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Re: Re: Speeding Up Realtime Visualization

• To: mathgroup at smc.vnet.net
• Subject: [mg84367] Re: Re: Speeding Up Realtime Visualization
• From: Will <secandum.oculi at gmail.com>
• Date: Thu, 20 Dec 2007 00:11:34 -0500 (EST)

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[
Thread[Rxn[c,p,#1,#2]]&,Range[2,length-1],Range[2,length-1]
];
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.