Hello,

I have an application built on top of the Moose framework, and I'm trying to debug a solve that is not converging. My linear solve converges very nicely. However, my non-linear solve does not, and the problem appears to be in the line search. Reading the PetSc FAQ, I see that the most common cause of poor line searches are bad Jacobians. However, I'm using a finite-differenced Jacobian; if I run -snes_type=test, I get "norm of matrix ratios" < 1e-15. Thus in this case the Jacobian should be accurate. I'm wondering then if my problem might be these (taken from the FAQ page):

The matrix is very ill-conditioned. Check the condition number.
- Try to improve it by choosing the relative scaling of components/boundary conditions.
- Try -ksp_diagonal_scale -ksp_diagonal_scale_fix.
- Perhaps change the formulation of the problem to produce more friendly algebraic equations.
The matrix is nonlinear (e.g. evaluated using finite differencing of a nonlinear function). Try different differencing parameters, ./configure --with-precision=__float128 --download-f2cblaslapack, check if it converges in "easier" parameter regimes.

I'm almost ashamed to share my condition number because I'm sure it must be absurdly high. Without applying -ksp_diagonal_scale and -ksp_diagonal_scale_fix, the condition number is around 1e25. When I do apply those two parameters, the condition number is reduced to 1e17. Even after scaling all my variable residuals so that they were all on the order of unity (a suggestion on the Moose list), I still have a condition number of 1e12. I have no experience with condition numbers, but knowing that perfect condition number is unity, 1e12 seems unacceptable. What's an acceptable upper limit on the condition number? Is it problem dependent? Having already tried scaling the individual variable residuals, I'm not exactly sure what my next method would be for trying to reduce the condition number.

I definitely have a nonlinear problem. Could I be having problems because I'm finite differencing non-linear residuals to form my Jacobian? I can see about using a different differencing parameter. I'm also going to consider trying quad precision. However, my hypothesis is that my condition number is the fundamental problem. Is that a reasonable hypothesis?

If it's useful, below is console output with -pc_type=svd

Time Step 1, time = 1e-10
                dt = 1e-10
    |residual|_2 of individual variables:
               potential:    8.12402e+07
               potentialliq: 0.000819748
               em:           49.206
               emliq:        3.08187e-11
               Arp:          2375.94

0 Nonlinear |R| = 8.124020e+07
      SVD: condition number 1.457087640207e+12, 0 of 851 singular values are (nearly) zero
      SVD: smallest singular values: 5.637144317564e-09 9.345415388433e-08 4.106132915572e-05 1.017339655185e-04 1.147649477723e-04
      SVD: largest singular values : 1.498505466947e+03 1.577560767570e+03 1.719172328193e+03 2.344218235296e+03 8.213813311188e+03
    0 KSP unpreconditioned resid norm 3.185019606208e+05 true resid norm 3.185019606208e+05 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 6.382886902896e-07 true resid norm 6.382761808414e-07 ||r(i)||/||b|| 2.003994511046e-12
Linear solve converged due to CONVERGED_RTOL iterations 1
      Line search: Using full step: fnorm 8.124020470169e+07 gnorm 1.097605946684e+01
    |residual|_2 of individual variables:
               potential:    8.60047
               potentialliq: 0.335436
               em:           2.26472
               emliq:        0.642578
               Arp:          6.39151

1 Nonlinear |R| = 1.097606e+01
      SVD: condition number 1.457473763066e+12, 0 of 851 singular values are (nearly) zero
      SVD: smallest singular values: 5.637185516434e-09 9.347128557672e-08 1.017339655587e-04 1.146760266781e-04 4.064422034774e-04
      SVD: largest singular values : 1.498505466944e+03 1.577544976882e+03 1.718956369043e+03 2.343692402876e+03 8.216049987736e+03
    0 KSP unpreconditioned resid norm 2.653715381459e+01 true resid norm 2.653715381459e+01 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 6.031179341420e-05 true resid norm 6.031183387732e-05 ||r(i)||/||b|| 2.272731819648e-06
Linear solve converged due to CONVERGED_RTOL iterations 1
      Line search: gnorm after quadratic fit 2.485190757827e+11
      Line search: Cubic step no good, shrinking lambda, current gnorm 2.632996340352e+10 lambda=5.0000000000000003e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 4.290675557416e+09 lambda=2.5000000000000001e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 4.332980055153e+08 lambda=1.2500000000000001e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.677118626669e+07 lambda=6.2500000000000003e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.024469780306e+05 lambda=3.1250000000000002e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 7.011543252988e+03 lambda=1.5625000000000001e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.750171277470e+03 lambda=7.8125000000000004e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 3.486970625406e+02 lambda=3.4794637057251714e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 7.830624839582e+01 lambda=1.5977866967992950e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 2.147529381328e+01 lambda=6.8049915671999093e-05
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.138950943123e+01 lambda=1.7575203052774536e-05
      Line search: Cubically determined step, current gnorm 1.095195976135e+01 lambda=1.7575203052774537e-06
    |residual|_2 of individual variables:
               potential:    8.59984
               potentialliq: 0.395753
               em:           2.26492
               emliq:        0.642578
               Arp:          6.34735

