def SchemeNonlinear(u, x, f, bc):
coef, offsets = Selling.Decomposition(D(x))
du = bc.DiffCentered(u, offsets)
d2u = bc.Diff2(u, offsets)
p = lp.dot_AV(lp.inverse(D(x)), np.sum(coef * du * offsets, axis=1))
return np.where(
bc.interior,
-1 / 2 * lp.dot_VV(omega(x), p)**2 - lp.dot_VV(coef, d2u) - f,
u - bc.grid_values,
)
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 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