Re: Speeding Up Realtime Visualization

• To: mathgroup at smc.vnet.net
• Subject: [mg84384] Re: Speeding Up Realtime Visualization
• From: Ray Koopman <koopman at sfu.ca>
• Date: Thu, 20 Dec 2007 16:50:33 -0500 (EST)
• References: <fkcugf\$5vk\$1@smc.vnet.net>

```On Dec 19, 9:27 pm, Will <secandum.oc... at gmail.com> wrote:
> Since I haven't mentioned this before, I'm trying to make a two species reaction diffusion simulator by approximating the associated PDE using finite difference integration. These 'rough' simulators, e.g.http://texturegarden.com/java/rd/index.html, are typically very fast. I'm going for something with this speed.
>
> In my code, I've compiled a function which performs the 'next-time-interval' finite difference calculation. Using the Timing function, I found this calculation takes essentially no time. Mathematica usually says 0. sec. However, when I perform this calculation across the entire 100x100 grid, i.e. ~99^2 calcuations, it takes 0.45 sec.
>
> I originally computed the entire grid using For loops, which required ~0.5-0.6 sec. Figuring that Mathematica doesn't like procedural programming, I tried implementing a Threading technique to apply the finite difference calculation across the grid, which takes 0.45. Of course, this is way to slow for a 'fast' simulator.
>
> Again, I can speed things up by lowering my grid size, or keeping the grid size and increasing the spatial interval in the integrator, but I'll be losing significant quality.
>
> I still think the sluggishness has something to do with the way I'm setting up my matrices, which are used to read/write during the finite difference calculation. I figure I'm forgetting something huge
>
> Ok, here's my code:
> Clear["Global`*"];
> N[
> L=100;
> ds=1;
> length=L/ds;
> dt=0.5;
> a=0.082;(*u diffusion*)
> b=0.041;(*v diffusion*)
> f=0.035;(*u feed/rxn rate*)
> k=0.0625;(*v death rate*)
> dtf=dt f;
> dtfk=dt(f+k);
> e=a dt/(ds)^2;
> g=b dt/(ds)^2;
>
> uP=Table[1,{i,1,length},{j,1,length}];(*previous u state*)
> uC=Table[0,{i,1,length},{j,1,length}];(*current u state*)
> vP=Table[0,{i,1,length},{j,1,length}];(*previous v state*)
> vC=Table[0,{i,1,length},{j,1,length}];(*current v state*)
> c=2;p=1;(*for double buffer*)
> u={uP,uC};(*u[[1]] is previous, u[[2]] is current*)
> v={vP,vC};(*v[[1]] is previous, v[[2]] is current*)
>
> (*initial conditions, small square of altered conc values*)
> Do[u[[1,i,j]]=0.35,{i,0.4length,0.6length},{j,0.40length,0.6length}];
> Do[v[[1,i,j]]= Sin[0.1i-0.5length]Cos[0.1j-0.5length]^2,{i,1,length},{j,1,length}];
> ];
>
> (*finite difference/rxn diffusion calculation*)
> Rxn=Compile[{{c,_Integer},{p,_Integer},{i,_Integer},{j,_Integer}},
> u[[c,i,j]]=u[[p,i,j]](1-4*e-dt*(v[[p,i,j]])^2-dtf)+dtf+e*(u[[p,i+1,j]]+u[[p,i-1,j]]+u[[p,i,j+1]]+u[[p,i,j-1]])//N;
>
> v[[c,i,j]]=v[[p,i,j]](1-4*g+dt*u[[p,i,j]]*v[[p,i,j]])-dtfk+
> g*(v[[p,i+1,j]]+v[[p,i-1,j]]+v[[p,i,j+1]]+v[[p,i,j-1]])//N;
> ];
>
> (*Rxn Diffusion*)
> While[True,
> Outer[
> ];
> If[p==1,c=1;p=2,c=2;p=1] ;  (*c/p inverter*)
> ugrid=u[[2]];
> vgrid=v[[2]];
> ]
>
> Then I Dynamically visualize either ugrid or vgrid using ArrayPlot or Graphics[Raster[]] , which seem take take the same time.
>
> Any ideas?
>
> Thanks.

Here's some untested code that works with arrays rather than scalars.

{p,c} = {1,2}; While[True,
u[[c, 2;;-2, 2;;-2]] = u[[p, 2;;-2, 2;;-2]](1 - 4*e - dt*
v[[p, 2;;-2, 2;;-2]]^2 - dtf) + dtf +
e*ListConvolve[{{0,1,0},{1,0,1},{0,1,0}},u[[p]]];
v[[c, 2;;-2, 2;;-2]] = v[[p, 2;;-2, 2;;-2]](1 - 4*g + dt*
u[[p, 2;;-2, 2;;-2]]*v[[p, 2;;-2, 2;;-2]]) -
dtfk + g*ListConvolve[{{0,1,0},{1,0,1},{0,1,0}},v[[p]]];
ugrid = u[[c]]; vgrid = v[[c]]; {p,c} = {c,p} ]

Or maybe update u & v simultaneously
and do away with the double-buffering:

While[True, {u[[2;;-2, 2;;-2]], v[[2;;-2, 2;;-2]]} =
{u[[2;;-2, 2;;-2]](1 - 4*e - dt*v[[2;;-2, 2;;-2]]^2 - dtf) +
dtf + e*ListConvolve[{{0,1,0},{1,0,1},{0,1,0}},u],
v[[2;;-2, 2;;-2]](1 - 4*g + dt*u[[2;;-2, 2;;-2]]*v[[2;;-2, 2;;-2]])-
dtfk + g*ListConvolve[{{0,1,0},{1,0,1},{0,1,0}},v]};
ugrid = u; vgrid = v ]

Is there any particular reason why you don't precompute
1-4*e-dtf  and  1-4*g ?

```

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