Re: request for algorithm
- From: ndros <ndros@xxxxxxxxxxxxxxx>
- Date: Thu, 12 Jan 2006 16:37:07 +0200
On 2006-01-12, loic-dev@xxxxxxx <loic-dev@xxxxxxx> wrote:
> Hi,
>
>
> I could eventually understand why you don't want to have a forward
> reference in time (you would have to solve a system of equations). But
> I can't really understand why you don't want to depend on the points
> U(x+1,y,t), U(x,y+1,t) for instance...
>
> What is the problem you are faced to, exactly?
>
> As already pointed out by Peter, such scheme can only work with
> particular equation, that has a flow of information coming from the
> points x-k, y-m, t-l with k,m,l >0.
>
> This can't be the case e.g. for elliptic equation like the Poisson
> equation, because by nature the flow of information comes from every
> direction (hence, you need value at x+k,y+m). Of course, you can always
> use an explicit discretization in time (e.g. the value at time t is
> computed with the values at previous times t-l), but you will need
> value of U(x+k, y+m, t)
>
First of all, thank you both for two very helpful responses, you've
certainly given me something to look into! Loic, you are quite right about
all your remarks, the problem is that my perspective on the subject is a
bit different: I've been studying computer parallelization techniques for
iterative algorithms exhibiting the properties I described (i.e. three-
or more dimensional loops, with nonnegative dependencies between data
points, so that only previously computed points are required for the
current computation), where I mostly used micro-kernels or parts of real
applications. My study has been focused on high performance issues,
from the computer science point of view; I wasn't targeted to solving
a specific physical problem.
Given this restrictions, I am now additionally looking for real, practical
problems, where these techniques might be applicable. It would really
be helpful because of all the reasons mentioned above, if all data
dependencies of the algorithm where nonnegative, that's why I am asking
for those *particular* cases, where this assumption holds. I have been
studying discretization techniques of boundary value problems (initial
value problems don't seem to help, as there is this flow of information
you describe), for which backward discretization seems to deliver a
differences equation consistent with the initial differential equation,
at least according to my poor theoretical knowledge on the subject :-)
For example, determination of the values of U, given that
laplacian(U(x,y,z))=F(x,y,z), domain [0,X]*[0,Y]*[0,Z], known values
U(x,y,z)=Gxyz on the "left" domain border, so that I have one-sided
flow of information, discretization parameters dx,dy,dz. The iterative
solution of this equation has lead me to following conclusions:
a) the convergence depends heavily on both the function F, as well as
the 3D problem domain.
b) generally, increasing dx, dy and dz reduces the approximation error,
which kind of contradicts my intuition. However, my thoughts on this
are not very clear, I have to look into it more thoroughly.
Once again, thank you very much fellows, you make up for the lack of
interdisciplinary cooperation I am facing in the University :-) Sorry
for the long post.
> HTH,
> Loic.
.
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