def test_nobackground():
#=======================================================================
"Check that dipolar kernel builds correctly without a background specified"
t = np.linspace(0, 3, 80)
r = np.linspace(2, 6, 100)
K1 = dipolarkernel(t, r)
K2 = dipolarkernel(t, r, bg=None)
assert np.all(abs(K2 - K1) < 1e-5)
def test_lambda():
#=======================================================================
"Check that dipolar kernel with modulation depth works"
t = np.linspace(0, 4, 150) # µs
r = np.linspace(1, 5, 150) # nm
lam = 0.4
Kref = ((1 - lam) + lam * dipolarkernel(t, r, integralop=False))
K = dipolarkernel(t, r, mod=lam, integralop=False)
assert np.all(abs(K - Kref) < 1e-14)
def test_r_scaling():
#=======================================================================
"""Check whether K matrix elements scale properly with t and r"""
t = 2 # nm
r = 3 # µs
c = 1.2 # distance scaling factor
K1 = dipolarkernel(t, r)
K2 = dipolarkernel(t * c**3, r * c)
assert np.max(K1 - K2) < 1e-15
def test_orisel_uni_integral():
#=======================================================================
"Check that orientation selection works for a uniform distribution"
t = 1
r = 1
Ptheta = lambda theta: np.ones_like(theta)
Kref = dipolarkernel(t, r, method='integral')
K = dipolarkernel(t, r, method='integral', orisel=Ptheta)
assert np.max(K - Kref) < 1e-10
def test_integralop():
#=======================================================================
"""Check that integral operator-nature of kernel can be disabled"""
t = np.linspace(0, 5, 101)
r = np.linspace(2, 5, 101)
dr = np.ones(len(r)) * np.mean(np.diff(r))
dr[0] = dr[0] / 2
dr[-1] = dr[-1] / 2
Kref = dipolarkernel(t, r, integralop=True) / dr
K = dipolarkernel(t, r, integralop=False)
assert np.max(K - Kref) < 1e-14
def test_negative_time_integral():
#=======================================================================
"Check that kernel is constructed properly for negative times using the integral method"
tneg = np.linspace(-5, 0, 50)
tpos = np.linspace(0, 5, 50)
r = np.linspace(2, 6, 10)
Kneg = dipolarkernel(tneg, r, method='integral')
Kpos = dipolarkernel(tpos, r, method='integral')
delta = abs(Kneg - np.flipud(Kpos))
assert np.all(delta < 1e-12)
def test_background():
#=======================================================================
"Check that dipolar kernel builds with the background included"
t = np.linspace(0, 3, 80)
r = np.linspace(2, 6, 100)
B = bg_exp(t, 0.5)
lam = 0.25
KBref = (1 - lam +
lam * dipolarkernel(t, r, integralop=False)) * B[:, np.newaxis]
KB = dipolarkernel(t, r, mod=lam, bg=B, integralop=False)
assert np.all(abs(KB - KBref) < 1e-5)
def test_excbandwidth_inf_integral():
#=======================================================================
"Check that specifying a bandwidth effectively changes the kernel (integral method)"
r = 2.5 # nm
excitewidth = 15 # MHz
t = np.linspace(0, 2, 501) # µs
# Reference kernel with infinite bandwidth
Kinf = dipolarkernel(t, r, excbandwidth=inf, method='integral')
# Kernel with limited bandwidth
Klim = dipolarkernel(t, r, excbandwidth=excitewidth, method='integral')
assert not np.array_equal(Kinf, Klim)
def Vnonlinear_fcn(*nonlin):
""" Non-linear part of the dipolar signal function """
# Make input arguments as array to access subsets easily
nonlin = np.atleast_1d(nonlin)
# Construct the basis function of the intermolecular contribution
if Bmodel is None:
Bfcn = np.ones_like(t)
elif hasattr(Bmodel, 'lam'):
Bfcn = lambda t, lam: Bmodel.nonlinmodel(
t, *np.concatenate([nonlin[Bsubset], [lam]]))
else:
Bfcn = lambda t, _: Bmodel.nonlinmodel(t, *nonlin[Bsubset])
# Construct the definition of the dipolar pathways
pathways = PathsModel.nonlinmodel(*nonlin[PathsSubset])
# Construct the dipolar kernel
Kdipolar = dipolarkernel(t,
r,
pathways=pathways,
bg=Bfcn,
excbandwidth=excbandwidth,
orisel=orisel,
g=g,
method=kernelmethod)
