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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 VCLOD Test for Coefficient 4
This script performs the main VCLOD test for this thesis with a specific diffusion coefficient. We investigate the energy error of the VCLOD dependent on the updated correctors. For this purpose, we update every corrector individually and compare it to the reference solution. This enables a good comparison between percentages. We desire to yield a fast decrease of the energy error of the VCLOD method since, due to the error indicator, we sort and update the element correctors in terms of the effect that comes with the perturbation.
Step1:  Result function
The 'result' function investigates the VCLOD for each percentage. The reference solution is computed by a standard FEM on the fine mesh. We compute the 'worst solution' that represents zero percentage updating and clearly has no computational cost at all. Afterwards, we compute the error indicator for the given patch size $k=4$ and use every value gradually. Furthermore we store the resulting energy error for the VCLOD as well as the optimal energy error that results from 100 percentage updating. Once again, we take advantage of the 'gridlod' module in order to compute the required matrices.
Step2:  Preparations
We use the same setting as we have already used before containing the 'buildcoef2d' class in order to construct the coefficient. We visualize the coefficient and store the information in an extern folder.
Step3:  Perturbations of the same entries
To keep comparability, we use the 'specific' perturbation function and use a random seed.
Step4:  Precomputations
Step5:  Change in value
Step6:  Disappearance
Step7:  Shift
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Python Code:
import os
import sys
import numpy as np
import scipy.sparse as sparse
import random
import csv
%matplotlib notebook
import matplotlib.pyplot as plt
from visualize import drawCoefficient
from data import * 
from gridlod import interp, coef, util, fem, world, linalg, femsolver
import pg_rand, femsolverCoarse, buildcoef2d
from gridlod.world import World
Explanation: VCLOD Test for Coefficient 4
This script performs the main VCLOD test for this thesis with a specific diffusion coefficient. We investigate the energy error of the VCLOD dependent on the updated correctors. For this purpose, we update every corrector individually and compare it to the reference solution. This enables a good comparison between percentages. We desire to yield a fast decrease of the energy error of the VCLOD method since, due to the error indicator, we sort and update the element correctors in terms of the effect that comes with the perturbation.
End of explanation
def result(pglod, world, A, R, f, k, String):
    print "-------------- " + String + " ---------------" 
    NWorldFine = world.NWorldFine
    NWorldCoarse = world.NWorldCoarse
    NCoarseElement = world.NCoarseElement
    
    boundaryConditions = world.boundaryConditions
    NpFine = np.prod(NWorldFine+1)
    NpCoarse = np.prod(NWorldCoarse+1)
        
    # new Coefficient
    ANew = R.flatten()
    Anew = coef.coefficientFine(NWorldCoarse, NCoarseElement, ANew)
    
    # reference solution
    f_fine = np.ones(NpFine)
    uFineFem, AFine, MFine = femsolver.solveFine(world, ANew, f_fine, None, boundaryConditions)
    
    # worst solution
    KFull = pglod.assembleMsStiffnessMatrix()
    MFull = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse)
    free  = util.interiorpIndexMap(NWorldCoarse)                                 
    
    bFull = MFull*f
    KFree = KFull[free][:,free]
    bFree = bFull[free]
    xFree = sparse.linalg.spsolve(KFree, bFree)
    
    basis = fem.assembleProlongationMatrix(NWorldCoarse, NCoarseElement)
    
    basisCorrectors = pglod.assembleBasisCorrectors()
    modifiedBasis = basis - basisCorrectors
    
    xFull = np.zeros(NpCoarse)
    xFull[free] = xFree
    uCoarse = xFull
    uLodFine = modifiedBasis*xFull
    
    uLodFineWorst = uLodFine
    
    # energy error
    errorworst = np.sqrt(np.dot(uFineFem - uLodFineWorst, AFine*(uFineFem - uLodFineWorst)))
    
    # tolerance = 0 
    vis, eps = pglod.updateCorrectors(Anew, 0, f, 1, clearFineQuantities=False, Computing=False)
    
    PotentialCorrectors = np.sum(vis)
    elemente = np.arange(np.prod(NWorldCoarse))
            
