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gtspring2009:gibson:localsolns [2009/07/20 07:09]
gibson
gtspring2009:gibson:localsolns [2009/07/20 07:26]
gibson
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 I started Arnoldi iteration on the phase-shift optimized versions the previously calculated solutions for 4pi,16pi. I didn't want to start these without symmetry optimization,​ since I need to examine symmetry to sort the eigenvalues,​ and it's a lot easier to detect the symmetry of the eigenfunction if you know the center of symmetry of the solution. That's running now. Looks like about an hour per solution at 100 Arnoldi iterations per solution. With two CPUs going and about fifty solutions (every other dot on the green curve), it'll take 24 hrs or so. It's clear from the first calculation that 100 Arnoldi iterations is not overkill. The first calc is at the 66th iteration, and the second zero eigenvalue is not yet resolved well enough to distinguish it clearly from the small non-zero eigenvalues. Arnoldi iteration generally requires more iterations to get accurate eigenvalues than GMRES does to get an sufficiently accurate solution. No great accuracy is required in GMRES solutions to Newton steps, since the Newton procedure is iterative itself. ​ I started Arnoldi iteration on the phase-shift optimized versions the previously calculated solutions for 4pi,16pi. I didn't want to start these without symmetry optimization,​ since I need to examine symmetry to sort the eigenvalues,​ and it's a lot easier to detect the symmetry of the eigenfunction if you know the center of symmetry of the solution. That's running now. Looks like about an hour per solution at 100 Arnoldi iterations per solution. With two CPUs going and about fifty solutions (every other dot on the green curve), it'll take 24 hrs or so. It's clear from the first calculation that 100 Arnoldi iterations is not overkill. The first calc is at the 66th iteration, and the second zero eigenvalue is not yet resolved well enough to distinguish it clearly from the small non-zero eigenvalues. Arnoldi iteration generally requires more iterations to get accurate eigenvalues than GMRES does to get an sufficiently accurate solution. No great accuracy is required in GMRES solutions to Newton steps, since the Newton procedure is iterative itself. ​
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 ===== 2009-07-20 ===== ===== 2009-07-20 =====
  
 Further progress on continuing the localized solutions 4pi,16pi and 4pi,24pi. 4pi,16pi appears to be headed to the upper branch of the Nagata solution. ​ Further progress on continuing the localized solutions 4pi,16pi and 4pi,24pi. 4pi,16pi appears to be headed to the upper branch of the Nagata solution. ​
 The Arnoldi iterations are still cooking. I had to restart them with a longer integration time (T=20) for the finite-time map u-f^T(u) in order to get better separation of the eigenvalues near zero. I've set up a bash script to do Arnoldi iteration on every other data point in the 4pi,16pi continuation;​ those should finish today, and then maybe I'll start it on the intervening points. Following that I'll have to solve the bookkeeping problem to track individual eigenvalues between continuation steps. In the meantime I'm going to work on some Poincare sections that have been stalled for some time... The Arnoldi iterations are still cooking. I had to restart them with a longer integration time (T=20) for the finite-time map u-f^T(u) in order to get better separation of the eigenvalues near zero. I've set up a bash script to do Arnoldi iteration on every other data point in the 4pi,16pi continuation;​ those should finish today, and then maybe I'll start it on the intervening points. Following that I'll have to solve the bookkeeping problem to track individual eigenvalues between continuation steps. In the meantime I'm going to work on some Poincare sections that have been stalled for some time...
 +
 +BTW, the limiting factor on these computations is going to be memory in the Arnoldi iteration. The algorithm requires an LU decomp of an M x N matrix, where M is the number of independent spectral coefficients in the discretization of the velocity field, and N is the number of Arnoldi iterations. For the 4pi,16pi box with 32 x 33 x 256 gridpoints, that's a 200,000 x 100 matrix, and the process size gets to 1.2 GB during the LU decomp. My machine has 4 GB DDR3 1600 RAM and 4 cores, so I can do a couple Arnoldis at a time without trouble. But I expect to hit my upper limit pretty soon (a factor of 2 or 4) in domain size or Reynolds, or even sooner with less smooth solutions, which will need finer resolution. The Nagata LB is unusually smooth. I could put in two more 2 GB sticks to move up to 8 GB RAM, but then I'd have to downclock the RAM to 1033. I think the PACE machines are all 4 GB. 
  
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gtspring2009/gibson/localsolns.txt ยท Last modified: 2010/02/02 07:55 (external edit)