# Compute the non-linear part of the distance distribution
Pnonlin = Pmodel.nonlinmodel(*[r] * Nconstants, *nonlin[Psubset])
# Forward calculation of the non-linear part of the dipolar signal
Vnonlin = Kdipolar @ Pnonlin
return Vnonlin
def test_gvalues():
#=======================================================================
"""Check whether K matrix elements scale properly g-values"""
t = 2 # nm
r = 3 # µs
# isotropic g values of two spins
g1 = 2.1
g2 = 2.4
ge = 2.00231930436256 # free-electron g factor (CODATA 2018 value)
K1 = dipolarkernel(t, r, g=[g1, g2])
K2 = dipolarkernel(t, r / (g1 * g2 / ge**2)**(1 / 3))
assert np.max(K1 - K2) < 1e-15
def test_multipath_harmonics():
#=======================================================================
"Check that multi-pathway kernels with higher harmonics are properly generated"
r = np.linspace(2, 6, 50)
t1 = np.linspace(0, 10, 300)
t2 = 0.3
tau1 = 4.24
tau2 = 4.92
t = (tau1 + tau2) - (t1 + t2)
prob = 0.8
lam = [prob**2, prob * (1 - prob)]
T0 = [0, tau2 - t2]
n = [2, 3]
paths = []
paths.append([1 - prob])
paths.append([prob**2, 0, 2])
paths.append([prob * (1 - prob), tau2 - t2, 3])
K = dipolarkernel(t, r, pathways=paths, integralop=False)
Kref = 1 - prob
for p in range(len(lam)):
Kref = Kref + lam[p] * elementarykernel(
n[p] * (t - T0[p]), r, 'fresnel', [], [], [ge, ge], None)
Kref = Kref
assert np.max(K - Kref) < 1e-3
def test_singledist():
#=======================================================================
"Check that one can generate a kernel with one single distance-domain point"
t = np.linspace(0, 5, 50)
r = 3.5
K = dipolarkernel(t, r)
assert np.shape(K)[1] == 1
def test_matrixsize_fresnel():
#=======================================================================
"Check non-square kernel matrix construction"
Ntime = 50
Ndist = 70
t = np.linspace(-0.5, 5, 50)
r = np.linspace(2, 6, 70)
K = dipolarkernel(t, r)
assert K.shape == (Ntime, Ndist)
def test_singletime():
#=======================================================================
"Check that one can generate a kernel with one single time-domain point"
t = 0.5
r = np.linspace(1, 5, 100)
K = dipolarkernel(t, r)
assert np.shape(K)[0] == 1
def test_complex_type_grid():
#=======================================================================
"Test that the values are complex-valued if requested."
# Generate kernel numerically
t = 1 # µs
r = 1 # nm
K = dipolarkernel(t, r, method='grid', complex=True)
assert isinstance(K[0, 0], np.complex128)
def test_nonuniform_r():
#=======================================================================
"Check that normalization is correct when using a non-uniform distance axis"
t = 0
r = np.sqrt(np.linspace(1, 7**2, 200))
P = dd_gauss(r, 3, 0.5)
K = dipolarkernel(t, r)
V0 = K @ P
assert np.round(V0, 3) == 1
def test_memory_limit():
# ======================================================================
"Check that the memory limiter works"
# Construct 12GB kernel (should fit in RAM)
t = np.linspace(0, 5, int(3e4))
r = np.linspace(1, 6, int(5e4))
with pytest.raises(MemoryError):
K = dipolarkernel(t, r)
# ======================================================================
def test_complex_value_grid_negative_time():
#=======================================================================
"Test whether comple-valued kernel matrix element (calculated using the grid method) is correct."
# Generate kernel numerically
t = -1 # µs
r = 1 # nm
K = dipolarkernel(t, r, method='grid', nKnots=2e6, complex=True)
# Kernel value for 1us and 1nm computed using Mathematica (FresnelC and FresnelS)
# and CODATA 2018 values for ge, muB, mu0, and h.
Kref = 0.0246978198952619 - 1j * 0.01380433055548875
assert abs(K - Kref) < 1e-6
def test_value_integral():
#=======================================================================
"Test whether kernel matrix element (calculated using the integal method) is correct."
