canhuaLiu
commented
1 year ago Dear Dr. Jakeman, After reading online documentation， I try to construct a polynomial chaos expansion (PCE) of a simple 2-D frame RE model with uncertain parameters using Leja sequences, the capacity of validation_samples is 200, the max_nsamples is 10000, the tolerance is 1e-10，however, the errors between pce_values and validation_values are not convergent and even increase. I would like to ask why does such a problem occur. The code is as follows： `import numpy as np from scipy import stats from pyapprox.variables import IndependentMarginalsVariable from pyapprox.interface.wrappers import evaluate_1darray_function_on_2d_array import math from warnings import simplefilter from pyapprox import analysis from pyapprox import surrogates import openseespy.opensees as ops simplefilter(action='ignore', category=FutureWarning)
def evaluate(NB): ops.wipe() ops.model('Basic', '-ndm', 2, '-ndf', 3)
h = 4
w = 3
ops.node(1, 0.0, 0.0)
ops.node(2, h, 0.0)
ops.node(3, 0.0, w)
ops.node(4, h, w)
ops.fix(1, 1, 1, 1)
ops.fix(2, 1, 1, 1)
ops.fix(3, 0, 0, 0)
ops.fix(4, 0, 0, 0)
ops.mass(3, NB[0], 0.0, 0.0)
ops.mass(4, NB[0], 0.0, 0.0)
ops.geomTransf('Linear', 1)
ops.element('elasticBeamColumn', 1, 1, 3, 0.25, NB[1], NB[2], 1)
ops.element('elasticBeamColumn', 2, 2, 4, 0.25, NB[1], NB[2], 1)
ops.element('elasticBeamColumn', 3, 3, 4, 0.25, NB[1], NB[2], 1)
ops.rayleigh(NB[3], 0, 0, 0) # RAYLEIGH damping
dt = 0.02
ops.timeSeries('Path', 200, '-dt', dt, '-filePath', 'EI.txt', '-factor', 10)
ops.pattern('UniformExcitation', 200, 1, '-accel', 200)
ops.constraints('Transformation')
ops.numberer('RCM')
ops.system('UmfPack')
ops.test('NormDispIncr', 0.000001, 1000)
ops.algorithm('KrylovNewton')
ops.integrator('Newmark', 0.55, 0.2765625)
ops.analysis('Transient')
tCurrent = ops.getTime()
tFinal = 52
time = [tCurrent]
us = [0.0]
ax = [0.0]
ok = 0
while tCurrent < tFinal:
while ok == 0 and tCurrent < tFinal:
ops.analysis('Transient')
ok = ops.analyze(1, .02)
if ok == 0:
tCurrent = ops.getTime()
time.append(tCurrent)
us.append(ops.nodeDisp(3, 1))
ax.append(ops.nodeAccel(3, 1))
usm = abs(max(us, key=abs))
return usmAvalues = [] def compute_l2_error(validation_samples, validation_values, pce, relative=True): pce_values = pce(validation_samples) Avalues.append(pce_values) error = np.linalg.norm(pce_values - validation_values, axis=0) if not relative: error /= np.sqrt(validation_samples.shape[1]) else: error /= np.linalg.norm(validation_values, axis=0)
return errornp.random.seed(1)
def trunNor(mu, sigma): lower, upper = mu - 2 sigma, mu + 2 sigma # 截断在[μ-3σ, μ+3σ] X = stats.truncnorm((lower - mu) / sigma, (upper - mu) / sigma, loc=mu, scale=sigma) return X
X1 = trunNor(20, 2) X2 = trunNor(2e5, 2e4) X3 = trunNor(5.21e-3, 5e-4) X4 = trunNor(0.05, 0.005) univariate_variables = [X1, X2, X3, X4] variable = IndependentMarginalsVariable(univariate_variables) nsamples = 150 validation_samples = variable.rvs(nsamples)
np.savetxt("SS.txt", validation_samples.T)
def pyapprox_fun_0(validation_samples): values = evaluate_1darray_function_on_2d_array(evaluate, validation_samples) return values
validation_values = pyapprox_fun_0(validation_samples) errors = [] num_samples = []
def callback(pce): error = compute_l2_error(validation_samples, validation_values, pce) errors.append(error) num_samples.append(pce.samples.shape[1])
max_num_samples = 200
opts = {"method": "leja", "options": {"max_nsamples": 1000, "tol": 1e-10, "callback": callback}} pce = surrogates.adaptive_approximate(pyapprox_fun_0, variable, "polynomial_chaos", opts).approx
res = analysis.gpc_sobol_sensitivities(pce.pce, variable) print(res.main_effects[:, 0])
S = np.size(errors) np.savetxt("R.txt", validation_values) Avalues = np.array(Avalues) Avalues = np.reshape(Avalues, (S, -1)) np.savetxt("V.txt", Avalues.T)
`
The EI file data is as follows：
-0.002865497 -0.022105263 -0.020672515 -0.018011696 -0.019444444 -0.024561404 -0.029064326 -0.026198831 -0.02251462 -0.01739766 -0.01739766 -0.026812866 -0.036023392 -0.039707601 -0.033157895 -0.029473683 -0.022105263 -0.016783625 -0.008596491 -0.013508771 -0.026812866 -0.038888887 -0.040116959 -0.013508771 0.006140351 0.028859649 -0.01002924 -0.026198831 -0.029473683 -0.041549706 -0.053216373 -0.066520467 -0.062631578 -0.035204679 -0.040321638 -0.033362571 -0.03356725 -0.01371345 0.005116959 0.030701753 0.048304094 0.051578948 0.068771925 0.094766079 0.100701753 0.085760235 0.073479533 0.055467835 0.048099414 0.069385966 0.084327484 0.108479529 0.130789473 0.149824551 0.133450284 0.122602338 0.081871344 0.081871344 0.012894737 -0.105409354 -0.161081861 -0.123421048 -0.099064329 -0.051169589 -0.012076023 