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Hello, something a bit different from my last question, since that didn't progress so well: I have a related model which generally produces a rectangular matrix A, so I am using LSQR to solve the system. The matrix A has two nonzeros (1, -1)
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<meta charset="UTF-8"></head><body><pre style="font-family: sans-serif; font-size: 100%; white-space: pre-wrap; word-wrap: break-word">Hello,
something a bit different from my last question, since that didn't
progress so well:
I have a related model which generally produces a rectangular matrix A,
so I am using LSQR to solve the system.
The matrix A has two nonzeros (1, -1) per row, with A^T A being similar
to a finite difference Poisson matrix if the rows were permuted randomly.
The problem is singular in that the solution is only specified up to a
constant from the matrix, with my target solution being a weighted zero
average one, which I can handle by adding a nullspace to my matrix.
However, I'd also like to pin (potentially many) DOFs in the future so I
also tried pinning a single value, and afterwards subtracting the
average from the KSP solution.
This leads to the KSP *sometimes* diverging when I use a preconditioner;
the target size of the matrix will be something like ([1,20] N) x N,
with N ~ [2, 1e6] so for the higher end I will require a preconditioner
for reasonable execution time.
For a smaller example system, I set up my application to dump the input
to the KSP when it breaks down and I've attached a simple python script
+ data using petsc4py to demonstrate the divergence for those specific
systems.
With `python3 lsdiv.py -pc_type lu -ksp_converged_reason` that
particular system shows breakdown, but if I remove the pinned DOF and
add the nullspace (pass -usens) it converges. I did try different PCs
but they tend to break down at different steps, e.g. `python3 lsdiv.py
-usenormal -qrdiv -pc_type qr -ksp_converged_reason` shows the breakdown
for PCQR when I use MatCreateNormal for creating the PC mat, but
interestingly it doesn't break down when I explicitly form A^T A (don't
pass -usenormal).
For the moment I can work by adding the nullspace but eventually the
need for pinning DOFs will resurface, so I'd like to ask where the
breakdown is coming from. What causes the breakdowns? Is that a generic
problem occurring when adding (dof_i = val) rows to least-squares
systems which prevents these preconditioners from being robust? If so,
what preconditioners could be robust?
I did a minimal sweep of the available PCs by going over the possible
inputs of -pc_type for my application while pinning one DOF. Excepting
unavailable PCs (not compiled for, other setup missing, ...) and those
that did break down, I am left with ( hmg jacobi mat none pbjacobi sor
svd ).
Best regards,
Marco
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