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POHow to fit a function to a 3D numpy array?
 

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  1. POHow to fit a function to a 3D numpy array?
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    <p>I'm currently working on a project that involves calculating electric fields and there gradients in 3D space. This requires numerically solving Laplace's equation and I have written a class to do this (below) which works but here it is just for background;</p> <pre><code>################################################################################ # class: ThreeDRectLaplaceSolver # # # # A class to solve the Laplace equation given the potential on the boundaries.# ################################################################################ class ThreeDCuboidLaplaceSolver: ############################################################################# # Store the init variables as class varibales, calculate the grid spacing to# # solve Laplace's equation on and create arrays to store the iterations and # # results in. # ############################################################################ def __init__(self, xmin, xmax, ymin, ymax, zmin, zmax, gridstep): self.xmin, self.xmax = xmin, xmax self.ymin, self.ymax = ymin, ymax self.zmin, self.zmax = zmin, zmax self.xpoints = int((xmax-xmin)/gridstep) + 1 self.ypoints = int((ymax-ymin)/gridstep) + 1 self.zpoints = int((zmax-zmin)/gridstep) + 1 self.dx = (xmax-xmin)/self.xpoints self.dy = (ymax-ymin)/self.ypoints self.dz = (zmax-zmin)/self.zpoints self.v = np.zeros((self.xpoints, self.ypoints, self.zpoints)) self.old_v = self.v.copy() self.timeStep = 0 ############################################################################ # Set constant values along the boundaries # # # # Top(bottom) is +ive(-ive) end of z-axis # # Right(left) is +ive(-ive) end of y-axis # # Front(back) is +ive(-ive) end of x-axis # ############################################################################ def setBC(self, frontBC, backBC, rightBC, leftBC, topBC, bottomBC): self.v[-1, :, :] = frontBC self.v[0 , :, :] = backBC self.v[: ,-1, :] = rightBC self.v[: , 0, :] = leftBC self.v[: , :,-1] = topBC self.v[: , :, 0] = bottomBC self.old_v = self.v.copy() def solve_slow(self, PercentageTolerance = 5): PercentageError = PercentageTolerance + 1 while PercentageError &gt; PercentageTolerance: self.Iterate() PercentageError = self.Get_LargestPercentageError() print "Completed iteration number %s \n Percentage Error is %s\n" % (str(self.timeStep), str(PercentageError)) return self.v def solve_quick(self, Tolerance = 2): AbsError = Tolerance + 1 while AbsError &gt; Tolerance: self.Iterate() AbsError = self.Get_LargestAbsError() print "Completed iteration number %s \nAbsolute Error is %s\n" % (str(self.timeStep), str(AbsError)) return self.v def Get_LargestAbsError(self): return np.sqrt((self.v - self.old_v)**2).max() def Get_LargestPercentageError(self): AbsDiff = (np.sqrt((self.v - self.old_v)**2)).flatten() v = self.v.flatten() vLength = len(v) Errors = [] i=0 while i &lt; vLength: if v[i]==0 and AbsDiff[i]==0: Errors.append(0) elif v[i]==0 and AbsDiff[i]!=0: Errors.append(np.infty) else: Errors.append(AbsDiff[i]/v[i]) i+=1 return max(Errors)*100 # Perform one round of iteration (ie the value at each point is iterated by one timestep) def Iterate(self): self.old_v = self.v.copy() print self.Get_vAt(0,5,0) self.v[1:-1,1:-1,1:-1] = (1/26)*(self.v[0:-2, 2:, 2: ] + self.v[0:-2, 1:-1, 2: ] + self.v[0:-2, 0:-2, 2: ] +\ self.v[1:-1, 2:, 2: ] + self.v[1:-1, 1:-1, 2: ] + self.v[1:-1, 0:-2, 2: ] +\ self.v[2: , 2:, 2: ] + self.v[2: , 1:-1, 2: ] + self.v[2: , 0:-2, 2: ] +\ self.v[0:-2, 2:, 1:-1] + self.v[0:-2, 1:-1, 1:-1] + self.v[0:-2, 0:-2, 1:-1] +\ self.v[1:-1, 2:, 1:-1] + self.v[1:-1, 0:-2, 1:-1] +\ self.v[2: , 2:, 1:-1] + self.v[2: , 1:-1, 1:-1] + self.v[2: , 0:-2, 1:-1] +\ self.v[0:-2, 2:, 0:-2] + self.v[0:-2, 1:-1, 0:-2] + self.v[0:-2, 0:-2, 0:-2] +\ self.v[1:-1, 2:, 0:-2] + self.v[1:-1, 1:-1, 0:-2] + self.v[1:-1, 0:-2, 0:-2] +\ self.v[2: , 2:, 0:-2] + self.v[2: , 1:-1, 0:-2] + self.v[2: , 0:-2, 0:-2]) self.timeStep += 1 # Iterate through a certain number of time steps def IterateSteps(self, timeSteps): i = 0 while i &lt; timeSteps: self.Iterate() i+=1 # Get the value of v at a point (entered as coordinates, NOT indices) def Get_vAt(self, xPoint, yPoint, zPoint): # Find the indices nearest to the coordinates entered diff = [np.sqrt((x-xPoint)**2) for x in np.linspace(self.xmin,self.xmax,self.xpoints)] xIndex = diff.index(min(diff)) diff = [np.sqrt((y-yPoint)**2) for y in np.linspace(self.ymin,self.ymax,self.ypoints)] yIndex = diff.index(min(diff)) diff = [np.sqrt((z-zPoint)**2) for z in np.linspace(self.zmin,self.zmax,self.zpoints)] zIndex = diff.index(min(diff)) # retun the value from of v at this point return self.v[xIndex, yIndex, zIndex] </code></pre> <p>So when I run a the following</p> <pre><code> solver = ThreeDCuboidLaplaceSolver(0, 20, 0, 10, 0, 20, 1) TwoDsolver = TwoDRectLaplaceSolver(0,20,0,10,1) TwoDsolver.setBC(1000,0,0,0) v_0 = np.zeros((21,11)) v_0[4:6,4:6] = 1000 v_0[14:15, 9:10] = 2000 solver.setBC(0,0,0,0,0,v_0) v = solver.solve_quick(0.00001) </code></pre> <p>I get a 3D numpy array with the value of the potential (V) at each point on my grid. However in order to do more useful things with this I would like to be able to approximate this array of values with a continuous function so I can calculate the field and its gradient at points not on my grid.</p> <p>A) Is this possibe? B) How would you go about doing this? I have seen basic scipy fitting functions but nothing that would handle 3D data in this way (or indeed fit an analogous function for a 2D array).</p> <p>Might be a long shot that someone has tried to do this before but any help would be really appreciated,</p> <p>Thanks </p>
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      1. POHow to fit a function to a 3D numpy array?
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    1. COCheck http://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.interpolation.map_coordinates.html
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      1. POHow to fit a function to a 3D numpy array?
      1. USarynaq
    2. COHey, thanks for the quick response. The link you gave (if I have understood it right, I have only had a quick play) is really useful for finding the value of the field at a point not on my grid. I noticed that you set an order for the spline interpolation which if I have understood correctly means you have to guess whether the field between grid points is linear, quadratic, cubic ect. Since I don't know this, is there any way to perform a similar operation that automatically calculates the best fit to all the data points?
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      1. POHow to fit a function to a 3D numpy array?
      1. USuser1562862
 

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