Re: Parallel Processing Toolkit
- To: mathgroup at smc.vnet.net
- Subject: [mg31048] Re: Parallel Processing Toolkit
- From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
- Date: Sat, 6 Oct 2001 03:32:12 -0400 (EDT)
- Organization: Universitaet Leipzig
- References: <9pecmf$sj4$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hi,
a parallel Runge-Kutta method ?
It is attached.
Regards
Jens
Blimbaum Jerry DLPC wrote:
>
> Does anyone have a notebook (that they could share) with examples of
> how they have used Mathematica's Parallel Processing Toolkit..
>
> thanks....jerry blimbaum NSWC Panama City, Fl
filename="ParallelDIRKSend.nb"
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When the steps of the DIRKmethod are evaluated we have to solve \
equations of the form\
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\t\t\ tau = t + alpha*h; \n\t\ti = 0; \n\t\t
While[\(i++\) < maxit, \n\t\t\targ = yn + beta*h*xi; \n\t\t\t
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\t\t\t\t\ {b1, b2, b3, b4} = {
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\ \ \n\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ tau*\((3\ + \ tau*\((\(-15\)/2\ + \ 6*tau)\))\),
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\n$Jacobian = Null;
\ \ \ \ \ \ \ \ \ \ \ \ (*\
D[f, y]\ of\ the\ system\ as\ pure\ function\ *) \n\n
DIRKSerial[t_, y0_List, h_, maxit_Integer, errPhi_?NumericQ,
errY_?NumericQ, yEstimate_List] := \n\t
Module[{xi1, xi2, xi3, xi4, psi3, psi4, y1, y1Estimate, hNext, tau}, \n
\t\t\ \ xi1 =
newtonSolve[t, h, y0, $RightHandSide, 1/2, 1/2, $Jacobian, maxit,
errPhi]; \n\t\t\ \
xi2 = newtonSolve[t, h, y0, $RightHandSide, 2/3, 2/3, $Jacobian,
maxit, errPhi]; \n\t\t\ \ psi3 = y0 + 5*h*\((\(-xi1\) + xi2)\)/2;
\n\t\t\ \ psi4 = y0 + h*\((\(-5\)*xi1 + 4*xi2)\)/3; \n\t\t\ \
xi3 = newtonSolve[t, h, psi3, $RightHandSide, 1/2, 1/2, $Jacobian,
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xi4 = newtonSolve[t, h, psi4, $RightHandSide, 1/3, 2/3, $Jacobian,
maxit, errPhi]; \n\t\t\ \
y1 = y0 + h*\((3*\((xi2 + xi4)\)/2 - xi1 - xi3)\); \n
\t\t (*\ Now\ we\ estimate\ the\ error\ from\ the\ prediction\ of\ the
\ last\ step\ ... *) \n\t\t\ \
hNext = DIRKErrorEstimate[h, y1, yEstimate, errY]; \n\t\t\ \
tau = 1. + hNext/h; \n
\t\t\t (*\ and\ predict\ the\ y - values\ for\ t + h + hNext\ *)
\t\t\t\t\n\ \ \ \ \ \ \
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\($DIRKLinkP2 = Null; \n\n
ParallelDIRKAlgorithmQ[DIRKParallel\ , lnk_] :=
If[Head[lnk] === LinkObject, True, False]\n\),
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\(ParallelDIRKAlgorithmQ[meth_Symbol, lnk_] :=
If[meth === DIRKParallel\ && \ Head[lnk] === LinkObject, True, False]\n
\),
\(ParallelDIRKAlgorithmQ[meth_Symbol, Null] := False\n\),
\(DIRKGetVector[lnk_]\ := \n\ \ \
