On 6/17/21 1:46 AM, Simon wrote:
One more question came up when I watched your video "What solver to use". In
there you mentioned that direct solvers in 2d have a complexity of O(N^2),
where N is the number of unknowns.
There is an approximation for N, i.e.
N \approx p^d *|Omega| / h^d.
So from that can one say that solving a linear system with UMFPACK has a
complexity of O(1/h^{4}) for a fixed degree p and d=2?
I ask that because I measured the times for solving two large linear systems
in 2d, one with 1 Million DoFs and the other one with 2 Million DoFs (reducing
h to h/2). The former took 11 seconds, the latter 110 seconds, i.e. they are
related by a factor of 10 and not 16.
Backcalculating this, the factor of 10 would follow from a complexity of
O(N^5/3}).
So is this is a plausible rate or is my linear system still not big enough to
see the rate of 16? Is the renumbering scheme, which does UMFPACK internally,
may the source why I see a lower rate?
The argument you make seems reasonable, but the rates I quote are for one
particular way of computing a sparse LU decomposition and, more importantly,
are *asymptotic*. You can't infer from just two data points what the rate is,
and it may well be possible that you can't ever observe the true rate of the
algorithm because you can't solve problems so large that the asymptotic rate
is attained.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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