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Re: Re: Numerical evaluation is Mathematica bottleneck?!

• To: mathgroup at smc.vnet.net
• Subject: [mg69107] Re: Re: Numerical evaluation is Mathematica bottleneck?!
• From: "Jens-Peer Kuska" <kuska at informatik.uni-leipzig.de>
• Date: Wed, 30 Aug 2006 06:33:17 -0400 (EDT)
• Organization: Uni Leipzig
• References: <ed0sb8$src$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hi,

and there is only the unreadable form below ??
usual I prefer equations in the traditional 
mathematical
way -- because from you original question it is 
clear
that one should use NDSolve[] directly, or build a
SparseMatrix[] object and not the discretisation 
you
give below.

Regards
  Jens

"gregorc" <gregor.cernivec at fe.uni-lj.si> schrieb 
im Newsbeitrag news:ed0sb8$src$1 at smc.vnet.net...
| Hello,
|
| The problem consists of three coupled nonlinear 
equations: the Poisson
| equation and the countiniuty equations for 
electrons and holes. The
| countiniuty equations consist of 
convection-diffusion type equations for
| electrons and hole current and the 
generation-recombination terms.
|
| The Poisson equation for non-uniform mesh at 
node 2:
|
| FV[2]=(2*(-((-V[[1]] + V[[2]])/h[[1]]) + 
(-V[[2]] +
| V[[3]])/h[[2]]))/(h[[1]] + h[[2]])-(q*(nt[2] + 
n[[2]] + NA[[2]] -
| ND[[2]] - p[[2]]-pt[2]))/eps[[2]]
|
| The electron and hole currents between nodes 1-2 
and 2-3 and the
| countinuity equations, respectively:
|
| Jn[1]=(Dn[[1]]*(-(B[(V[1] - 
V[2])/UT[[1]]]*n[[1]]) + B[(-V[1] +
| V[2])/UT[[1]]]*n[[2]]))/h[[1]]
| Jn[2]=(Dn[[2]]*(-(B[(V[2] - 
V[3])/UT[[2]]]*n[[2]]) + B[(-V[2] +
| V[3])/UT[[2]]]*n[[3]]))/h[[2]]
| FJn[2]=-2/(h[[2]]+h[[1]])*(Jn[1]-Jn[2])-R+GL
|
| Jh[1]=(Dp[[1]]*(B[(-V[1] + 
V[2])/UT[[1]]]*p[[1]] - B[(V[1] -
| V[2])/UT[[1]]]*p[[2]]))/h[[1]]
| Jh[2]=(Dp[[2]]*(B[(-V[2] + 
V[3])/UT[[2]]]*p[[2]] - B[(V[2] -
| V[3])/UT[[2]]]*p[[3]]))/h[[2]]
| FJh[2]=2/(h[[2]]+h[[1]])*(Jp[1]-Jp[2])-R+GL
|
| ,where V[[2]] is the calculation variable for 
the electrical potential
| at node 2, n[[2]] and p[[2]] are the calculation 
varibles for electron
| and hole concentrations, respectively. The rest 
is material data, built
| in and/or trapped charge, 
recombination-generation terms, which might be
| also set to zero (R,GL) and constants. B is the 
Bernoulli function,
| which has to be approximated with Taylor series 
to avoid singularity
| with argument 0 - zero field (V[[1]]=V[[2]]). 
h[[1]], h[[2]]... is the
| discretization step.
|
| Functions:FJn[2], FV[2], FJh[2]
| 
variables:X={n[[1]],V[[1]],p[[1]],n[[2]],V[[2]],p[[2]],n[[3]],V[[3]],p[[3]]}
|
| The Jacobian:
|
| JFV=Map[D[FV[2],#]&,X]
| JFJn=Map[D[FJn[2],#]&,X]
| JFJp=Map[D[FJp[2],#]&,X]
|
| Functions and the coresponding Jacobians are 
compiled:
|
| 
JFVnum=Compile[{{n,_Real,1},{V,_Real,1},{p,_Real,1},{consts},...},Evaluate[JFV]].
|
| Numerical evaluation:
|
| 
JFVvar=MapThread[JFVnum,{var`n,var`V,var`p,...consts, 
mat. data}]
|
| ,where var`n=Partition[n,3,1] and same for the V 
and p and the rest of
| the material data and constants.
|
| The numerical function is Threaded over the 
overlapping triplets of
| calculation variables and data. When the 
numerical function and the
| coresponding jacobians are calculated, the 
sparse array is formed with
| the SparseArray function and the system is 
solved with the LinearSolve.
| The majority of the computational time takes the 
MapThread of the
| compiled function on the partitioned data and 
varibles. The functions
| and jacobians are compiled for sure, since I 
have checked it with
| compiled_function[[-3]], which returns all real 
numbers.
|
| Regards,
|
| gregor.
|
| Jens-Peer Kuska wrote:
|
| >Hi,
| >
| >the C/C++ version may be faster -- or not, it 
depend on
| >your programming skills and the numerical 
methods you are
| >using.
| >But as long as you don't tell more about the 
problem and
| >how you have solved it in Mathematica, there is 
not defined answer.
| >
| >Regards
| >   Jens
| >
| >gregorc wrote:
| >
| >
| >>Dear mathgroup members,
| >>
| >>I am solving a non-linear semiconductor PDE 
problem with finite
| >>difference discretization. The Compiled 
algebraic equation and its
| >>Jacobian  take as input central point and two 
adjacent neighbors and
| >>they are Threaded over the whole mesh. A 
LinearSolve and some damping
| >>schemes are used to obtain next step towards 
the solution. Typical mesh
| >>is about 300 points and the solution is 
usually found in 7 iterations,
| >>which takes about 7 seconds of computational 
time.This is already too
| >>slow, but since I plan to solve the problem in 
2D this is much too
| >>slow!!! The bottleneck is the numerical 
evaluation of the compiled
| >>equation and its Jacobian. I am thinking of 
using MathLink and write the
| >>numerical evaluation routine in C++, which 
would take a list of data and
| >>return evaluated function and the Jacobian.
| >>
| >>Is this the right way? Could the link itself 
be a timing bottleneck? Any
| >>other suggestions?
| >>
| >>Regards,
| >>
| >>Gregor
| >>
| >>
| >>
| >
| >
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