dolfin team mailing list archive

["Johan Hoffman" <jhoffman@xxxxxxxxxxx>]

> On Tue, Nov 08, 2005 at 09:58:46AM +0100, Johan Hoffman wrote:
>
> ...
>
>>
>> I believe you Johan estimated the work for a matrix-free approach based
>> on
>> the action of f(u), to be favourable to forming the matrix A and using
>> matrix-vector multiplication Ax? In general?
>>
>> /Johan
>>
>
> Well, I speculated that it would be favorable :). I made a simple test
> with the form for linear elasticty once, and I ran it again now also
> with a mass form to compare. The mesh is a unit cube with 162k cells
> and I perform each operation (assemble or multiply) 10 times. "Matrix"
> is assembly of a matrix, "Action" is the assembly of the action vector
> and "Mult" is matrix-vector multiplication with the assembled
> matrix. Here are the results:
>
> Elasticity:
>
> Matrix: 31s
> Action: 10s
> Mult:   0.53s
>
> Mass:
>
> Matrix: 28s
> Action: 3.1s
> Mult:   0.53s
>
>
> I'm not sure why the Matrix time only goes up 3 seconds while the
> Action time goes up 6 seconds between the mass and elasticity forms,
> that should be investigated. The extra work going from mass to
> elasticity should be the same for both cases. Perhaps there is room
> for optimization there.
>
> But we can clearly see that the sparse matrix overhead dominates the
> matrix assembly for these forms. But that overhead becomes less
> relevant for more complicated forms.
>
> If you need to re-assemble the matrix for every matrix-vector
> multiplication (i.e. every iteration of some algorithm), then it's
> clearly favorable to assemble the action instead. But if you are
> allowed to reuse the matrix for several iterations, then it seems to
> be faster to assemble the matrix and then do matrix-vector
> multiplication.
>
> But I think it's pretty interesting that assembling the action of the
> mass form is only a factor 6 slower than performing the matrix-vector
> multiplication (which assumes you already have the matrix). We could
> probably make the assembly even faster, but I doubt the matrix-vector
> multiplication time could be significantly reduced.
>
>   Johan
>

Interesting. How come that assembling the matrix is 3-10 times as slow as
"assembling" the action? Is it only due to the overhead of managing the
sparse structure?

/Johan