# Generate kernel numerically
t = 1 # µs
r = 1 # nm
K = dipolarkernel(t, r, method='integral')
# Kernel value for 1us and 1nm computed using Mathematica (FresnelC and FresnelS)
# and CODATA 2018 values for ge, muB, mu0, and h.
Kref = 0.024697819895260188
assert abs(K - Kref) < 1e-7
def test_complex_value_fresnel():
#=======================================================================
"Test whether comple-valued kernel matrix element (calculated using Fresnel integrals) is correct."
# Generate kernel numerically
t = 1 # µs
r = 1 # nm
K = dipolarkernel(t, r, method='fresnel', complex=True)
# Kernel value for 1us and 1nm computed using Mathematica (FresnelC and FresnelS)
# and CODATA 2018 values for ge, muB, mu0, and h.
Kref = 0.0246978198952619 + 1j * 0.01380433055548875
assert abs(K - Kref) < 1e-14
def test_orisel_value_integral():
#=======================================================================
"Check that orientation selection works for a uniform distribution"
t = 1
r = 1
thetamean = pi / 4
sigma = pi / 3
Ptheta = lambda theta: 1 / sigma / np.sqrt(2 * pi) * np.exp(-(
theta - thetamean)**2 / 2 / sigma**2)
# Kernel value for 1us and 1nm computed using Mathematica
# and CODATA 2018 values for ge, muB, mu0, and h
Kref = 0.02031864642707554
K = dipolarkernel(t, r, method='integral', orisel=Ptheta)
assert abs(K - Kref) < 1e-4
def _hom3dex(t, conc, rex, lam):
# Conversion: umol/L -> mol/L -> mol/m^3 -> spins/m^3
conc = conc * 1e-6 * 1e3 * Nav
# Excluded volume
Vex = 4 * np.pi / 3 * (rex * 1e-9)**3
# Averaging integral
z = np.linspace(0, 1, 1000)[np.newaxis, :]
Dt = D * t[:, np.newaxis] * 1e-6
Is = 4 * np.pi / 3 * np.trapz(Dt * (1 - 3 * z**2) * sici(
(Dt * (1 - 3 * z**2)) / ((rex * 1e-9)**3))[0],
z,
axis=1)
# Background function
C_k = -Vex + Is + np.squeeze(Vex *
(dipolarkernel(t, rex, integralop=False)))
B = np.exp(-lam * conc * C_k)
return B
def _hom3dex_phase(t, conc, rex, lam):
# Conversion: umol/L -> mol/L -> mol/m^3 -> spins/m^3
conc = conc * 1e-6 * 1e3 * Nav
# Excluded volume
Vex = 4 * np.pi / 3 * (rex * 1e-9)**3
# Averaging integral
ξ = 8 * pi**2 / 9 / np.sqrt(3) * (np.sqrt(3) +
np.log(2 - np.sqrt(3))) / np.pi * D
z = np.linspace(0, 1, 1000)[np.newaxis, :]
Dt = D * t[:, np.newaxis] * 1e-6
Ic = -ξ * (t * 1e-6) + 4 * np.pi / 3 * np.trapz(Dt * (1 - 3 * z**2) * sici(
(Dt * np.abs(1 - 3 * z**2)) / ((rex * 1e-9)**3))[1],
z,
axis=1)
# Background function
C_k = -Ic - np.squeeze(
Vex * (dipolarkernel(t, rex, integralop=False, complex=True)).imag)
B = np.exp(-1j * lam * conc * C_k)
return B
def test_multipath_background():
#=======================================================================
"Check that multi-pathway kernels are properly generated with background"
r = np.linspace(2, 6, 50)
t1 = np.linspace(0, 10, 300)
t2 = 0.3
tau1 = 4.24
tau2 = 4.92
t = (tau1 + tau2) - (t1 + t2)
prob = 0.8
lam = [prob**2, prob * (1 - prob)]
T0 = [0, tau2 - t2]
conc = 50
Bmodel = lambda t, lam: bg_hom3d(t, conc, lam)
# Reference
Kref = 1 - prob
for p in range(len(lam)):
Kref = Kref + lam[p] * elementarykernel(t - T0[p], r, 'fresnel', [],
[], [ge, ge], None)
Kref = Kref
Bref = 1
for p in range(len(lam)):