    # identify tolerances
    epsnozero = filter(lambda x: x!=0, eps)
    
    assert(np.size(epsnozero) != 0)
    
    mini = np.min(epsnozero)
    minilog = int(round(np.log10(mini)-0.49))
    epsnozero.append(10**(minilog))
    ToleranceListcomplete = []
    for i in range(0,int(np.size(epsnozero))):
        ToleranceListcomplete.append(epsnozero[i])
    ToleranceListcomplete.sort()
    ToleranceListcomplete = np.unique(ToleranceListcomplete)
    # with tolerance
    errorplotinfo = []
    tolerancesafe = []
    errorBest = []
    errorWorst = []
    recomputefractionsafe = []
    recomputefraction = 0
    Correctors = 0
    leng = np.size(ToleranceListcomplete)
    for k in range(leng-1,-1,-1):
        tol = ToleranceListcomplete[k]
        print " --- "+ str(-k+leng) + "/" + str(leng)+ " --- Tolerance: " + str(round(tol,5)) + " in "+ String +" ---- ", 
        vistol = pglod.updateCorrectors(Anew, tol, f, clearFineQuantities=False, Testing=True)
        
        Correctors += np.sum(vistol)
        
        recomputefraction += float(np.sum(vistol))/PotentialCorrectors * 100
        recomputefractionsafe.append(recomputefraction)
        
        KFull = pglod.assembleMsStiffnessMatrix()
        MFull = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse)
        free  = util.interiorpIndexMap(NWorldCoarse)                                 
        bFull = MFull*f
        KFree = KFull[free][:,free]
        bFree = bFull[free]
        xFree = sparse.linalg.spsolve(KFree, bFree)
        basis = fem.assembleProlongationMatrix(NWorldCoarse, NCoarseElement)
        basisCorrectors = pglod.assembleBasisCorrectors()
        modifiedBasis = basis - basisCorrectors
        xFull = np.zeros(NpCoarse)
        xFull[free] = xFree
        uCoarse = xFull
        uLodFine = modifiedBasis*xFull
        
        #energy error
        errortol = np.sqrt(np.dot(uFineFem - uLodFine, AFine*(uFineFem - uLodFine)))
        
        errorplotinfo.append(errortol)
        tolerancesafe.append(tol)
    
    # 100% updating
    uLodFinebest = uLodFine
    errorbest = np.sqrt(np.dot(uFineFem - uLodFinebest, AFine*(uFineFem - uLodFinebest)))
    
    for k in range(leng-1,-1,-1):
        errorBest.append(errorbest)
        errorWorst.append(errorworst)
    return vis, eps, PotentialCorrectors, recomputefractionsafe, errorplotinfo, errorWorst, errorBest
Explanation: Result function
The 'result' function investigates the VCLOD for each percentage. The reference solution is computed by a standard FEM on the fine mesh. We compute the 'worst solution' that represents zero percentage updating and clearly has no computational cost at all. Afterwards, we compute the error indicator for the given patch size $k=4$ and use every value gradually. Furthermore we store the resulting energy error for the VCLOD as well as the optimal energy error that results from 100 percentage updating. Once again, we take advantage of the 'gridlod' module in order to compute the required matrices.
End of explanation
bg = 0.05       #background
val = 1         #values
#fine World
NWorldFine = np.array([256, 256])
NpFine = np.prod(NWorldFine+1)                                                                               
#coarse World
NWorldCoarse = np.array([16,16])
NpCoarse = np.prod(NWorldCoarse+1)
#ratio between Fine and Coarse
NCoarseElement = NWorldFine/NWorldCoarse
boundaryConditions = np.array([[0, 0],
                               [0, 0]])
world = World(NWorldCoarse, NCoarseElement, boundaryConditions)
#righthandside
f = np.ones(NpCoarse)
#Coefficient 4
CoefClass = buildcoef2d.Coefficient2d(NWorldFine, 
                        bg                  = bg, 
                        val                 = val, 
                        thick               = 1, 
                        space               = 0, 
                        probfactor          = 1, 
                        right               = 0, 
                        down                = 0, 
                        diagr1              = 1, 
                        diagr2              = 0, 
                        diagl1              = 1, 
                        diagl2              = 0, 
                        LenSwitch           = [4,5,6,7,8], 
                        thickSwitch         = None, 
                        equidistant         = None, 
                        ChannelHorizontal   = None, 
                        ChannelVertical     = None,
                        BoundarySpace       = None) 
A = CoefClass.BuildCoefficient()
ABase = A.flatten()
ROOT = '../test_data/Coef4'
#safe NworldFine
with open("%s/NWorldFine.txt" % ROOT, 'wb') as csvfile:
    writer = csv.writer(csvfile)
    for val in NWorldFine:
        writer.writerow([val])
#safe NworldCoarse
with open("%s/NWorldCoarse.txt" % ROOT, 'wb') as csvfile:
    writer = csv.writer(csvfile)
    for val in NWorldCoarse:
        writer.writerow([val])
#ABase
with open("%s/OriginalCoeff.txt" % ROOT, 'wb') as csvfile:
    writer = csv.writer(csvfile)
    for val in ABase:
        writer.writerow([val])
#fine-fem
f_fine = np.ones(NpFine)
uFineFem, AFine, MFine = femsolver.solveFine(world, ABase, f_fine, None, boundaryConditions)
#fine solution
with open("%s/finescale.txt" % ROOT, 'wb') as csvfile:
    writer = csv.writer(csvfile)
    for val in uFineFem:
        writer.writerow([val])
        