0.027426899 0.063040932 0.102134504 0.145321633 0.203654973 0.249707605 0.313157878 0.296783609 0.237426903 0.191374265 0.182573089 0.189532156 0.171725146 0.184415197 0.203245616 0.247660821 0.0671345 -0.30292396 -0.423684195 -0.407309926 -0.41549706 -0.372514604 -0.354093551 -0.358187118 -0.358187118 -0.37046782 -0.333625715 -0.276315792 -0.223099418 -0.160058469 -0.087807018 -0.003479532 0.073684209 0.16067251 0.237426903 0.327485364 0.401169575 0.493274839 0.558771916 0.622222209 0.654970747 0.699999987 0.577192969 0.474853786 -0.245614038 -0.485087704 -0.335672498 -0.382748522 -0.225146201 -0.154122809 -0.035409355 0.023128655 0.109093562 0.18318713 0.243567254 0.360233902 0.11789473 -0.538304079 -0.317251445 -0.354093551 -0.206725148 -0.118508773 0.048508772 -0.1371345 -0.405263142 -0.335672498 -0.345906416 -0.30292396 -0.251754389 -0.204678365 -0.153713444 -0.107046779 -0.055467835 -0.009005848 0.038479531 -0.019444444 -0.088625728 -0.17152047 -0.194649115 -0.146549701 -0.122602338 -0.068362574 -0.022105263 0.037865495 0.085964911 0.137748541 -0.0198538 -0.07614035 -0.008187134 0.002251462 0.070409357 0.115643272 0.180730996 0.231286552 0.278362576 0.04482456 0.049327486 0.139795325 0.141023391 0.270175441 0.276315792 0.417543844 -0.190555548 -0.268128658 -0.141637416 -0.111754381 0.014736841 0.138157892 -0.21900585 -0.304970743 -0.21900585 -0.237426903 -0.155964902 -0.114415206 -0.044005846 -0.025789474 -0.137953216 -0.06631579 -0.068976609 -0.02230994 0.003479532 0.061198829 0.099883036 0.12444444 0.045438597 -0.006549707 -0.050146197 0.015760233 0.043187135 0.116257305 0.169064321 0.247660821 0.30292396 0.356140335 0.086169586 0.005935673 0.053011694 0.059970757 -0.01125731 -0.030087718 0.029269006 0.042163743 0.102134504 0.132017541 0.195877183 0.231286552 0.296783609 0.333625715 0.399122791 0.380701738 0.405263142 0.362280686 0.255847956 -0.247660821 -0.110935672 -0.078596492 -0.063654969 -0.229239769 -0.339766065 -0.503508757 -0.41549706 -0.376608171 -0.270175441 -0.196491224 -0.066520467 0.031520466 0.167017538 0.270175441 0.372514604 -0.011871345 -0.034590642 0.058333332 0.091491227 0.201198833 0.290643258 0.378654955 0.503508757 0.345906416 -0.282456143 -0.204473686 -0.223099418 -0.185643265 -0.095994153 -0.255847956 -0.43187133 -0.331578931 -0.345906416 -0.268128658 -0.227192985 -0.158216376 -0.104385962 -0.11134503 -0.245614038 -0.247660821 -0.237426903 -0.235380119 -0.146754375 -0.111754381 0.013099415 -0.164561403 -0.333625715 -0.175818713 -0.196695899 -0.081052626 -0.030087718 0.065292394 0.132631582 0.179298238 0.096608186 0.040526314 -0.005526316 0.059766083 0.091081868 0.16067251 0.210818716 0.276315792 0.329532147 0.380701738 0.261988307 0.130994149 0.041754384 0.064269005 0.076345026 0.101520463 0.048099414 -0.017192981 -0.034385963 -0.023128655 -0.046871344 -0.05076023 -0.032134501 -0.014122807 0.030087718 0.0775731 0.118508773 0.052192981 -0.008391813 -0.087602336 -0.027222223 0.019444444 0.047076022 -0.026403507 -0.010233918 0.016374269 0.042982455 0.077777777 0.104385962 0.032134501 -0.006549707 -0.022719298 0.001023392 0.015555555 0.007163743 -0.019444444 -0.007368421 -0.003274854 0.007777777 0.01739766 -0.011461988 -0.062222221 -0.086169586 -0.049941519 -0.048304094 -0.036228071 -0.026403507 -0.00368421 0.041549706 -0.022105263 -0.018625731 -0.006959064 -0.021695907 -0.022719298 -0.020263156 -0.000409357 0.01494152 0.048099414 0.072660817 0.144298243 0.159444444 0.037660817 -0.053830406 -0.025380115 -0.008596491 0.032543858 0.009824562 -0.04482456 -0.095584794 -0.087602336 -0.044210527 -0.00880117 0.032543858 0.065497075 0.085760235 0.025175439 -0.032748536 -0.041754384 -0.016783625 -0.042163743 -0.028040934 -0.01125731 0.010847953 0.027426899 0.054444443 0.047485381 0.01616959 -0.001637427 0.040935671 0.089035085 0.100701753 0.039093568 0.018830409 -0.004502924 -0.004298245 0.010643274 0.019035088 0.052192981 0.075321634 0.107456137 0.110730989 0.086988301 0.081461985 0.114415206 0.154736836 0.074707601 0.084122802 0.020058479 -0.041754384 -0.050964911 -0.082894736 -0.084532161 -0.096403504 -0.088625728 -0.093742687 -0.011666666 0.036432747 -0.042573096 -0.100701753 -0.108479529 -0.074093566 -0.082894736 -0.063040932 -0.064678361 -0.054239765 -0.054239765 -0.055058478 -0.070614034 -0.063245613 -0.044415205 -0.015964912 0.017807017 0.057514618 0.063450291 0.073274851 0.069795316 0.073274851 0.058742691 0.062426899 0.022923975 0.043801168 0.027836258 0.078596492 -0.176228062 -0.276315792 -0.274269009 -0.276315792 -0.243567254 -0.212865499 -0.169678362 -0.133245607 -0.090877194 -0.052807014 -0.012280701 -0.018625731 -0.037251462 -0.030087718 0.01739766 0.033362571 