2 Nonlinear |R| = 1.095196e+01
      SVD: condition number 1.457459214030e+12, 0 of 851 singular values are (nearly) zero
      SVD: smallest singular values: 5.637295371943e-09 9.347057884198e-08 1.017339655949e-04 1.146738253493e-04 4.064421554132e-04
      SVD: largest singular values : 1.498505466946e+03 1.577543742603e+03 1.718948052797e+03 2.343672206864e+03 8.216128082047e+03
    0 KSP unpreconditioned resid norm 2.653244141805e+01 true resid norm 2.653244141805e+01 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 4.480869560737e-05 true resid norm 4.480686665183e-05 ||r(i)||/||b|| 1.688757771886e-06
Linear solve converged due to CONVERGED_RTOL iterations 1
      Line search: gnorm after quadratic fit 2.481752147885e+11
      Line search: Cubic step no good, shrinking lambda, current gnorm 2.631959989642e+10 lambda=5.0000000000000003e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 4.289110800463e+09 lambda=2.5000000000000001e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 4.332043942482e+08 lambda=1.2500000000000001e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.677933337886e+07 lambda=6.2500000000000003e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.027980597206e+05 lambda=3.1250000000000002e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 7.054113639063e+03 lambda=1.5625000000000001e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.771258630210e+03 lambda=7.8125000000000004e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 3.517070127496e+02 lambda=3.4519087020105563e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 7.844350966118e+01 lambda=1.5664532891249369e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 2.114833995101e+01 lambda=6.5367917100814859e-05
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.144636844292e+01 lambda=1.6044984646715980e-05
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.095640770627e+01 lambda=1.6044984646715980e-06
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.095196729511e+01 lambda=1.6044984646715980e-07
      Line search: Cubically determined step, current gnorm 1.095195451041e+01 lambda=2.3994454223607641e-08
    |residual|_2 of individual variables:
               potential:    8.59983
               potentialliq: 0.396107
               em:           2.26492
               emliq:        0.642578
               Arp:          6.34733

3 Nonlinear |R| = 1.095195e+01
      SVD: condition number 1.457474387942e+12, 0 of 851 singular values are (nearly) zero
      SVD: smallest singular values: 5.637237413167e-09 9.347057670885e-08 1.017339654798e-04 1.146737961973e-04 4.064420550524e-04
      SVD: largest singular values : 1.498505466946e+03 1.577543716995e+03 1.718947893048e+03 2.343671853830e+03 8.216129148438e+03
    0 KSP unpreconditioned resid norm 2.653237816527e+01 true resid norm 2.653237816527e+01 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 8.525213442515e-05 true resid norm 8.527696332776e-05 ||r(i)||/||b|| 3.214071607022e-06
Linear solve converged due to CONVERGED_RTOL iterations 1
      Line search: gnorm after quadratic fit 2.481576195523e+11
      Line search: Cubic step no good, shrinking lambda, current gnorm 2.632005412624e+10 lambda=5.0000000000000003e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 4.289212002697e+09 lambda=2.5000000000000001e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 4.332196637845e+08 lambda=1.2500000000000001e-02
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.678040222943e+07 lambda=6.2500000000000003e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.027868984884e+05 lambda=3.1250000000000002e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 7.010733464460e+03 lambda=1.5625000000000001e-03
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.751519860441e+03 lambda=7.8125000000000004e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 3.497889916171e+02 lambda=3.4753778542938795e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 7.932631084466e+01 lambda=1.5879606741873878e-04
      Line search: Cubic step no good, shrinking lambda, current gnorm 2.194608479634e+01 lambda=6.5716583192912669e-05
      Line search: Cubic step no good, shrinking lambda, current gnorm 1.117190149691e+01 lambda=1.1541218569257328e-05
      Line search: Cubically determined step, current gnorm 1.093879875464e+01 lambda=1.1541218569257329e-06
    |residual|_2 of individual variables:
               potential:    8.59942
               potentialliq: 0.403326
               em:           2.26505
               emliq:        0.714844
               Arp:          6.3169

4 Nonlinear |R| = 1.093880e+01