plt.figure("Original")
drawCoefficient(NWorldFine, ABase,greys=True)
plt.title("Original coefficient")
plt.show()
Explanation: Preparations
We use the same setting as we have already used before containing the 'buildcoef2d' class in order to construct the coefficient. We visualize the coefficient and store the information in an extern folder.
End of explanation
# random seed
random.seed(20)
# decision
valc = np.shape(CoefClass.ShapeRemember)[0]
numbers = []
decision = np.zeros(100)
decision[0] = 1
for i in range(0,valc):
    a = random.sample(decision,1)[0]
    if a == 1:
        numbers.append(i)
value1 = 3
C1 = CoefClass.SpecificValueChange(ratio=value1,
                                    Number = numbers,
                                    probfactor=1,
                                    randomvalue=None,
                                    negative=None,
                                    ShapeRestriction=True,
                                    ShapeWave=None,
                                    ChangeRight=1,
                                    ChangeDown=1,
                                    ChangeDiagr1=1,
                                    ChangeDiagr2=1,
                                    ChangeDiagl1=1,
                                    ChangeDiagl2=1,
                                    Original = True,
                                    NewShapeChange = True)
V = CoefClass.SpecificVanish(Number = numbers,
                                probfactor=1,
                                PartlyVanish=None,
                                ChangeRight=1,
                                ChangeDown=1,
                                ChangeDiagr1=1,
                                ChangeDiagr2=1,
                                ChangeDiagl1=1,
                                ChangeDiagl2=1,
                                Original = True)
M1 = CoefClass.SpecificMove(probfactor=1,
                            Number = numbers,
                            steps=1,
                            randomstep=None,
                            randomDirection=None,
                            ChangeRight=1,
                            ChangeDown=1,
                            ChangeDiagr1=1,
                            ChangeDiagr2=1,
                            ChangeDiagl1=1,
                            ChangeDiagl2=1,
                            Right=1,
                            BottomRight=0,
                            Bottom=0,
                            BottomLeft=0,
                            Left=0,
                            TopLeft=0,
                            Top=0,
                            TopRight=0,
                            Original = True)
Explanation: Perturbations of the same entries
To keep comparability, we use the 'specific' perturbation function and use a random seed.
End of explanation
k = 4
NWorldFine = world.NWorldFine
NWorldCoarse = world.NWorldCoarse
NCoarseElement = world.NCoarseElement
boundaryConditions = world.boundaryConditions
NpFine = np.prod(NWorldFine+1)
NpCoarse = np.prod(NWorldCoarse+1)
#interpolant
IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix(i, N, NWorldCoarse, NCoarseElement, boundaryConditions)
#old Coefficient
ABase = A.flatten()
Aold = coef.coefficientFine(NWorldCoarse, NCoarseElement, ABase)
pglod = pg_rand.VcPetrovGalerkinLOD(Aold, world, k, IPatchGenerator, 0)
pglod.originCorrectors(clearFineQuantities=False)
Explanation: Precomputations
End of explanation
vis, eps, PotentialUpdated, recomputefractionsafe, errorplotinfo, errorworst, errorbest = result(pglod ,world, A, C1, f, k, 'Specific value change' + str(value1))
safeChange(ROOT, C1, vis, eps, PotentialUpdated, recomputefractionsafe, errorplotinfo, errorworst, errorbest)
Explanation: Change in value
End of explanation
vis, eps, PotentialUpdated, recomputefractionsafe, errorplotinfo, errorworst, errorbest = result(pglod ,world, A, V, f, k, 'Vanish')
safeVanish(ROOT, V, vis, eps, PotentialUpdated, recomputefractionsafe, errorplotinfo, errorworst, errorbest)
Explanation: Disappearance
End of explanation
vis, eps, PotentialUpdated, recomputefractionsafe, errorplotinfo, errorworst, errorbest = result(pglod ,world, A, M1, f, k, 'One Step Move')
safeShift(ROOT, M1, vis, eps, PotentialUpdated, recomputefractionsafe, errorplotinfo, errorworst, errorbest)
Explanation: Shift
End of explanation
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