0.010233918 0.054035086 0.119122806 0.17745613 0.245614038 0.3479532 0.227192985 -0.225146201 -0.074912275 -0.091081868 -0.048304094 -0.196491224 -0.134268999 -0.122192982 -0.1371345 -0.112982455 -0.005526316 0.077368418 0.21900585 0.341812849 0.193830399 0.083508769 0.136520459 0.027017542 -0.019444444 -0.106432746 -0.169268996 -0.235380119 -0.235380119 -0.164356727 -0.075526317 0.005935673 0.111549705 0.24152047 0.329532147 -0.055263156 0.006959064 -0.011461988 0.004093567 0.029883041 0.10991228 0.163333335 -0.041959063 -0.120760231 -0.034590642 -0.035818712 -0.005730994 0.015146199 0.078187133 0.116052631 0.154122809 0.163947362 0.12116959 0.062222221 0.004707602 0.013099415 -0.08309941 -0.092309936 -0.01616959 0.034385963 0.116052631 0.019035088 -0.01125731 0.009005848 -0.025175439 -0.057719299 -0.089444443 -0.072046784 -0.052192981 -0.022719298 0.041959063 0.106228071 0.174795322 0.233333336 0.150029243 0.048508772 -0.075321634 -0.055467835 -0.044415205 -0.178684212 -0.199152049 -0.120555555 -0.068771925 0.015760233 0.053011694 0.103976603 0.073888884 0.016578948 -0.011461988 -0.042777776 -0.064883042 -0.048713447 -0.076959059 -0.112573097 -0.147777767 -0.164356727 -0.107046779 -0.069590642 -0.002251462 0.013304093 -0.007573099 -0.001023392 -0.034385963 -0.083918128 -0.016374269 0.01616959 0.076549708 0.12587719 0.136111109 0.051988303 -0.011666666 -0.097017545 -0.072865492 -0.049736839 -0.009824562 0.025789474 0.0775731 0.049327486 -0.046461989 -0.087602336 -0.138976608 -0.135292391 -0.120760231 -0.104999995 -0.083508769 -0.063245613 -0.054444443 -0.110730989 -0.128538007 -0.185847955 -0.227192985 -0.180321629 -0.157602335 -0.119122806 -0.096812863 -0.068157891 -0.040730993 0.004093567 0.043187135 0.088421052 0.125467832 0.156988294 0.190964914 0.21900585 0.231286552 0.243567254 0.255847956 0.272222225 0.32543858 0.368421037 0.417543844 0.253801172 0.090467835 -0.028654971 -0.136315783 -0.113596488 -0.141842109 -0.201403511 -0.255847956 -0.24152047 -0.214912283 -0.18830409 -0.152076025 -0.165584795 -0.173976604 -0.176023388 -0.176637429 -0.178684212 -0.177660821 -0.181140347 -0.10991228 0.010643274 0.044005846 0.050146197 0.118713448 0.064269005 0.048304094 0.099269003 0.120555555 0.107456137 0.072660817 0.040321638 0.040730993 0.100701753 0.070204675 0.058947365 0.088421052 0.048918127 0.018011696 0.015760233 -0.030292398 -0.015760233 -0.003888889 0.015350877 0.009005848 -0.029678361 -0.064678361 -0.049327486 -0.005730994 0.037251462 0.087192977 0.089853802 0.104795321 0.095380112 0.098040937 0.039502922 0.045438597 0.05608187 0.080438593 0.103157896 0.118099414 0.120350872 0.168245604 0.163128645 0.194239764 0.070614034 0.009210526 -0.025175439 -0.071023393 -0.087192977 -0.085146194 -0.056286548 -0.055263156 0.015146199 0.087602336 -0.047280702 -0.079210525 -0.016988304 0.028450291 0.091081868 0.005526316 -0.142660808 -0.16292397 -0.05137427 -0.027631579 0.01616959 -0.023538012 -0.05137427 -0.068157891 -0.055058478 -0.061608186 -0.040935671 -0.01371345 -0.007777777 0.021491228 0.060584794 0.070409357 0.195877183 0.183801172 0.036637425 -0.074093566 -0.203450295 -0.165175428 -0.152280701 -0.110321639 -0.067543858 -0.026198831 0.006345029 0.030292398 0.103976603 -0.004502924 -0.10008772 -0.073274851 -0.141432742 -0.10561403 -0.075935667 0.018011696 0.129356725 0.172134495 0.261988307 0.284502927 0.243567254 0.153713444 0.04605263 -0.018011696 -0.046461989 0.015146199 0.037046784 0.11134503 0.081666668 0.009210526 -0.016783625 -0.037865495 -0.004093567 0.00122807 -0.023947367 -0.042982455 -0.062017541 -0.104795321 -0.148801159 -0.118508773 -0.054444443 -0.036432747 0.008187134 0.020058479 0.028040934 0.045233919 0.089444443 0.018625731 -0.112163738 -0.113596488 -0.049736839 -0.016578948 0.051169589 0.083918128 0.037251462 -0.005526316 -0.049736839 -0.003070175 0.050555556 0.09865497 0.16026316 0.127309941 0.067748533 -0.002865497 -0.039912279 -0.050555556 -0.043391813 -0.02251462 0.010233918 0.049327486 -0.006959064 -0.044210527 -0.096403504 -0.074298242 -0.039912279 -0.00368421 0.03479532 -0.016374269 0.001023392 0.047076022 0.076549708 0.123011689 0.10561403 0.088421052 0.070409357 0.10336257 0.133654974 0.139795325 0.035204679 -0.03479532 -0.107865496 -0.135906432 -0.079210525 -0.045438597 -0.006754386 0.024356723 -0.026198831 -0.0718421 -0.105204679 -0.06856725 -0.04461988 -0.00245614 0.029064326 0.014327485 -0.012894737 -0.024561404 -0.065906433 -0.070818708 -0.018625731 0.01494152 0.063245613 0.096608186 0.123421048 0.11789473 0.067543858 -0.01494152 -0.159035077 -0.12444444 -0.08964912 -0.042777776 0.006345029 0.071637426 0.059970757 0.024766082 0.069181283 0.064883042 