Module[{pkt, \ i}, \n\ \ \ \ \ \ \ \ i\ = \ 0; \ \n\ \ \ \ \ \ \ \ \
While[\(i++\)\ < \ $IterationLimit, \n\t\ \ \ \ \
pkt\ = \ LinkRead[lnk]; \n\t\ \ \ \ \
If[VectorQ[pkt, \ NumericQ], \ Return[pkt]]; \n
\ \ \ \ \ \ \ \ \ \ \ \ \ \ Pause[1]; \n\ \ \ \ \ \ \ \ \ \ \ \ ];
\n\ \ ]\n\n (*\
A\ condition\ is\ added\ to\ the\ function\ to\ prevent\ the\ evaluation
\ inside\ DIRKParallel . \n\t\ \ the\ DIRKNDSolve[]\ function\ will
\ export\ $DIRKLinkP2 =
Null\ to\ P2\ and\ than\ set\ \n\t\t\ \ the\ value\ of\ $DIRKLinkP2\
to\ a\ \(LinkObject[] . \)\ *) \n\n\),
\(DIRKParallelOnP2[t_?NumericQ, y0_List, h_?NumericQ, maxit_Integer,
errPhi_?NumericQ] /; \ $DIRKLinkP2 === Null := \n\t\ \
Module[{xi1, xi2, psi4, xi4}, \t\t\ \n\t\t\t\
xi2 = newtonSolve[t, h, y0, $RightHandSide, 2/3, 2/3, $Jacobian,
maxit, errPhi]; \n\t\t\t\ \ LinkWrite[$ParentLink, xi2]; \n
\ \ \ \ \ \ \ \ \ xi1 = DIRKGetVector[$ParentLink]; \n\t\t\ \ \ \
psi4 = y0 + h*\((\(-5\)*xi1 + 4*xi2)\)/3; \n\t\t\ \ \ \
xi4 = newtonSolve[t, h, psi4, $RightHandSide, 1/3, 2/3, $Jacobian,
maxit, errPhi]; \n\ \ \ \ \ \ \ \ \ LinkWrite[$ParentLink, xi4];
\n\t\t\ \ ]\n\n\),
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errY_?NumericQ, yEstimate_List] := \n\t
Module[{xi1, xi2, xi3, xi4, psi3, y1, y1Estimate, hNext, tau}, \n
\t\t\ \ LinkWrite[$DIRKLinkP2,
EvaluatePacket[DIRKParallelOnP2[t, y0, h, maxit, errPhi]]]; \n
\t\t\ \ xi1 =
newtonSolve[t, h, y0, $RightHandSide, 1/2, 1/2, $Jacobian, maxit,
errPhi]; \n\t\t\txi2 = DIRKGetVector[$DIRKLinkP2]; \n\t\t\ \
LinkWrite[$DIRKLinkP2, xi1]; \n\t\t\ \
psi3 = y0 + 5*h*\((\(-xi1\) + xi2)\)/2; \n\t\t\ \
xi3 = newtonSolve[t, h, psi3, $RightHandSide, 1/2, 1/2, $Jacobian,
maxit, errPhi]; \n\t\t\ \ xi4 = DIRKGetVector[$DIRKLinkP2]; \n
\t\t\ \ While[LinkReadyQ[$DIRKLinkP2], \ (*\ Clean\ the\ link\ *) \n
\t\t\t\ \ \ \ \ LinkRead[$DIRKLinkP2]\n\t\t\t\ \ ]; \n\t\t\ \
y1 = y0 + h*\((3*\((xi2 + xi4)\)/2 - xi1 - xi3)\); \n
\t\t (*\ Now\ we\ estimate\ the\ error\ from\ the\ prediction\ of\ the
\ last\ step\ ... *) \n\t\t\ \
hNext = DIRKErrorEstimate[h, y1, yEstimate, errY]; \n\t\t\ \
tau = 1. + hNext/h; \n
\t\t\t (*\ and\ predict\ the\ y - values\ for\ t + h + hNext\ *)
\t\t\t\t\n\ \ \ \ \ \ \
y1Estimate = DIRKPredictY[y0, h, tau, xi1, xi2, xi3, xi4]; \n
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Cell[CellGroupData[{
Cell[BoxData[{
\(Clear[DIRKIntegrate]\n\t\),
\(DIRKIntegrate[tsym_, t0_, t1_, yInit_, \n\ \ \ \ hInit_, errImplicit_,
errY_, \n\t\ maxIterations_Integer, \n\tmaxSteps_, \n\tdirkmethod_\n
\ \ \ ] := \n\t
Module[{posneg, t, y, yEstimate, h, hNext, i, outsave}, \n\t\t\ \
posNeg = Sign[t1 - t0]; \n\t\t\ \ t = t0; \n\t\t\ \ y = yInit; \n
\t\t\ \ yEstimate = {}; \n\t\t\ \ h = \(hNext = hInit*posNeg\); \n
\t\t\ i = 0; \n\t\t\ outsave = {{t0, yInit}}; \n\t\t\
While[True, \n\t\t\t\ \ \
If[\((t - t1)\)*posNeg + $MachineEpsilon > 0.0, \ Break[]]; \n
