Finish paper

[sampotter]

Numerical experiments

• [ ] Mesh compressibility test:
  • For octree and Fiedler tree:
  • For each $h$ and $p$ mesh:
    • For each, fixed number of eigenvectors, $m$:
    • Stream and butterfly factorize $m$ eigenvectors
    • Store the following:
      • Size of uncompressed matrix
      • Size of compressed matrix
      • "Stream eigenvectors" time
      • "Butterfly factorize" time
      • Uncompressed MVP time
      • Butterfly MVP time
• [ ] Sphere covariance test:
  • Use $h$ and/or $p$ refinement to squash discretization error
  • Compute analytic covariance at node locations
  • For each butterfly tolerance $\epsilon$ and each Chebyshev order $p$:
  • Operator norm error vs analytic
• [ ] Non-sphere covariance test:
  • Use one or two reasonably-sized meshes
  • Compute reference covariance at node locations
  • For each butterfly tolerance $\epsilon$ and each Chebyshev order $p$:
  • Estimated operator norm error compared to reference
• [ ] Less accurate ARPACK?
  • Might be worth running ARPACK with reduced accuracy.
  • If our SVD truncation level is 1e-2, then running ARPACK with e.g. 1e-15 tolerance probably excessive.
  • Experiment with running ARPACK with reduced accuracy for possible speed boost.

Todo

• [x] Sam: add size_t-converted MFEM sphere discretizations (stiffness and mass matrices in CSR format) to Dropbox
• [ ] Sam: debug MFEM sphere discretizations so that bf_lbo runs on them
• [x] Sam: footnotes 1 in https://github.com/sampotter/butterfly/issues/16
• [ ] Sam: footnotes 2 in https://github.com/sampotter/butterfly/issues/16
• [ ] Sam: dump index array for MFEM meshes (high-order, eventually) so we can test Fiedler tree
• [ ] Paul: set up compression test scripts
• [ ] Paul: set up analytic covariance comparison test script
• [ ] Sam: set up twisted torus geometry in MFEM
• [ ] Sam: write up spectrum covering algorithm
• [ ] Sam: write up complexity analysis
• [ ] Sam: make refinement figures (TODO: ???)
• [ ] Sam: write up Fiedler tree
• [ ] Sam: run Paul's final scripts to collect timings

[sampotter]

@pbeckman I wrote your PDF up here to make it easier to iterate on this

[pbeckman]

We discussed today that the compression test should really be:

• Fix number of desired eigenfunctions $m$
• For octree and Fiedler tree:
  • For each $h$ and $p$ refined mesh:
  • Compute and compress $m$ eigenfunctions
  • Store the following:
    • Size of uncompressed matrix
    • Size of compressed matrix
    • "Stream eigenvectors" time
    • "Butterfly factorize" time

[sampotter]

Updated. Not sure how to make it so that you can edit my to-do list...

[sampotter]

@pbeckman Footnote 1 was actually already fixed. This is was what --rowTreeInitDepth is for. This option lets you specify an offset (from the root level == 0; e.g., level == 1 in an octree = 8 nodes) into the row/space tree to use when doing the adaptive butterfly. Basically, if I want to not waste my time compressing the first level, assuming it won't work very well (possibly because I know I'm compressing columns from an orthogonal matrix, which is not what's going on with a generalized eigenvalue problem...), then I would pass e.g. --rowTreeInitDepth=1 to start on the first non-root level.