Discussed large eigenvalues in the projection method when using LOI.
- Sage's Tasks
- Code
- Writing
- Large Eigenvalues
- Meeting Followup
- References
Sage's Tasks
Code
- MLS approximation in 2D
- MLS approximation of derivative
- MLS on tangent-plane
- Implement time-stepping.
- Become familiar with Varun's repo.
Writing
- Thesis
- Write Paper
Large Eigenvalues
The following table shows the real parts of the eigenvalue with largest real part for the finite difference matrices generated by each method. The surface was the sphere, the point sets were the spiral nodes (my implementation, which yeilds different points than the SpherePts implementation). The stencil size was 100 points, the SVD tolerance and LOI tolerance were each 1e-12. The RBF was the 10th degree PHS.
deg = 2
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | -1.11861E-13 | -7.34968E-14 | -8.97060E-14 | 4.40402E-12 |
| 4000 | 2.73894E-13 | 5.49425E+03 | -4.71225E-13 | -1.59098E-10 |
| 8000 | -1.40109E-13 | 2.72130E-14 | -1.88455E-13 | -1.26133E-09 |
| 16000 | 2.66574E-13 | -2.25597E-13 | -1.03028E-12 | 1.12420E-09 |
deg = 3
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | 1.34143E-13 | 4.81706E-14 | -4.02254E-14 | 1.41685E-11 |
| 4000 | -1.16327E-13 | -1.60050E-13 | -1.38820E-13 | -2.66511E-10 |
| 8000 | -2.62410E-13 | -1.92069E-13 | -3.60526E-13 | -5.58063E-10 |
| 16000 | 1.06928E-13 | 5.30727E+04 | 8.32907E-13 | 1.95616E-09 |
deg = 4
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | -1.54670E-13 | 2.74753E-14 | 1.05127E+03 | 3.58774E+04 |
| 4000 | 1.68937E-13 | -1.15904E-14 | 3.97904E-13 | -7.48135E-10 |
| 8000 | -1.33913E-13 | 1.18622E-13 | 9.67115E-14 | 3.93966E-08 |
| 16000 | -1.32622E-12 | -5.12570E-14 | -8.31335E-13 | 4.02871E-08 |
deg = 5
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | -2.04674E-14 | 8.61533E-14 | 6.84410E-14 | 9.78416E-09 |
| 4000 | -2.46142E-13 | -1.61575E-13 | 5.01099E-14 | 1.44964E-08 |
| 8000 | 3.47409E-13 | 9.24960E+04 | 7.81597E-14 | -1.05611E-08 |
| 16000 | -1.17231E-13 | 4.14012E+06 | 3.38719E-10 | -1.57737E-08 |
The following table shows the same, but for the minimum energy nodes.
LOI tolerance: 1e-12
deg = 2
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | 3.78482E-14 | 2.45898E-14 | -2.25576E-13 | 1.71565E-11 |
| 4000 | -3.41548E-14 | 4.13313E+03 | -2.74660E-13 | 1.62625E-10 |
| 7921 | 1.82763E-13 | -8.27243E-15 | 1.07195E-16 | 3.75277E+01 |
| 16384 | -6.93451E-14 | 4.26602E+05 | -2.21361E-13 | 2.53941E-09 |
deg = 3
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | 4.44200E-14 | 1.02301E-14 | -1.71611E-14 | 3.62219E-11 |
| 4000 | -1.09144E-13 | 1.45141E-13 | 4.57109E-14 | 3.78532E-10 |
| 7921 | -4.29638E-14 | 6.78498E-14 | 2.04503E-13 | 1.11148E-09 |
| 16384 | -1.41131E-12 | 3.12246E+04 | -2.77274E-13 | -4.73564E-08 |
deg = 4
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | -6.23868E-14 | -4.83480E-15 | 2.23195E-13 | 7.13818E+04 |
| 4000 | 2.36222E-14 | 1.10884E-13 | 1.14815E-13 | 1.91252E-09 |
| 7921 | 1.86528E-13 | 7.30597E-14 | 2.60988E-13 | 8.45011E-08 |
| 16384 | -1.24131E-12 | 1.62978E-12 | -2.23821E-13 | 1.02253E-07 |
deg = 5
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | -6.07371E-14 | 4.49265E-14 | -2.73363E-14 | 1.28433E-09 |
| 4000 | 1.12428E-13 | 2.47735E+07 | 4.68958E-13 | -6.31353E-09 |
| 7921 | 1.58929E-13 | 4.53077E+06 | 2.22477E-13 | -7.75160E-09 |
| 16384 | 3.32588E-13 | 2.11288E+03 | -1.62150E-11 | -6.67094E-09 |
LOI tolerance: 1e-6
deg = 2
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | -1.32998E-13 | -7.53363E-15 | -5.87975E-14 | 1.71645E-11 |
| 4000 | -3.18564E-14 | 4.13313E+03 | 7.48433E-14 | 1.62819E-10 |
| 7921 | -4.00711E-14 | -1.43731E-13 | -8.32070E-14 | 3.75277E+01 |
| 16384 | -8.95912E-13 | 4.26602E+05 | -4.74586E-13 | 2.53920E-09 |
deg = 3
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | 3.43228E-15 | -3.00376E-14 | 8.73213E-14 | 3.62578E-11 |
| 4000 | 1.61685E-14 | 2.15058E-13 | -1.30039E-13 | 3.78507E-10 |
| 7921 | -1.03080E-13 | 2.62373E-13 | -5.95711E-13 | 1.11152E-09 |
| 16384 | -3.14237E-13 | 3.12246E+04 | -6.02410E-13 | -4.73562E-08 |
deg = 4
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | -3.65028E-13 | -4.94107E-14 | 1.04112E-12 | 7.13818E+04 |
| 4000 | -1.85860E-15 | 6.88688E-14 | -2.29123E-13 | 1.91254E-09 |
| 7921 | -1.10850E-13 | -9.83608E-14 | -7.65318E-14 | 8.45011E-08 |
| 16384 | 2.33565E-13 | -2.71683E-13 | 4.33431E-13 | 1.02253E-07 |
deg = 5
| TPM | Proj-LOI | Hermite | Proj-SVD | |
| 2000 | 6.85859E-14 | 8.01145E-14 | 1.00073E-13 | 1.28428E-09 |
| 4000 | -9.47916E-14 | 4.06090E-14 | -2.08264E-13 | -6.31361E-09 |
| 7921 | -3.37002E-13 | 1.48758E-13 | -6.75016E-14 | -7.75189E-09 |
| 16384 | -5.28041E-13 | 1.50207E-13 | -1.70139E-11 | -6.67057E-09 |
I also ran 16,384 points using an LOI tolerance of 1e-3 where $\Re(\lambda_1) = 4.940E+05$. Sadly it seems that raising the LOI tolerance doesn't correct all of the bad eigenvalues.
Meeting Followup
References