 0.i<br>
0.909567 + 0.i<br>
0.909714 + 0.i<br>
0.916938 + 0.i<br>
0.920716 + 0.i<br>
0.926377 + 0.i<br>
0.926776 + 0.i<br>
0.928557 + 0.i<br>
0.928572 + 0.i<br>
0.939078 + 0.i<br>
0.939392 + 0.i<br>
0.940033 + 0.i<br>
0.941661 + 0.i<br>
0.942081 + 0.i<br>
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7.93419 + 0.i<br>
8.58745 + 0.i<br>
8.58793 + 0.i<br>
8.65657 + 0.i<br>
8.65693 + 0.i<br>
8.65703 + 0.i<br>
8.65703 + 0.i<br>
8.76218 + 0.i<br>
8.76323 + 0.i<br>
8.76344 + 0.i<br>
8.76345 + 0.i<br>
8.76364 + 0.i<br>
8.76379 + 0.i<br>
8.84516 + 0.i<br>
8.84531 + 0.i<br>
8.84538 + 0.i<br>
8.84542 + 0.i<br>
8.84542 + 0.i<br>
8.84542 + 0.i<br>
9.96128 + 0.i<br>
10.0422 + 0.i<br>
10.3582 + 0.i<br>
11.1981 + 0.i<br>
11.5961 + 0.i<br>
11.5961 + 0.i<br>
11.5961 + 0.i<br>
11.5962 + 0.i<br>
11.5962 + 0.i<br>
11.5962 + 0.i<br>
11.5962 + 0.i<br>
11.5963 + 0.i<br>
16.048 + 0.i<br>
17.1999 + 0.i<br>
17.2057 + 0.i<br>
18.1546 + 0.i<br>
18.1548 + 0.i<br>
18.1548 + 0.i<br>
18.155 + 0.i<br>
18.155 + 0.i<br>
18.155 + 0.i<br>
53.4484 + 0.i<br>
285.167 + 0.i<br>
285.353 + 0.i<br>
</blockquote></div><br><br clear="all"><div><br></div>-- <br><div class="gmail_signature">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener</div>
</div></div>