Dear Wayne,
I am a bit surprised by your numbers and find them rather high, at least
with the chosen problem sizes. I would expect the matrix-free solver to
run in less than a second for 111,000 unknowns on typical computers, not
almost 10 seconds. I need to honestly say that I do not have a good
explanation at this point. I did not write this tutorial program, but I
know more or less what should happen. Let me ask a basic question first:
Did you record the timings with release mode? The numbers would make
more sense if they are based on the debug mode.
Best,
Martin
On 19.10.22 12:08, 'yy.wayne' via deal.II User Group wrote:
Thanks for your reply Peter,
The matrix-free run is basic same as in step-75 except I substitute
coarse grid solver. For fe_degree=6 without GMG and fe_degree in each
level decrease by 1 for pMG, the solve_system() function runtime is
24.1s. It's decomposed to *MatrixFree MG operators
construction*(1.36s), MatrixFree MG transfers(2.73s), KLU coarse grid
solver(5.7s), *setting smoother_data and compute_inverse_diagonal for
level matrices*(3.4s) CG iteration(9.8s).
The two bold texts cost a lot more(133s and 62s, respectively) in
matrix-based multigrid case. I noticed just as in step-16, the finest
level matrix is assembled twice(one for system_matrix and one for
mg_matrices[maxlevel]) so assembling time cost more.
Best,
Wayne
在2022年10月19日星期三 UTC+8 17:10:27<[email protected]> 写道：
Hi Wayne,
your numbers make totally sense. Don't forget that you are running
for high order: degree=6! The number of non-zeroes per
element-stiffness matrix is ((degree + 1)^dim)^2 and the cost of
computing the element stiffness matrix is even ((degree +
1)^dim)^3 if I am not mistaken (3 nested loop: i, j and q). Higher
orders are definitely made for matrix-free algorithms!
Out of curiosity: how large is the setup cost of MG in the case of
the matrix-free run? As a comment: don't be surprised that the
setup costs are relatively high compared to the solution process:
you are probably setting up a new Triangulation-, DoFHander-,
MatrixFree-, ... -object per level. In many simulations, you can
reuse these objects, since you don't perform AMR every time step.
Peter
On Wednesday, 19 October 2022 at 10:38:34 UTC+2 yy.wayne wrote:
Hello everyone,
I modified step-75 a little bit and try to test it's runtime.
However the result is kind of inexplainable from my point of
view, especially on *disproportionate assemble time and solve
time*. Here are some changes:
1. a matrix-based version of step75 is contructed to compare
with matrix-free one.
2. no mesh refinement and no GMG, and fe_degree is constant
across all cells within every cycle. Fe_degree adds one after
each cycle. I make this setting to compare runtime due to
fe_degree.
3. a direct solver on coareset grid. I think it won't affect
runtime since coarest grid never change
For final cycle it has fe_degree=6 and DoFs=111,361.
For matrix-based method, overall runtime is 301s where setup
system(84s) and solve system(214s) take up most. In step-75
solve system actually did both multigrid matrices assembling,
smoother construction, and CG solving. Runtime of this case is
shown:
matrix-based.png
On each level I print time assembling level matrix. *The solve
system is mostly decomposed to MG matrices
assembling(83.9+33.6+...=133s), smoother set up(65s), coarse
grid solve(6s) and CG solve(2.56).* My doubt is why actual CG
solve only takes 2.56 out of 301 seconds for this problem? The
time spent on assembling and smoother construction account too
much that they seems a burden.
For matrix-free method however, runtime is much smaller
without assembling matrices. Besides, CG solve cost more
because of more computation required by matrix-free I guess.
But *smoother construction time reduces significantly* as well
is out of my expectation.
matrix-free.png
Matrix-free framework saves assembling time but it seems too
efficient to be real. The text in bold are my main confusion.
May someone share some experience on matrix-free and multigrid
methods' time consumption？
Best,
Wayne
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