\t\t\t\ \ \ If[\((t + h - t1)\)*posNeg > 0.0, \ h = t1 - t];
\ (*\ cut\ if\ overshoot\ the\ end\ point\ *) \n
\t\t\t\ \ \ {y, hNext, yEstimate} = \n\t\t\t\t\ \ \ \ \ \
dirkmethod[t, y, h, \n\t\t\t\t\t\ \ \ \ \ \ \ \ maxIterations,
errImplicit, errY, yEstimate]; \n\t\t\t\ \ \ \ t += h; \n
\t\t\t\ \ \ \ h = hNext; \n\t\t\t\ \ \
outsave = Join[outsave, {{t, y}}]; \n\t\t\t\ \ \
If[\(++i\)\ > \ maxSteps, \n\t\t\t\t\ \ \ \
Message[DIRKNDSolve::mxst, maxSteps, tsym, t]; \n\t\t\t\t\ \ \ \
Break[]; \n\t\t\t\t]\n\t\t\ \ \ ]; \n\t\t\ \ outsave\n\t\t]\)}],
"Input"],
Cell[BoxData[
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"Possible spelling error: new symbol name \"\!\(hInit\)\" is similar to \
existing symbol \"\!\(yInit\)\"."\)], "Message"],
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}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[{
\(\nClear[dirkMakeFun]\n
\n (*\ We\ must\ wrap\ a\ Hold[]\ around\ the\ Part[new, _]\ to\ \ \n
\t\t\tprevent\ that\ the\ part\ of\ a\ symbol\ is\ serached\ *) \n\t\t
\),
\(dirkMakeVars[y_List, new_Symbol] := \n\t
Module[{x, i}, \n\t\ \
Thread[y -> Table[Hold[new[\([x]\)]]\ /. \ x -> i, {i, Length[y]}]]\n
\t\t]\n\),
\(DIRKNDSolve::noexpl =
"\<No explicit solution found for equations `1`.\>"\n\n\t\),
\(dirkMakeFun[eqn : {_Equal .. }, y_List, t_Symbol, compile_: True] := \n
\tModule[{sol, f, jac, rules, newsym, fj}, \n\t\t\ \
sol = \ Solve[eqn, \(D[#, t]\ &\)\ /@\ y]; \n\t\t\
If[sol == {}, \n\t\t\t\ \ Message[DIRKNDSolve::noexpl, eqn]; \n
\t\t\t\ \ Return[$Failed]; \n\t\t\t]; \n\t\t\ \
f = D[y, t]\ /. \ sol; \ (*\ explicit\ solution\ *) \n\t\t\
jac = \(Outer[D[#1, #2]\ &, #, y]\ &\)\ /@\ f; \n\t\t
newsym = Unique[]; \n\t\trules = \ dirkMakeVars[y, newsym]; \n\t\t
fj = Transpose[{f, jac}] /. \ \ rules; \n\t\t
If[compile, \n
\t\t\t (*\
at\ first\ we\ add\ the\ function\ arguments\ for\ compile\n
\t\t\t\t\t\twith\ an\ evaluated\ newsym, \
than\ Hold[]\ is\ wrapped\n\t\t\t\t\t\tand\ the\ Hold[part[_]]\
are\ removed\ with\ the\ last\n\t\t\t\t\t\tReplace[]\ *) \n\t\t
fj = Map[Hold,
Map[{{{t, _Real}, {newsym, _Real, 1}}, #}\ &\ , fj, {2}]\ , {
2}] /. \ Hold[a_Part] :> \ a; \n
\t\t\t (*\
Now\ we\ can\ start\ with\ compile . \ To\ prevent\ a\ early\n
\t\t\t\t\t\tevaluation\ of\ the\ Part[]\ function\ in\ the\
Apply[]\ call\ an\ Unevaluated[]\ is\ needed\ *) \n\t\t\t
fj\ /. \ Hold[l_]\ :> \ Apply[Compile, Unevaluated[l]]\n
\t\t\t (*\ Else\ *) , \n\t\t\t
ReleaseHold[Map[Function[Evaluate[{t, newsym}], #]\ \ &, fj, {2}]]\n
\t\t\t]\n\t]\)}], "Input"],
Cell[BoxData[
\("No explicit solution found for equations `1`."\)], "Output"],
Cell[BoxData[
\(General::"spell1" \( : \ \)
"Possible spelling error: new symbol name \"\!\(compile\)\" is similar \
to existing symbol \"\!\(Compile\)\"."\)], "Message"]
}, Open ]],
Cell[BoxData[
