V-cycle errors

[Julian-Wyatt]

I've managed to successfully pass the jacobi tests for the coursework (ie the test_two_grid function), however my v-cycle code produces increasing errors on more refined grids which look like:

[1.18167568e-05, 1.99099355e-06, 2.68594848e-07, 1.28529688e-01, 3.97952194e+04],
on grid refinements 1-5, when solved with LU decomposition at the coarsest level (running jacboi here instead just performs slightly worse).
When calculating the W-cycle, I get very similar errors.

I'm not sure what I could be missing here, would you be able to give any advice or insight?

[annereinarz]

There's a few things that could be causing issues/that you can try:

• increase the number of smoothing steps: does the convergence continue further?
• change the alpha scaling for the Jacobi iterations.
• can you check if the error is highest at the boundaries? Try scaling both sides of the equations by the 3D volume element.
• check the restriction/prolongation operators you are using. Injection as a restriction might not work particularly well.

[Julian-Wyatt]

Thanks - this is for the mms convergence test on the 1D Poisson solver with the preset parameters. I've changed the parameters as suggested - for results below

1.

Altering the pre and post smoothing steps from 1 to 10 and trying post smoothing = 2x pre smoothing, I found that the overall results were of the same magnitude (ie grid refinements 4 and 5 still perform poorly).
Increasing pre and post iters to 40 gives errors of:
V cycle:
[1.73367637e-05 3.07660989e-06 4.58071118e-07 1.54047477e-07 8.96348653e-01]
W cycle:
[1.73212064e-05 3.13555370e-06 5.65832698e-07 1.79710076e-07 4.76815470e+01]

2.

Updated alpha to 0.94 (with pre and post smoothing iters kept at 1) gives better error of:
V cycle:
[1.72803171e-05 2.99676583e-06 4.58914367e-07 2.12145402e-05 2.63942233e+01]
W cycle:
[1.73402772e-05, 3.14048444e-06, 4.05319078e-07, 1.15997288e-08, 8.92409541e+00]

3.

For 4 degrees of freedom (ie 1d grid of size 161), and after 10 iterations of the loop in MGSolver.solve, the error (expectation - x) is 0.005 (at x_{10}), increasing linearly to 0.07 (at x_{140}), which then decreases to 0.015 at x_{157}, so it doesn't look like the majority of the error is at the boundary. As the iterations tends to 60+, the error generally stays of +- same order of magnitude, until about 70-80 where the error increases past 1.

[annereinarz]

It looks like you've found a better alpha value, that's good. I'd try to stick with somewhere in the range of 3-5 smoothing iterations. For 3 it doesn't look like that's the issue, but just multiplying both sides of the equation with the volume element might still help, so maybe give that a go.
In the 1D test case there is a factor of h (the 1D volume element) on both sides. In 3D you need h^3 instead.

[Julian-Wyatt]

I couldn't really improve those errors further with that smoothing range and adjusting the alpha.

When you say multiply both sides of the equation by the volume element, am I correct in saying this is already done in the test case rhs function with the H_x component in:

barray[i] = numpy.pi ** 2 / 4 * numpy.sin(numpy.pi / 2 * x) * Hx

and for the left hand side with:

# Equation is scaled by volume element (Hx) for 
# compatibility with the multigrid transfer operators 
# PETSc creates (hence 2/Hx rather than 2/Hx**2), and the
# right hand side is scaled appropriately too.
yarray[i] = 2 / Hx * xarray[iloc]

So therefore the 1D test function should work considering it is already scaled?

[annereinarz]

Sorry, I missed that you were already working with the 1D case, yes the scaling is already introduced there.

How coarse is your coarsest level and how fine is the finest level?

[Julian-Wyatt]

It is worth mentioning - after fixing a silly error in my w-cycle code, that now passes the test with errors of:
[1.73237317e-05 3.09196226e-06 5.13490925e-07 7.88679903e-08 5.63582524e-09], but the V-cycle is still struggling on the final refinement

[annereinarz]

I can't see any obvious issues, I would try to use the transpose of the prolongation operator as the restriction instead of using injection. Considering that you only have issues with the finest levels that might be causing the issues.

[Julian-Wyatt]

Having made that change and fiddling with the parameters of the solve function to: alpha=0.96 & pre iters = post iters = 2, the test passes with errors:
[1.73474246e-05 3.06599435e-06 5.42076939e-07 9.58209022e-08 1.84443642e-08]

Thanks very much!

Should I consider making this change for W cycle (even though that works)?

[annereinarz]

Probably yes, injection is probably the simplest form of restriction, but it does "throw away" a lot of information that could be used.

[annereinarz]

Added the restriction issue to the above and marked as answer