[petsc-users] Issue using multi-grid as a pre-conditioner with KSP for a Poisson problem

[Jason Lefley]

Hello,

We are attempting to use the PETSc KSP solver framework in a fluid dynamics simulation we developed. The solution is part of a pressure projection and solves a Poisson problem. We use a cell-centered layout with a regular grid in 3d. We started with ex34.c from the KSP tutorials since it has the correct calls for the 3d DMDA, uses a cell-centered layout, and states that it works with multi-grid. We modified the operator construction function to match the coefficients and Dirichlet boundary conditions used in our problem (we’d also like to support Neumann but left those out for now to keep things simple). As a result of the modified boundary conditions, our version does not perform a null space removal on the right hand side or operator as the original did. We also modified the right hand side to contain a sinusoidal pattern for testing. Other than these changes, our code is the same as the original ex34.c

With the default KSP options and using CG with the default pre-conditioner and without a pre-conditioner, we see good convergence. However, we’d like to accelerate the time to solution further and target larger problem sizes (>= 1024^3) if possible. Given these objectives, multi-grid as a pre-conditioner interests us. To understand the improvement that multi-grid provides, we ran ex45 from the KSP tutorials. ex34 with CG and no pre-conditioner appears to converge in a single iteration and we wanted to compare against a problem that has similar convergence patterns to our problem. Here’s the tests we ran with ex45:

mpirun -n 16 ./ex45 -da_grid_x 129 -da_grid_y 129 -da_grid_z 129
	time in KSPSolve(): 7.0178e+00
	solver iterations: 157
	KSP final norm of residual: 3.16874e-05

mpirun -n 16 ./ex45 -da_grid_x 129 -da_grid_y 129 -da_grid_z 129 -ksp_type cg -pc_type none
	time in KSPSolve(): 4.1072e+00
	solver iterations: 213
	KSP final norm of residual: 0.000138866

mpirun -n 16 ./ex45 -da_grid_x 129 -da_grid_y 129 -da_grid_z 129 -ksp_type cg
	time in KSPSolve(): 3.3962e+00
	solver iterations: 88
	KSP final norm of residual: 6.46242e-05

mpirun -n 16 ./ex45 -da_grid_x 129 -da_grid_y 129 -da_grid_z 129 -pc_type mg -pc_mg_levels 5 -mg_levels_ksp_type richardson -mg_levels_ksp_max_it 1 -mg_levels_pc_type bjacobi
	time in KSPSolve(): 1.3201e+00
	solver iterations: 4
	KSP final norm of residual: 8.13339e-05

mpirun -n 16 ./ex45 -da_grid_x 129 -da_grid_y 129 -da_grid_z 129 -pc_type mg -pc_mg_levels 5 -mg_levels_ksp_type richardson -mg_levels_ksp_max_it 1 -mg_levels_pc_type bjacobi -ksp_type cg 
	time in KSPSolve(): 1.3008e+00
	solver iterations: 4
	KSP final norm of residual: 2.21474e-05

We found the multi-grid pre-conditioner options in the KSP tutorials makefile. These results make sense; both the default GMRES and CG solvers converge and CG without a pre-conditioner takes more iterations. The multi-grid pre-conditioned runs are pretty dramatically accelerated and require only a handful of iterations.

We ran our code (modified ex34.c as described above) with the same parameters:

mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128
	time in KSPSolve(): 5.3729e+00
	solver iterations: 123
	KSP final norm of residual: 0.00595066

mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 -ksp_type cg -pc_type none
	time in KSPSolve(): 3.6154e+00
	solver iterations: 188
	KSP final norm of residual: 0.00505943

mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 -ksp_type cg
	time in KSPSolve(): 3.5661e+00
	solver iterations: 98
	KSP final norm of residual: 0.00967462

mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 -pc_type mg -pc_mg_levels 5 -mg_levels_ksp_type richardson -mg_levels_ksp_max_it 1 -mg_levels_pc_type bjacobi
	time in KSPSolve(): 4.5606e+00
	solver iterations: 44
	KSP final norm of residual: 949.553

mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 -pc_type mg -pc_mg_levels 5 -mg_levels_ksp_type richardson -mg_levels_ksp_max_it 1 -mg_levels_pc_type bjacobi -ksp_type cg
	time in KSPSolve(): 1.5481e+01
	solver iterations: 198
	KSP final norm of residual: 0.916558

We performed all tests with petsc-3.7.6.

The trends with CG and GMRES seem consistent with the results from ex45. However, with multi-grid, something doesn’t seem right. Convergence seems poor and the solves run for many more iterations than ex45 with multi-grid as a pre-conditioner. I extensively validated the code that builds the matrix and also confirmed that the solution produced by CG, when evaluated with the system of equations elsewhere in our simulation, produces the same residual as indicated by PETSc. Given that we only made minimal modifications to the original example code, it seems likely that the operators constructed for the multi-grid levels are ok.

We also tried a variety of other suggested parameters for the multi-grid pre-conditioner as suggested in related mailing list posts but we didn’t observe any significant improvements over the results above.

Is there anything we can do to check the validity of the coefficient matrices built for the different multi-grid levels? Does it look like there could be problems there? Or any other suggestions to achieve better results with multi-grid? I have the -log_view, -ksp_view, and convergence monitor output from the above tests and can post any of it if it would assist.

Thanks