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performance tuning - Is there any way to speed up this code that's Maximizing a function got from numerical integration?


I have this code below, which is calculating the Binding energy of an electron in a Quantum Well Wire with a hydrogenic impurity in it. Well you don't have to care much about what kind of calculation it does, because it is returning the right number, my only problem is, that it's taking about 20 minutes for it to return a single value for the Eb function (you can try Eb[0.7, 1, 0.01]). I'm wondering, if there's a way to make this code run faster. As you can see I have written everything almost the same way as one would write on paper. I've searched and tried many different approaches to make it faster, but nothing has helped so far.


e = 4.803*10^-10;
m = 0.067*9.109*10^-28;
h = 1.054*10^-27;
c = 2.997*10^10;
e0 = 13.18;


O1[Om_] = 10^13*Om;

oH[h0_] = (e*10000*h0)/(m*c);

Oc[h0_, Om_] = Sqrt[oH[h0]^2 + 4*O1[Om]^2];

aH[h0_, Om_] = Sqrt[h/(m*Oc[h0, Om])];

r0[rho_, phi_, z_, rhoi_] = 
  Sqrt[rho^2 + rhoi^2 - 2*rho*rhoi*Cos[phi] + z^2];


Psi[rho_, h0_, Om_] = E^(-(1/2)*(rho^2*aH[1, Om]^2)/aH[h0, Om]^2);

MGamma[rho_, phi_, z_, rhoi_, lambda_, Om_] = 
  E^(-lambda*r0[rho, phi, z, rhoi]);

CPhi[rho_, phi_, z_, rhoi_, lambda_, h0_, Om_] = 
  Psi[rho, h0, Om]*MGamma[rho, phi, z, rhoi, lambda, Om];

intCPhiCPhi[rhoi_, lambda_, h0_, Om_] := 

  NIntegrate[
   Abs[CPhi[rho, phi, z, rhoi, lambda, h0, Om]]^2*rho, {rho, 0, 
    Infinity}, {phi, 0, 2*\[Pi]}, {z, -Infinity, +Infinity}];

leftover[rho_, phi_, z_, rhoi_, lambda_, h0_, Om_] = 
  CPhi[rho, phi, z, rhoi, lambda, h0, Om]*
    h^2/(2*m*(aH[1, Om])^2)*(Psi[rho, h0, Om]/rho*
       D[MGamma[rho, phi, z, rhoi, lambda, Om], rho] + 
      2*D[MGamma[rho, phi, z, rhoi, lambda, Om], rho]*
       D[Psi[rho, h0, Om], rho] + 

      Psi[rho, h0, Om]*
       D[MGamma[rho, phi, z, rhoi, lambda, Om], {rho, 2}] + 
  Psi[rho, h0, Om]/rho^2*
   D[MGamma[rho, phi, z, rhoi, lambda, Om], +{phi, 2}] + 
  Psi[rho, h0, Om]*
   D[MGamma[rho, phi, z, rhoi, lambda, Om], {z, 2}]) + (e^2*
  Abs[CPhi[rho, phi, z, rhoi, lambda, h0, Om]]^2)/(e0*
  r0[rho, phi, z, rhoi]*aH[1, Om]);

IntLeftover[rhoi_?NumericQ, lambda_?NumericQ, h0_?NumericQ, 

  Om_?NumericQ] := (2*e0^2*h^2)/(e^4*
     m)*((1/intCPhiCPhi[rhoi, lambda, h0, Om] )*
    NIntegrate[
     leftover[rho, phi, z, rhoi, lambda, h0, Om]*rho, {rho, 0, 
      Infinity}, {phi, 0, 2*\[Pi]}, {z, -Infinity, +Infinity}])

Eb[rhoi_, h0_, Om_] := 
  FindMaximum[IntLeftover[rhoi, lambda, h0, Om], lambda];

Answer



One way to speed up multidimensional integrals is to reduce the PrecisionGoal. That will be acceptable in some cases but not in others.



