def ConstrainedMaximize(Q, l, m):
dim = l.shape[0]
if dim == 1:
return (l[0] + np.sqrt(Q[0, 0])) / m[0]
# Discard infinite values, handled afterwards
pos_bad = l.min(axis=0) == -np.inf
L = l.copy()
L[:, pos_bad] = 0
# Solve the quadratic equation
A = lp.inverse(Q)
lAl = lp.dot_VAV(L, A, L)
lAm = lp.dot_VAV(L, A, m)
mAm = lp.dot_VAV(m, A, m)
delta = lAm**2 - (lAl - 1.) * mAm
pos_bad = np.logical_or(pos_bad, delta <= 0)
delta[pos_bad] = 1.
mu = (lAm + np.sqrt(delta)) / mAm
# Check the positivity
# v = dot_AV(A,mu*m-L)
rm_ad = np.array
v = lp.dot_AV(rm_ad(A), rm_ad(mu) * rm_ad(m) - rm_ad(L))
pos_bad = np.logical_or(pos_bad, np.any(v < 0, axis=0))
result = mu
result[pos_bad] = -np.inf
# Solve the lower dimensional sub-problems
# We could restrict to the bad positions, and avoid repeating computations
for i in range(dim):
axes = np.full((dim), True)
axes[i] = False
res = ConstrainedMaximize(Q[axes][:, axes], l[axes], m[axes])
result = np.maximum(result, res)
return result
def Scheme(a, b, d2u, stencil):
delta = d2u - lp.dot_VAV(
np.expand_dims(stencil.V1, (2, 3)),
np.expand_dims(a, 2),
np.expand_dims(stencil.V1, (2, 3)),
)
spad_sum(b)
spad_sum(delta)
# For now, replace `b` with one when it is zero, to prevent errors during automatic
# differentiation.
b_zero = b == 0
b = np.where(b_zero, 1, b)
residue = -np.inf
for i in range(stencil.V3.shape[2]):
residue = np.maximum(
residue,
H3(
stencil.Q[:, :, i, np.newaxis, np.newaxis],
stencil.w[:, i, np.newaxis, np.newaxis],
b,
delta[stencil.V3_indices[:, i]],
),
)
for i in range(stencil.V2.shape[2]):
residue = np.maximum(
residue,
H2(
stencil.omega0[i, np.newaxis, np.newaxis],
stencil.omega1[:, i, np.newaxis, np.newaxis],
stencil.omega2[:, i, np.newaxis, np.newaxis],
b,
delta[stencil.V2_indices[:, i]],
),
)
# Reset residue to minus infinity where `b` should have been zero.
residue = np.where(b_zero, -np.inf, residue)
for i in range(stencil.V1.shape[1]):
residue = np.maximum(
residue, H1(stencil.V1[:, i, np.newaxis, np.newaxis], delta[i]))
return residue
def NextAngleAndSuperbase(theta, sb, D):
pairs = np.stack([(1, 2), (2, 0), (0, 1)], axis=1)
scals = lp.dot_VAV(np.expand_dims(sb[:, pairs[0]], axis=1),
np.expand_dims(D, axis=-1),
np.expand_dims(sb[:, pairs[1]], axis=1))
phi = np.arctan2(scals[2], scals[1])
cst = -scals[0] / np.sqrt(scals[1]**2 + scals[2]**2)
theta_max = np.pi * np.ones(3)
mask = cst < 1
theta_max[mask] = (phi[mask] - np.arccos(cst[mask])) / 2
theta_max[theta_max <= 0] += np.pi
theta_max[theta_max <= theta] = np.pi
k = np.argmin(theta_max)
i, j = (k + 1) % 3, (k + 2) % 3
return (theta_max[k],
np.stack([sb[:, i], -sb[:, j], sb[:, j] - sb[:, i]], axis=1))
def MinimizeTrace(u, alpha, bc, sqrt_relax=1e-16):
# Compute the tensor decompositions
D = MakeD(alpha)
theta, sb = AnglesAndSuperbases(D)
theta = np.array([theta[:-1], theta[1:]])
# Compute the second order differences in the direction orthogonal to the superbase
sb_rotated = np.array([-sb[1], sb[0]])
d2u = bc.Diff2(u, sb_rotated)
d2u[..., bc.not_interior] = 0. # Placeholder values to silent NaNs
# Compute the coefficients of the tensor decompositions
sb1, sb2 = np.roll(sb, 1, axis=1), np.roll(sb, 2, axis=1)
sb1, sb2 = (e.reshape((2, 3, 1) + sb.shape[2:]) for e in (sb1, sb2))
D = D.reshape((2, 2, 1, 3, 1) + D.shape[3:])
# Axes of D are space,space,index of superbase element, index of D, index of superbase, and possibly shape of u
scals = lp.dot_VAV(sb1, D, sb2)
# Compute the coefficients of the trigonometric polynomial
scals, theta = (bc.as_field(e) for e in (scals, theta))
coefs = -lp.dot_VV(scals, np.expand_dims(d2u, axis=1))
# Optimality condition for the trigonometric polynomial in the interior
value = coefs[0] - np.sqrt(
np.maximum(coefs[1]**2 + coefs[2]**2, sqrt_relax))
coefs_ = np.array(coefs) # removed AD information
angle = np.arctan2(-coefs_[2], -coefs_[1]) / 2.
angle[angle < 0] += np.pi
# Boundary conditions for the trigonometric polynomial minimization
mask = np.logical_not(np.logical_and(theta[0] <= angle, angle <= theta[1]))
t, c = theta[:, mask], coefs[:, mask]
value[mask], amin_t = ad.min_argmin(c[0] + c[1] * np.cos(2 * t) +
c[2] * np.sin(2 * t),
axis=0)
# Minimize over superbases
value, amin_sb = ad.min_argmin(value, axis=0)
# Record the optimal angles for future use
angle[mask] = np.take_along_axis(t, np.expand_dims(amin_t, axis=0),
axis=0).squeeze(axis=0) # Min over bc
angle = np.take_along_axis(angle, np.expand_dims(amin_sb, axis=0),
axis=0) # Min over superbases
return value, angle
def SchemeLinear(u, x, f, bc):
coef, offsets = Selling.Decomposition(D(x))
# coef_min = np.min(coef)
# offsets_norm2 = lp.dot_VV(offsets, offsets)
# offsets_max2 = np.max(np.where(coef < 1e-13, 0, offsets_norm2))
# print(f"h: {bc.gridscale}, c: {coef_min}, e2: {offsets_max2}")
du = bc.DiffCentered(u, offsets)
d2u = bc.Diff2(u, offsets)
return np.where(
bc.interior,
-lp.dot_VAV(omega(x), lp.inverse(D(x)),
np.sum(coef * du * offsets, axis=1)) -
lp.dot_VV(coef, d2u) - f,
u - bc.grid_values,
)
def F(g):
return lp.dot_VAV(g - omega, D, g - omega)