0.051988303 0.042163743 0.040526314 0.035614033 0.004298245 -0.029473683 -0.070204675 -0.069385966 -0.029678361 -0.005730994 0.03479532 -0.019649123 -0.052192981 -0.057105262 -0.079415202 -0.049532164 -0.044005846 -0.037251462 -0.035614033 -0.007777777 -0.005526316 -0.037865495 -0.025175439 0.017807017 0.070204675 0.14225146 0.186257306 0.174590645 0.155555551 0.104999995 0.038070176 0.003070175 -0.038888887 -0.030906433 -0.01494152 0.004298245 0.026403507 0.044005846 0.004912281 -0.025380115 -0.067339182 -0.106228071 -0.144912284 -0.118508773 -0.094561403 -0.062836258 -0.029678361 -0.001842105 -0.036842104 -0.065087716 -0.095175436 -0.080029235 -0.070614034 -0.064678361 -0.089035085 -0.100497071 -0.097222219 -0.085964911 -0.073888884 -0.056695907 -0.052807014 -0.028450291 -0.013918129 0.103771929 0.147777767 0.179707604 0.160058469 0.156578943 0.089853802 0.016374269 0.002660819 -0.025789474 -0.003070175 0.006140351 0.021286548 0.021286548 0.039502922 0.041959063 0.015146199 -0.011461988 -0.014736841 0.014327485 0.021695907 0.030087718 -0.001842105 -0.032543858 -0.038274854 -0.001432748 0.031725145 0.021491228 -0.023538012 -0.061812866 -0.063245613 -0.019444444 -0.011871345 0.000818713 0.004093567 0.010233918 0.011666666 0.0198538 0.027426899 0.036228071 0.04461988 0.053421051 0.061812866 0.070818708 0.079005843 0.097017545 0.080438593 0.048713447 0.023538012 -0.01616959 -0.025380115 0.011052631 0.005526316 -0.051169589 -0.115847946 -0.128947366 -0.120964906 -0.084532161 -0.013918129 0.055672515 0.056695907 -0.004298245 -0.012280701 -0.02251462 -0.045233919 -0.085146194 -0.106228071 -0.045438597 0.006140351 0.01616959 0.028450291 0.035 0.051783622 0.066111108 0.080029235 0.03356725 -0.027836258 -0.066111108 -0.059561402 -0.058742691 -0.062222221 -0.069385966 -0.050146197 -0.015555555 0.025584796 0.076959059 0.082280701 0.050146197 0.031929825 -0.008187134 -0.03131579 -0.059152045 -0.064678361 -0.022719298 0.019239765 0.06856725 0.11789473 0.086783627 0.029269006 -0.001432748 -0.027631579 -0.055263156 -0.069795316 -0.073070176 -0.081052626 -0.082280701 -0.099883036 -0.098245611 -0.08309941 -0.083304092 -0.0718421 -0.038274854 -0.011666666 0.009005848 -0.003888889 -0.014736841 -0.034590642 -0.023538012 0.025789474 0.073274851 0.133859649 0.146549701 0.155964902 0.15125731 0.128538007 0.099064329 0.054035086 -0.009005848 -0.058947365 -0.078596492 -0.100701753 -0.087602336 -0.085146194 -0.056491227 -0.010643274 0.048508772 0.087192977 0.12362573 0.092514619 0.058128654 0.025789474 -0.011052631 -0.056286548 -0.086578944 -0.035818712 0.000204678 0.051988303 0.098450287 0.131198824 0.113391814 0.084532161 0.037660817 -0.009824562 -0.062017541 -0.108684206 -0.144912284 -0.189941523 -0.176637429 -0.12915204 -0.076959059 0.017807017 0.063245613 0.120555555 0.125672514 0.078801168 0.0718421 0.063654969 0.050350878 0.003888889 -0.040526314 -0.032339182 -0.003070175 0.020263156 0.05853801 0.083508769 0.115233913 0.108684206 0.064269005 0.033771928 -0.004912281 -0.038684209 -0.056491227 -0.075935667 -0.09210526 -0.109298247 -0.098859646 -0.0775731 -0.060584794 -0.040116959 -0.037660817 -0.032543858 -0.024356723 -0.010847953 0.000409357 0.012076023 -0.004707602 -0.022923975 -0.041959063 -0.065906433 -0.079415202 -0.069795316 -0.058742691 -0.0671345 -0.083304092 -0.099678362 -0.115233913 -0.131812865 -0.113596488 -0.091286545 -0.001637427 0.051783622 0.084122802 0.131812865 0.118508773 0.097017545 0.078596492 0.078801168 0.069590642 0.073070176 0.001637427 -0.051988303 -0.094152044 -0.096403504 -0.045438597 -0.013304093 0.03356725 0.072660817 0.103157896 0.075935667 0.05730994 0.032339182 0.008391813 -0.005526316 0.004298245 0.009005848 0.020263156 0.027631579 0.019239765 0.011871345 0.004502924 -0.006345029 -0.006345029 0.000818713 -0.005116959 -0.025380115 -0.048099414 -0.08309941 -0.108479529 -0.143479526 -0.066111108 -0.010643274 0.019239765 0.0671345 0.097836254 0.104181288 0.073274851 0.070000001 0.062631578 0.058333332 0.053830406 0.034385963 -0.001023392 -0.043391813 -0.074707601 -0.063450291 -0.060789474 -0.05730994 -0.048508772 -0.054444443 -0.063040932 -0.074912275 -0.073070176 -0.063040932 -0.039502922 0.003888889 0.040116959 0.032748536 0.026403507 0.028654971 0.02251462 0.022105263 0.018830409 0.018216373 0.003888889 -0.026812866 -0.050555556 -0.089239761 -0.088421052 -0.061403507 -0.039298246 -0.009824562 0.0198538 0.034385963 0.030292398 0.035409355 0.015964912 -0.011871345 -0.044005846 -0.047894735 0.012280701 0.053625732 0.055058478 0.017192981 -0.008391813 -0.046461989 -0.015555555 0.006345029 0.037251462 0.063450291 0.098040937 0.093947369 0.033976609 -0.011871345 -0.081052626 -0.090877194 -0.049327486 -0.020672515 0.044415205 0.053421051 0.024766082 0.001637427 -0.034385963 -0.078187133 -0.115847946 -0.159853795 -0.126695906 -0.052397659 0.009005848 0.093333328 0.16026316 0.225146201 0.195058482 0.10008772 0.024970758 -0.079210525 -0.173157887 -0.251754389 -0.176842105 -0.126081865 -0.046666663 0.030701753 0.097631578 0.066111108 0.05076023 0.019444444 0.046461989 0.069181283 0.10336257 0.121578947 0.106228071 0.112982455 0.121783623 0.126286549 0.121374264 0.095380112 0.032339182 -0.015760233 -0.047076022 -0.062017541 -0.067339182 -0.074502925 -0.100292395 -0.135087717 -0.14798246 -0.160467836 -0.173157887 -0.115847946 -0.042777776 0.026198831 0.077982451 0.099883036 0.074298242 0.051169589 0.032339182 0.077163742 0.121578947 0.128947366 0.075730993 0.044415205 0.008187134 0.004707602 -0.012690058 -0.05853801 -0.077777777 -0.093538011 -0.10336257 -0.159035077 -0.128538007 -0.038684209 0.038684209 0.104181288 0.152280701 0.153304093 0.115029239 0.093128652 0.107456137 0.129970758 0.149619877 0.174181278 0.189736832 0.1995614 0.190146197 0.166608187 0.054035086 -0.059152045 -0.184415197 -0.278362576 -0.175818713 -0.095789471 0.016578948 0.121578947 0.168450294 0.113801163 0.061608186 -0.002251462 -0.077777777 -0.071023393 -0.004093567 0.036842104 0.129561399 0.206725148 0.229239769 0.227192985 0.192397657 0.164766078 0.12444444 0.103976603 0.042163743 -0.041140351 -0.117076023 -0.208771932 -0.272222225 -0.251754389 -0.247660821 -0.206725148 -0.155964902 -0.112163738 -0.10008772 -0.082894736 -0.059561402 -0.036023392 -0.028859649 -0.020058479 -0.010233918 0.003888889 0.016374269 0.007163743 -0.005935673 -0.01248538 -0.006959064 -0.002660819 0.015555555 0.041959063 0.066929825 0.09210526 0.118099414 0.115029239 0.094970761 0.061403507 0.021900583 -0.000818713 -0.006754386 -0.014122807 -0.020672515 -0.041140351 -0.038479531 0.005116959 0.027426899 0.049736839 0.054444443 0.054239765 0.031725145 -0.003888889 -0.019444444 -0.044210527 -0.034181287 -0.024561404 -0.013099415 -0.026403507 -0.033362571 -0.039502922 -0.049532164 -0.048304094 -0.035818712 -0.025380115 -0.037865495 -0.054239765 -0.066111108 -0.068771925 -0.092923978 -0.088011693 -0.068362574 -0.043596488 -0.014122807 0.006140351 0.000614035 -0.019035088 -0.018216373 -0.030701753 -0.03356725 -0.048713447 -0.066111108 -0.086169586 -0.093538011 -0.081257309 -0.071432749 -0.052807014 -0.035204679 -0.004093567 0.031929825 0.058128654 0.074093566 0.072456141 0.055058478 0.020672515 -0.009210526 -0.025584796 -0.050146197 -0.046871344 -0.025789474 -0.013918129 0.00368421 0.019035088 0.040935671 0.05853801 0.074707601 0.063654969 0.037046784 0.004912281 -0.031929825 -0.065292394 -0.04482456 -0.024152047 0.003070175 0.03131579 0.060994149 0.049736839 0.027836258 0.020877191 0.004912281 -0.002046784 -0.004707602 -0.007368421 -0.009824562 -0.018216373 -0.031725145 -0.030292398 -0.015555555 0 -0.004093567 -0.030292398 -0.04605263 -0.076549708 -0.074707601 -0.05137427 -0.03356725 -0.002865497 0.03131579 0.061812866 0.080438593 0.08309941 0.078801168 0.067339182 0.037456139 0.025584796 0.014327485 -0.000204678 -0.013918129 -0.026608187 -0.026198831 -0.023538012 -0.020877191 -0.017192981 -0.028859649 -0.041959063 -0.055058478 -0.071637426 -0.073479533 -0.060789474 -0.046461989 0.00368421 0.020263156 0.041549706 0.052397659 0.040730993 0.032748536 0.021900583 0.031929825 0.040526314 0.049532164 0.028859649 0.01125731 -0.013508771 -0.0198538 -0.005321637 0.003274854 0.020467836 0.019239765 0.007982456 0 -0.013508771 -0.020263156 -0.02230994 -0.023538012 -0.026812866 -0.034385963 -0.040730993 -0.021286548 -0.002046784 0.01125731 0.026403507 0.040935671 0.040730993 0.034590642 0.029269006 0.02230994 0.026198831 0.031111109 0.028040934 0.023947367 0.0198538 0.012280701 -0.005730994 -0.010233918 -0.017192981 -0.036432747 -0.065701749 -0.085964911 -0.106023387 -0.096608186 -0.080643276 -0.059356724 -0.02251462 0.010233918 0.027017542 0.032134501 0.036228071 0.039912279 0.052192981 0.067748533 0.0671345 0.051578948 0.035409355 0.009210526 -0.01739766 -0.038888887 -0.046871344 -0.062222221 -0.056695907 -0.046461989 -0.035614033 -0.024766082 -0.025789474 -0.026403507 -0.016374269 -0.005321637 0.007982456 0.013304093 0.01125731 0.010233918 0.013099415 0.025789474 0.036637425 0.049736839 0.062836258 0.060994149 0.05137427 0.044210527 0.033362571 0.039298246 0.047894735 0.057719299 0.061608186 0.040116959 0.021695907 -0.007982456 -0.033362571 -0.065906433 -0.06856725 -0.04482456 -0.030292398 -0.002046784 0.002251462 -0.010847953 -0.020672515 -0.025789474 -0.029269006 -0.026403507 -0.021286548 -0.014532163 -0.00368421 0.006754386 0.01739766 