Skip to main content

plotting - Mathematica taking forever to compute


I want to simulate the fresnel diffraction through circular aperture.But mathematica is taking forever to calculate.Here is my code.


  f[x_?NumericQ, y_?NumericQ, d_?NumericQ, p_?NumericQ, t_?NumericQ] := 
 NIntegrate[(\[Zeta]*(Exp[
       I*20000 ((d^2 + \[Zeta]^2)^(0.5) + ((p^2 + (x - \[Zeta]*
                  Cos[t])^2 + (y - \[Zeta]*
                  Sin[t])^2)^(0.5)))]))/(d^2 + \[Zeta]^2)^(0.5), {\
\[Zeta], 0, 0.008}]

w[x_?NumericQ, y_?NumericQ, d_?NumericQ, p_?NumericQ] := 
 NIntegrate[f[x, y, d, p, t], {t, 0, 2*Pi}]
g[x_, y_, d_, p_] := (Abs[w[x, y, d, p]])^2
Plot3D[g[x, y, 0.004, 20], {x, -100, 100}, {y, -100, 100}, 
 PlotRange -> Full]

What's the way out? *Specs:i5 7th gen intel processor,12 gb ram



Answer



Let's compile the integrand into a Listable CompiledFunction:


cintegrand = Block[{x, y, d, p, ζ, t},

   With[{code = 
      N[(ζ (Exp[I 20000 ((d^2 + ζ^2)^(1/2) + ((p^2 + (x - ζ Cos[t])^2 + (y - ζ Sin[t])^2)^(1/2)))]))/(d^2 + ζ^2)^(1/2)]
     },
    Compile[{{x, _Real}, {y, _Real}, {d, _Real}, {p, _Real}, {ζ, _Real}, {t, _Real}},
     code,
     CompilationTarget -> "C",
     RuntimeAttributes -> {Listable},
     Parallelization -> True
     ]
    ]

   ];

Next, pick a Gauss quadrature rule for high order polynomials (the integrand is very smooth but quite oscillatory):


{pts, weights, errweights} = NIntegrate`GaussRuleData[9, $MachinePrecision];

Divide the integration domain into $m \times n$ rectangles and set up the quadrature points and weights:


m = 20;
n = 20;
ζdata = Partition[Subdivide[0., 0.008, m], 2, 1];
tdata = Partition[Subdivide[0., 2 Pi, n], 2, 1];

{ζ, t} = Transpose[Tuples[{Flatten[ζdata.{1. - pts, pts}], Flatten[tdata.{1. - pts, pts}]}]];
ω = Flatten[KroneckerProduct[
    Flatten[KroneckerProduct[Differences /@ ζdata, weights]],
    Flatten[KroneckerProduct[Differences /@ tdata, weights]]
    ]];

Define the mapping that parameterizes the graph of OP's function w:


W = {x, y} \[Function] {x, y, Abs[cintegrand[x, y, 0.004, 20., ζ, t].ω]^2};

Compute the values of W on a $101 \times 101$ grid:



R = 2.;
data = Outer[W, Subdivide[-R, R, 100], Subdivide[-R, R, 100]]; // AbsoluteTiming // First


28.2668



Plot the result


ListPlot3D[Flatten[data, 1], PlotRange -> All, AxesLabel -> {"x", "y", "w"}]

enter image description here



Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more