Modified OP's code:


Clear[e, m, h, c, e0]  (* I moved the parameter initialization to later (unimportant *)

O1[Om_] = 10^13*Om;
oH[h0_] = (e*10000*h0)/(m*c);
Oc[h0_, Om_] = Sqrt[oH[h0]^2 + 4*O1[Om]^2];
aH[h0_, Om_] = Sqrt[h/(m*Oc[h0, Om])];
r0[rho_, phi_, z_, rhoi_] = Sqrt[rho^2 + rhoi^2 - 2*rho*rhoi*Cos[phi] + z^2];
Psi[rho_, h0_, Om_] = E^(-(1/2)*(rho^2*aH[1, Om]^2)/aH[h0, Om]^2);
MGamma[rho_, phi_, z_, rhoi_, lambda_, Om_] = E^(-lambda*r0[rho, phi, z, rhoi]);

CPhi[rho_, phi_, z_, rhoi_, lambda_, h0_, Om_] = 
  Psi[rho, h0, Om]*MGamma[rho, phi, z, rhoi, lambda, Om];

intCPhiCPhi[rhoi_, lambda_, h0_, Om_] := 
  NIntegrate[Abs[CPhi[rho, phi, z, rhoi, lambda, h0, Om]]^2*rho, (* NB: Abs[..]^2=(..)^2 *)
   {phi, 0, 2*Ï€}, {rho, 0, Infinity}, {z, -Infinity, +Infinity},
   PrecisionGoal -> 3];

leftover[rho_, phi_, z_, rhoi_, lambda_, h0_, Om_] = 
  CPhi[rho, phi, z, rhoi, lambda, h0, Om]*

    h^2/(2*m*(aH[1, Om])^2)*(Psi[rho, h0, Om]/rho*
       D[MGamma[rho, phi, z, rhoi, lambda, Om], rho] + 
      2*D[MGamma[rho, phi, z, rhoi, lambda, Om], rho]*D[Psi[rho, h0, Om], rho] + 
      Psi[rho, h0, Om]*D[MGamma[rho, phi, z, rhoi, lambda, Om], {rho, 2}] + 
      Psi[rho, h0, Om]/rho^2*D[MGamma[rho, phi, z, rhoi, lambda, Om], +{phi, 2}] + 
      Psi[rho, h0, Om]*D[MGamma[rho, phi, z, rhoi, lambda, Om], {z, 2}]) +
      (e^2*Abs[CPhi[rho, phi, z, rhoi, lambda, h0, Om]]^2) / 
       (e0*r0[rho, phi, z, rhoi]*aH[1, Om]);

IntLeftover[rhoi_?NumericQ, lambda_?NumericQ, h0_?NumericQ, 

  Om_?NumericQ] := (2*e0^2*h^2)/(e^4*
     m)*((1/intCPhiCPhi[rhoi, lambda, h0, Om])*
    NIntegrate[leftover[rho, phi, z, rhoi, lambda, h0, Om]*rho,
     {phi, 0, 2*Ï€}, {rho, 0, Infinity}, {z, -Infinity, +Infinity},
      PrecisionGoal -> 3]);

Eb[rhoi_, h0_, Om_] := FindMaximum[IntLeftover[rhoi, lambda, h0, Om], lambda];

OP's example:


PrintTemporary@Dynamic@Clock@{0, Infinity};  (* running timer (helps preserve my sanity *)

Block[{  (* parameter initialization *)
  e = 4.803*10^-10,
  m = 0.067*9.109*10^-28,
  h = 1.054*10^-27,
  c = 2.997*10^10,
  e0 = 13.18},
 Eb[0.7, 1, 0.01] // AbsoluteTiming
 ]



FindMaximum::lstol: The line search decreased the step size to within the tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient increase in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances.



(*  {11.0647, {1.16175, {lambda -> 2.28525}}}  *)

Playing with it, I sometimes did not get a FindMaximum::lstol warning, but I always got 2.28525 or 2.28526 for an answer. I suspect that the noise from the numerical integration is the source.


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