0.032339182 0.048918127 0.065292394 0.069795316 0.065087716 0.043596488 0.01494152 0.001842105 -0.003070175 -0.009619882 -0.015350877 -0.024766082 -0.031929825 -0.02374269 -0.011461988 -0.00122807 -0.000614035 -0.000614035 0.002865497 0.009415204 0.014736841 0.01760234 0.020058479 0.025380115 0.030087718 0.035204679 0.040935671 0.052397659 0.064883042 0.058742691 0.047280702 0.021491228 -0.000204678 -0.002251462 -0.007368421 -0.010847953 -0.016988304 -0.010643274 -0.001432748 0.007573099 0.019649123 0.031725145 0.041959063 0.029269006 0.01494152 0.004298245 -0.004502924 -0.014327485 -0.020467836 -0.015350877 -0.011052631 -0.005935673 -0.003070175 0 -0.000204678 -0.00122807 -0.002251462 -0.002046784 -0.000614035 0.000204678 0.003274854 0.010847953 0.01760234 0.025789474 0.031520466 0.026812866 0.020877191 0.011461988 0.00122807 -0.008187134 -0.020058479 -0.019649123 -0.009415204 -0.001432748 0.006345029 0.009210526 0.013918129 0.010643274 0.009210526 0.00368421 -0.000409357 -0.005935673 -0.005730994 -0.003479532 -0.00122807 -0.001023392 -0.003479532 -0.004502924 -0.007368421 -0.006549707 -0.001432748 0.002865497 0.008391813 0.013304093 0.012894737 0.010643274 0.00122807 -0.009210526 -0.015555555 -0.013099415 -0.013304093 -0.021900583 -0.032953217 -0.035614033 -0.023947367 -0.014327485 -0.007368421 -0.005116959 -0.000409357 0.000818713 0.007368421 0.016783625 0.023538012 0.014327485 0.007777777 -0.002046784 -0.005116959 -0.006959064 -0.008391813 -0.009210526 -0.013304093 -0.024561404 -0.023947367 -0.022105263 -0.020058479 -0.016578948 -0.015555555 -0.02251462 -0.028245614 -0.033976609 -0.029473683 -0.026608187 -0.022719298 -0.022105263 -0.020058479 -0.020058479 -0.012894737 -0.000409357 0.011052631 0.02251462 0.031111109 0.040321638 0.03479532 0.028654971 0.021081872 0.013099415 0.002046784 -0.01371345 -0.021491228 -0.024970758 -0.028245614 -0.033976609 -0.044005846 -0.055058478 -0.05608187 -0.044415205 -0.036432747 -0.02374269 -0.01248538 0 0.011052631 0.020263156 0.029473683 0.038274854 0.047485381 0.053625732 0.052192981 0.046461989 0.035 0.02599415 0.011052631 -0.009619882 -0.030497074 -0.042982455 -0.051988303 -0.051578948 -0.050146197 -0.04625731 -0.036637425 -0.027836258 -0.016783625 -0.002865497 0.009005848 0.018216373 0.0198538 0.020467836 0.020467836 0.021286548 0.027836258 0.024766082 0.020467836 0.014736841 0.003888889 -0.005730994 -0.016988304 -0.026812866 -0.038888887 -0.039707601 -0.028450291 -0.020467836 -0.009005848 -0.009005848 -0.008187134 -0.009210526 -0.01002924 -0.007982456 -0.004298245 -0.004502924 -0.003274854 -0.003274854 -0.00245614 -0.00245614 0.005730994 0.013918129 0.023538012 0.019649123 0.011461988 0.003479532 -0.006345029 -0.015964912 -0.018830409 -0.018830409 -0.019444444 -0.018830409 -0.01760234 -0.010643274 -0.004502924 0.002865497 0.007777777 0.006959064 0.006959064 0.005935673 0.008187134 0.011052631 0.01371345 0.01739766 0.017807017 0.015760233 0.014327485 0.01248538 0.015350877 0.017807017 0.020877191 0.022719298 0.018421052 0.015350877 0.010847953 0.012690058 0.01616959 0.0198538 0.024152047 0.027631579 0.021491228 0.015760233 0.009005848 0.00245614 -0.004298245 -0.008596491 -0.013918129 -0.017807017 -0.024561404 -0.032543858 -0.040526314 -0.047690055 -0.052807014 -0.055263156 -0.056900581 -0.058742691 -0.056900581 -0.055263156 -0.052807014 -0.05137427 -0.04482456 -0.035409355 -0.026403507 -0.02230994 -0.021491228 -0.019444444 -0.018625731 -0.014736841 -0.012076023 -0.008187134 -0.004707602 0.00368421 0.007163743 0.008596491 0.009824562 0.00880117 0.00880117 0.007777777 0.007368421 0.006549707 0.009619882 0.012894737 0.01739766 0.02251462 0.027426899 0.032748536 0.037046784 0.035409355 0.034385963 0.032543858 0.030087718 0.022105263 0.013918129 0.006549707 0.001842105 0.003888889 0.005730994 0.008187134 0.011052631 0.014122807 0.019444444 0.023947367 0.029269006 0.029269006 0.025584796 0.024970758 0.025380115 0.025789474 0.026608187 0.027222223 0.021695907 0.01616959 0.009415204 0.004093567 0.001023392 -0.002251462 -0.005116959 -0.002251462 0.00368421 0.00880117 0.015555555 0.021081872 0.027222223 0.020672515 0.012690058 0.00368421 -0.005526316 -0.016374269 -0.018216373 -0.011052631 -0.006140351 0.00122807 0.00245614 0.004502924 0.005526316 0.006549707 0.009824562 0.014122807 0.018216373 0.022923975 0.027222223 0.028654971 0.028040934 0.028654971 0.027631579 0.028450291 0.020877191 0.006140351 -0.006549707 -0.012894737 -0.01739766 -0.022923975 -0.032134501 -0.040321638 -0.038479531 -0.037251462 -0.03479532 -0.031929825 -0.029064326 -0.026403507 -0.025584796 -0.024561404 -0.02374269 -0.023333332 -0.021695907 -0.013099415 -0.004298245 0.000409357 -0.000818713 -0.000818713 -0.002865497 -0.003888889 -0.001637427 0.002251462 0.005526316 0.010233918 0.010847953 0.008596491 0.006140351 0.002251462 -0.001023392 -0.004912281 -0.004093567 -0.001432748 0.000818713 0.003888889 0.006754386 0.009824562 0.011666666 0.013304093 0.015146199 0.016578948 0.018625731 0.02599415 0.034590642 0.029473683 0.021491228 0.014327485 0.007163743 0.000409357 -0.006549707 -0.013099415 -0.019444444 -0.018625731 -0.016988304 -0.01494152 -0.010438596 -0.006140351 -0.000818713 0.006754386 0.010643274 0.009005848 0.008187134 0.006140351 0.004093567 0.002046784 0.000204678 -0.002046784 -0.002865497 0.000409357 0.003274854 0.006140351 0.004707602 0.003274854 0.003274854 0.005321637 0.006549707 0.009210526 0.009415204 0.004298245 -0.000614035 -0.006140351 -0.012280701 -0.017807017 -0.02374269 -0.021900583 -0.019239765 -0.015964912 -0.013918129 -0.012894737 -0.011666666 -0.010233918 -0.008391813 -0.006754386 -0.004912281 -0.002865497 -0.001023392 0.000818713 0.002660819 0.004298245 0.000204678 -0.00368421 -0.008187134 -0.013099415 -0.017807017 -0.02251462 -0.026403507 -0.030906433 -0.033771928 -0.035614033 -0.037660817 -0.039707601 -0.04134503 -0.043596488 -0.040526314 -0.035409355 -0.027631579 -0.013304093 -0.002865497 0.004093567 0.01125731 0.016783625 0.018216373 0.020877191 0.019035088 0.016374269 0.01248538 0.006754386 0.000614035 -0.004912281 -0.011666666 -0.009210526 -0.002660819 0.002660819 0.006754386 0.009824562 0.011052631 0.011871345 0.012690058 0.013304093 0.014122807 0.014736841 0.015350877 0.015760233 0.01494152 0.014122807 0.013099415 0.011871345 0.011052631 0.009824562 0.01248538 0.01616959 0.019649123 0.023333332 0.025789474 0.028450291 0.026812866 0.025175439 0.023333332 0.021081872 0.017807017 0.014736841 0.011461988 0.008187134 0.004912281 0.001432748 0.000818713 0.001637427 0.001842105 0.004707602 0.008596491 0.010643274 0.01248538 0.014736841 0.01616959 0.018216373 0.019444444 0.019444444 0.019035088 0.018216373 0.005116959 0 -0.001023392 -0.004093567 -0.004912281 -0.006754386 -0.009824562 -0.013099415 -0.016374269 -0.0198538 -0.023333332 -0.026198831 -0.021695907 -0.018216373 -0.013508771 -0.00880117 -0.003888889 -0.002251462 -0.001842105 -0.001023392 -0.000818713 0.001023392 0.002865497 0.004912281 0.007163743 0.00880117 0.009005848 0.007982456 0.002660819 -0.001637427 -0.007368421 -0.009619882 -0.01125731 -0.012690058 -0.013918129 -0.01494152 -0.016374269 -0.015350877 -0.013099415 -0.005730994 0.001023392 0.009415204 0.013304093 0.012280701 0.012280701 0.011052631 0.01002924 0.009005848 0.007982456 0.005935673 0 -0.005321637 -0.010643274 -0.008596491 -0.007573099 -0.005116959 -0.003070175 -0.000409357 0.000409357 0 -0.000204678 -0.000818713 -0.001432748 -0.00122807 -0.000204678 0.000614035 0.001432748 -0.000204678 -0.001432748 -0.003070175 -0.004912281 -0.005935673 -0.005730994 -0.005730994 -0.005116959 -0.004912281 -0.004298245 -0.005526316 -0.009005848 -0.011871345 -0.015555555 -0.017192981 -0.016783625 -0.016988304 -0.015350877 -0.010233918 -0.005730994 -0.000409357 0.00368421 0.00368421 0.004298245 0.004298245 0.004298245 0.006345029 0.00880117 0.01002924 0.01002924 0.01002924 0.009824562 0.009619882 0.010643274 0.01371345 0.01616959 0.019444444 0.0198538 0.019649123 0.019649123 0.019035088 0.018625731 0.018216373 0.017807017 0.017192981 0.016783625 0.016374269 0.015760233 0.012690058 0.009619882 0.007163743 0.007368421 0.006959064 0.007163743 0.007368421 0.006549707 0.003888889 0.003070175 0.002865497 0.00245614 0.00245614 0.002251462 0 -0.002046784 -0.004093567 -0.005116959 -0.005935673 -0.006754386 -0.007573099 -0.009824562 -0.012894737 -0.015760233 -0.018830409 -0.021900583 -0.024970758 -0.028245614 -0.029269006 -0.024561404 -0.021081872 -0.015964912 -0.01371345 -0.012894737 -0.011871345 -0.011461988 -0.010847953 -0.010233918 -0.009824562 -0.009210526 -0.008596491 -0.007982456 -0.006345029 -0.005321637 -0.002251462 0.002046784 0.006345029 0.009415204 0.009824562 0.011052631 0.01125731 0.012894737 0.017192981 0.020672515 0.017192981 0.014532163 0.010847953 0.006754386 0.005730994 0.005935673 0.005730994 0.005526316 0.00245614 -0.000204678 -0.002046784 -0.002251462 -0.003274854 -0.003070175 -0.006345029 -0.013918129 -0.020263156 -0.020877191 -0.021491228 -0.021695907 -0.020263156 -0.018216373 -0.01616959 -0.015146199 -0.014736841 -0.013918129 -0.01371345 -0.012894737 -0.008187134 -0.002660819 0.002660819 0.00880117 0.014532163 0.020263156 0.01760234 0.012894737 0.008391813 0.002660819 -0.002865497 -0.007163743 -0.007573099 -0.009005848 -0.009415204 -0.01002924 -0.010438596 -0.011052631 -0.011461988 -0.011871345 -0.009415204 -0.005935673 -0.002660819 