\(\(DIRKMakeInterpolation[tsym_Symbol, l : {{_, _List} .. }] := \n\t\ \
Module[{t, y}, \n\t\ \ \ \ {t, y} = \ Transpose[l]; \n\t\t\ \
\(\(Interpolation[Transpose[{t, #}]]\)[tsym]\ &\)\ /@\ Transpose[y]\n
\ \ ]\t\t\)\)], "Input"],
Cell[CellGroupData[{
Cell[BoxData[{
\(\nOptions[DIRKNDSolve] = {MaxSteps -> 4000, MaxIterations -> 20,
ImplicitError -> Automatic, Compiled -> True, LocalError -> 0.1*^-5,
\n\t\ \ Method -> DIRKSerial, DIRKSlave -> Null}; \n\n
\n$DIRKMethods = {DIRKSerial, DIRKParallel}; \n\n
DIRKNDSolve::noninit =
"\<Initial values `3` are not numbers at `1`==`2`.\>"\),
\(DIRKNDSolve::nonrhs =
"\<Right hand side of `1` does not evaluate to numbers at `2`==`3`.\>"\
\),
\(DIRKNDSolve::ivmeth =
"\<Invalid solution method `1` given. Method must be one of `2`.\>"\),
\(DIRKNDSolve::invp2 =
"\<DIRKSlave must be a LinkObject[] for parallel evaluation instead of \
`1`. Proceed with serial algorithm.\>"\n\),
\(DIRKNDSolve[deqn : {_Equal .. },
y_List, {t_Symbol, t0_?NumericQ, t1_?NumericQ}, opts___] := \n\t
Module[{compile, maxit, maxsteps, y0, fj, fun, jac, errPhi, locError,
hinit, ndsdata, meth, p2slave}, \n
\t\t\ \ {compile, maxit, maxsteps, errPhi, locError} =
\({Compiled, MaxIterations, MaxSteps, ImplicitError, LocalError} /.
\ {opts}\) /. \ Options[DIRKNDSolve]; \n
\t\t\ \ {meth, p2slave} =
\({Method, DIRKSlave}\ /. \ {opts}\)\ /. \ Options[DIRKNDSolve];
\n\t\t\tIf[\(! MemberQ[$DIRKMethods, meth]\), \n\t\t\t\t\ \
Message[DIRKNDSolve::ivmeth, meth, $DIRKMethods]; \n\t\t\t\t\ \
Return[$Failed]; \n\t\t\t\t]; \n\t\t\
If[Automatic === errPhi, \n\t\t\t\ \ \(errPhi = locError/10. ; \)\n
\t\t\t]; \n\t\t\ \ y0 = y\ /. \ t -> \ t0; \n\t\t\ \
y0 = N[y0 /. Solve[deqn, y0]]; \n\t\t\
If[And\ @@\((\(\(! VectorQ[#, NumericQ]\)\ &\)/@\ y0)\), \n\t\t\t\ \
Message[DIRKNDSolve::noninit, t, t0, y0]; \n\t\t\t\ \
Return[$Failed]; \n\t\t\t]; \n\t\t\
fj = dirkMakeFun[deqn, y, t, compile]; \n\t\t\
If[\(! VectorQ[
Flatten[Function[z, \ Map[#[t0, z]\ &, fj, {2}]]\ /@\ y0],
NumericQ]\), \n\t\t\t\ \
Message[DIRKNDSolve::nonrhs, deqn, t, t0]; \n\t\t\t\ \
Return[$Failed]; \n\t\t\t]; \n\t\t{fun, jac} = fj[\([1]\)]; \n
\t\t\ $RightHandSide = fun; \ (*\ Assign\ the\ globals\ *) \n
\t\t\ \ $Jacobian = jac; \n\t\t\t
If[ParallelDIRKAlgorithmQ[meth, p2slave], \n\t\t\t\ \
ExportEnvironment[{newtonSolve, DIRKGetVector,
DIRKParallelOnP2, $RightHandSide, $Jacobian, $DIRKLinkP2}]; \n
\t\t\t\ \ While[LinkReadyQ[p2slave], LinkRead[p2slave]]; \n
\t\t\t\ \ $DIRKLinkP2 = p2slave, \n\t\t\t (*\ Else\ *) \n\t\t\t\ \
If[meth === DIRKParallel\ && \ Head[p2slave] =!= LinkObject, \n
\t\t\t\t\ \ $DIRKLinkP2 = Null; \n\t\t\t\ \ \ \
Message[DIRKNDSolve::invp2, p2slave]; \n\t\t\t\t\ \
meth = DIRKSerial; \n\t\t\t\t]\n\t\t\t\ ]; \n\t\t
y0 = y0[\([1]\)]; \n\t\t\ hinit = N[\((t1 - t0)\)/maxsteps]; \n\t\t
ndsdata =
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