Bref = Bref * Bmodel((t - T0[p]), lam[p])
KBref = Kref * Bref[:, np.newaxis]
paths = []
paths.append([1 - prob])
paths.append([prob**2, 0])
paths.append([prob * (1 - prob), tau2 - t2])
# Output
KB = dipolarkernel(t, r, pathways=paths, bg=Bmodel, integralop=False)
assert np.all(abs(KB - KBref) < 1e-3)
def test_excbandwidth():
#=======================================================================
"""Check kernel with limited excitation bandwidth is numerically correct"""
r = 2.5 # nm
excitewidth = 15 # MHz
t = np.linspace(0, 2, 501) # µs
# Kernel using numerical integrals and approx. bandwidth treatment
K = dipolarkernel(t, r, excbandwidth=excitewidth, method='grid')
# Manual calculation using numerical powder average (reference)
nKnots = 5001
costheta = np.linspace(0, 1, nKnots)
wdd = 2 * pi * 52.04 / r**3 # Mrad/s
q = 1 - 3 * costheta**2
D_ = 0
for orientation in q:
w = wdd * orientation
D_ = D_ + np.cos(w * abs(t)) * np.exp(-w**2 / excitewidth**2)
K0 = D_ / nKnots
K0 = K0.reshape((len(t), 1))
assert np.max(K0 - K) < 1e-4
"Check that the model has correct parameter names"
model = dipolarmodel(t,r,dd_gauss,bg_hom3d,npathways=1)
parameters = ['mean','width','conc','mod','reftime','scale']
for param in parameters:
assert hasattr(model,param)
# ======================================================================
t = np.linspace(-0.5,5,100)
r = np.linspace(2,5,50)
Bfcn = lambda t,lam: bg_hom3d(t,50,lam)
Bfcn_pheno = lambda t,_: bg_exp(t,0.1)
Pr = dd_gauss(r,3,0.2)
V1path = 1e5*dipolarkernel(t,r,mod=0.3,bg=Bfcn)@Pr
V1path_noB = 1e5*dipolarkernel(t,r,mod=0.3)@Pr
V1path_phenoB = 1e5*dipolarkernel(t,r,mod=0.3,bg=Bfcn_pheno)@Pr
V2path = 1e5*dipolarkernel(t,r,pathways=[[0.6],[0.3,0],[0.1,2]],bg=Bfcn)@Pr
V3path = 1e5*dipolarkernel(t,r,pathways=[[0.5],[0.3,0],[0.1,2],[0.1,5]],bg=Bfcn)@Pr
# ======================================================================
def test_call_positional():
"Check that the model called via positional arguments responds correctly"
Vmodel = dipolarmodel(t,r,dd_gauss,bg_hom3d,npathways=1)
Vsim = Vmodel(0.3,0.0,50,3,0.2,1e5)
assert np.allclose(Vsim,V1path)
model = dipolarmodel(t, r, dd_gauss, bg_hom3d, npathways=1)
parameters = ['mean', 'width', 'conc', 'mod', 'reftime', 'scale']
for param in parameters:
assert hasattr(model, param)
# ======================================================================
t = np.linspace(-0.5, 5, 100)
r = np.linspace(2, 5, 50)
Bfcn = lambda t, lam: bg_hom3d(t, 50, lam)
Bfcn_pheno = lambda t, _: bg_exp(t, 0.1)
Pr = dd_gauss(r, 3, 0.2)
V1path = 1e5 * dipolarkernel(t, r, mod=0.3, bg=Bfcn) @ Pr
V1path_noB = 1e5 * dipolarkernel(t, r, mod=0.3) @ Pr
V1path_phenoB = 1e5 * dipolarkernel(t, r, mod=0.3, bg=Bfcn_pheno) @ Pr
V2path = 1e5 * dipolarkernel(
t, r, pathways=[[0.6], [0.3, 0], [0.1, 2]], bg=Bfcn) @ Pr
V3path = 1e5 * dipolarkernel(
t, r, pathways=[[0.5], [0.3, 0], [0.1, 2], [0.1, 5]], bg=Bfcn) @ Pr
# ======================================================================
def test_call_positional():
"Check that the model called via positional arguments responds correctly"
Vmodel = dipolarmodel(t, r, dd_gauss, bg_hom3d, npathways=1)
Vsim = Vmodel(0.3, 0.0, 50, 3, 0.2, 1e5)