0.001432748 0.003479532 0.003888889 0.005116959 0.005116959 0.007163743 0.010233918 0.013099415 0.016374269 0.019239765 0.0198538 0.020877191 0.021491228 0.022105263 0.02251462 0.021286548 0.020263156 0.019035088 0.01760234 0.01616959 0.014736841 0.012894737 0.01125731 0.009415204 0.007777777 0.006345029 0.005526316 0.004298245 0.003479532 0.00245614 0.001432748 0.000614035 0.001637427 0.002660819 0.003888889 0.004502924 0.001842105 -0.000614035 -0.003479532 -0.006345029 -0.009619882 -0.010847953 -0.009824562 -0.00880117 -0.007777777 -0.005116959 -0.001842105 0.001432748 0.003479532 0.004502924 0.007163743 0.01248538 0.016988304 0.022923975 0.021900583 0.017807017 0.014532163 0.009619882 0.005526316 0.000818713 -0.003274854 -0.004912281 -0.006754386 -0.009415204 -0.014122807 -0.017192981 -0.018011696 -0.019035088 -0.019444444 -0.0198538 -0.020263156 -0.020672515 -0.020877191 -0.020058479 -0.019444444 -0.018421052 -0.017807017 -0.018216373 -0.018625731 -0.019035088 -0.019649123 -0.0198538 -0.018011696 -0.016578948 -0.014532163 -0.013918129 -0.013918129 -0.01371345 -0.01371345 -0.013508771 -0.013508771 -0.013099415 -0.011461988 -0.010233918 -0.00880117 -0.006959064 -0.004707602 -0.002660819 -0.000409357 0.002046784 0.003479532 0.001842105 0.000614035 -0.001023392 -0.002865497 -0.001432748 0.00122807 0.003274854 0.002660819 0.00245614 0.001637427 0.001023392 0 0.000409357 0.001842105 0.002865497 0.004298245 0.004502924 0.002046784 0 -0.00245614 -0.004912281 -0.004912281 -0.003888889 -0.003274854 -0.00368421 -0.004502924 -0.005526316 -0.006140351 -0.005321637 -0.004502924 -0.004502924 -0.006549707 -0.008391813 -0.010438596 -0.009415204 -0.00880117 -0.007777777 -0.006549707 -0.005321637 -0.004093567 -0.003888889 -0.00368421 -0.003070175 0.002660819 0.008391813 0.01371345 0.014327485 0.015555555 0.015964912 0.016374269 0.016578948 0.017192981 0.015964912 0.01248538 0.009824562 0.006345029 0.005935673 0.006140351 0.006549707 0.007163743 0.007573099 0.004912281 0.001432748 -0.001637427 -0.005730994 -0.007163743 -0.004502924 -0.002660819 0.000204678 0.000614035 0.000204678 -0.000204678 -0.001023392 -0.001637427 -0.002251462 -0.003070175 -0.00245614 -0.001637427 -0.000818713 0.000204678 0.001023392 0.001637427 0.001637427 0.001842105 0.002046784 0.004502924 0.006754386 0.009415204 0.012076023 0.014736841 0.01616959 0.012690058 0.010233918 0.006549707 0.003274854 -0.000818713 0 0.006140351 0.010643274 0.017807017 0.016783625 0.006345029 -0.000409357 -0.012690058 -0.005730994 0.011871345 0.026608187 0.045233919 0.037865495 0.019239765 0.018625731 0.015760233 0.018421052 0.003274854 -0.017192981 -0.040935671 -0.030292398 -0.001432748 0.021900583 0.020263156 0.009824562 0.000614035 -0.014122807 -0.021900583 -0.003070175 0.014122807 0.028245614 0.01371345 0.004093567 0.007163743 0.018421052 0.01760234 0.010643274 0.004912281 0.003274854 0.002660819 -0.002251462 -0.006959064 -0.01371345 -0.011052631 -0.004093567 0.002865497 0.009824562 0.0198538 0.011871345 -0.009824562 -0.024356723 -0.019444444 -0.012280701 -0.004912281 0.006549707 0.007777777 0.002865497 -0.000204678 -0.001432748 0.00122807 0.002046784 0.004912281 0.002660819 0.000818713 -0.001637427 -0.005526316 -0.01125731 -0.017192981 -0.019035088 -0.01248538 -0.007368421 -0.000409357 0.006754386 0.007573099 -0.002865497 -0.009210526 -0.021081872 -0.015760233 -0.004707602 0.006345029 0.008391813 0.005321637 0.002660819 0.003070175 0.01248538 0.01494152 0.015146199 0.015555555 0.01494152 0.014532163 0.011666666 0.007777777 0.004298245 -0.000409357 -0.006754386 -0.013099415 -0.013918129 -0.010438596 -0.011052631 -0.011052631 -0.011666666 -0.006754386 0 0.006754386 0.014327485 0.019649123 0.01739766 0.016374269 0.013918129 0.015350877 0.015760233 0.01739766 0.01248538 0.007573099 0.001432748 -0.000204678 0.000818713 0.001023392 -0.002046784 -0.005935673 -0.009415204 -0.014122807 -0.013304093 -0.009005848 -0.005730994 -0.001432748 -0.004707602 -0.006754386 -0.010438596 -0.012280701 -0.011461988 -0.011052631 -0.010233918 -0.013508771 -0.018625731 -0.023333332 -0.028654971 -0.028245614 -0.028654971 -0.024766082 -0.019649123 -0.015146199 -0.009415204 -0.006959064 -0.008596491 -0.009210526 -0.01125731 -0.012894737 -0.014736841 -0.016374269 -0.01494152 -0.010643274 -0.007573099 -0.005321637 -0.002865497
Dear Dr. Jakeman, please ask whether the pce module in the pyapprox library is only valid for continuous functions with expressions. For practical problems, I can only obtain the input variables and output results. Can I effectively use the library you developed. If it is convenient, I would be grateful if you could provide a simple example for reference.