def __init__(self,
y,
d,
sigma=None,
eps=1.0,
phi=phs3,
order=1,
extrapolate=True):
y = np.asarray(y)
assert_shape(y, (None, None), 'y')
nobs, dim = y.shape
d = np.asarray(d)
assert_shape(d, (nobs, ), 'd')
if sigma is None:
# if sigma is not specified then it is zeros
sigma = np.zeros(nobs)
elif np.isscalar(sigma):
# if a float is specified then use it as the uncertainties for
# all observations
sigma = np.repeat(sigma, nobs)
else:
sigma = np.asarray(sigma)
assert_shape(sigma, (nobs, ), 'sigma')
phi = get_rbf(phi)
# form block consisting of the RBF and uncertainties on the
# diagonal
K = phi(y, y, eps=eps)
Cd = scipy.sparse.diags(sigma**2)
# form the block consisting of the monomials
pwr = powers(order, dim)
P = mvmonos(y, pwr)
# create zeros vector for the right-hand-side
z = np.zeros((pwr.shape[0], ))
# solve for the RBF and mononomial coefficients
phi_coeff, poly_coeff = PartitionedSolver(K + Cd, P).solve(d, z)
self._y = y
self._phi = phi
self._order = order
self._eps = eps
self._phi_coeff = phi_coeff
self._poly_coeff = poly_coeff
self._pwr = pwr
self.extrapolate = extrapolate
return out
def clear_rbf_caches():
'''
Clear the caches of numerical functions for all the RBF instances
'''
for inst in RBF._INSTANCES:
if inst() is not None:
inst().clear_cache()
## Instantiate some common RBFs
#####################################################################
_phs8_limits = {}
_phs8_limits.update((tuple(i), 0) for i in powers(7, 1))
_phs8_limits.update((tuple(i), 0) for i in powers(7, 2))
_phs8_limits.update((tuple(i), 0) for i in powers(7, 3))
phs8 = RBF((_EPS * _R)**8 * sympy.log(_EPS * _R),
tol=1e-10,
limits=_phs8_limits)
_phs7_limits = {}
_phs7_limits.update((tuple(i), 0) for i in powers(6, 1))
_phs7_limits.update((tuple(i), 0) for i in powers(6, 2))
_phs7_limits.update((tuple(i), 0) for i in powers(6, 3))
phs7 = RBF((_EPS * _R)**7, tol=1e-10, limits=_phs7_limits)
_phs6_limits = {}
_phs6_limits.update((tuple(i), 0) for i in powers(5, 1))
_phs6_limits.update((tuple(i), 0) for i in powers(5, 2))
def weights(x, s, diffs, coeffs=None, phi=phs3, order=None, eps=1.0):
'''
Returns the weights which map a functions values at `s` to an approximation
of that functions derivative at `x`. The weights are computed using the
RBF-FD method described in [1]. In this function `x` is a single point in
D-dimensional space. Use `weight_matrix` to compute the weights for multiple
point.
Parameters
----------
x : (D,) array
Target point. The weights will approximate the derivative at this point.
s : (N, D) array
Stencil points. The derivative will be approximated with a weighted sum of
values at this point.
diffs : (D,) int array or (K, D) int array
Derivative orders for each spatial dimension. For example `[2, 0]`
indicates that the weights should approximate the second derivative with
respect to the first spatial dimension in two-dimensional space. diffs can
also be a (K, D) array, where each (D,) sub-array is a term in a
differential operator. For example the two-dimensional Laplacian can be
represented as `[[2, 0], [0, 2]]`.
coeffs : (K,) array, optional
Coefficients for each term in the differential operator specified with
`diffs`. Defaults to an array of ones. If `diffs` was specified as a (D,)
array then `coeffs` should be a length 1 array.
phi : rbf.basis.RBF instance or str, optional
Type of RBF. Select from those available in `rbf.basis` or create your own.
order : int, optional
Order of the added polynomial. This defaults to the highest derivative
order. For example, if `diffs` is `[[2, 0], [0, 1]]`, then order is set to
2.
eps : float or (N,) array, optional
Shape parameter for each RBF, which have centers `s`. This only makes a
difference when using RBFs that are not scale invariant. All the predefined
RBFs except for the odd order polyharmonic splines are not scale invariant.
Returns
-------
out : (N,) array
RBF-FD weights
Examples
--------
Calculate the weights for a one-dimensional second order derivative.
>>> x = np.array([1.0])
>>> s = np.array([[0.0], [1.0], [2.0]])
>>> diff = (2,)
>>> weights(x, s, diff)
array([ 1., -2., 1.])
Calculate the weights for estimating an x derivative from three points in a
two-dimensional plane
>>> x = np.array([0.25, 0.25])
>>> s = np.array([[0.0, 0.0],
[1.0, 0.0],
[0.0, 1.0]])
>>> diff = (1, 0)
>>> weights(x, s, diff)
array([ -1., 1., 0.])
References
----------
[1] Fornberg, B. and N. Flyer. A Primer on Radial Basis
Functions with Applications to the Geosciences. SIAM, 2015.
'''
x = np.asarray(x, dtype=float)
assert_shape(x, (None, ), 'x')
s = np.asarray(s, dtype=float)
assert_shape(s, (None, x.shape[0]), 's')
diffs = np.asarray(diffs, dtype=int)
diffs = _reshape_diffs(diffs)
if coeffs is None:
coeffs = np.ones(diffs.shape[0], dtype=float)
else:
coeffs = np.asarray(coeffs, dtype=float)
assert_shape(coeffs, (diffs.shape[0], ), 'coeffs')
phi = get_rbf(phi)
# stencil size and number of dimensions
size, dim = s.shape
# get the maximum polynomial order allowed for this stencil size
max_order = _max_poly_order(size, dim)
if order is None:
# If the polynomial order is not specified, make it equal to the derivative
# order, provided that the stencil size is large enough.
order = _default_poly_order(diffs)
order = min(order, max_order)
if order > max_order:
raise ValueError('Polynomial order is too high for the stencil size')
# center the stencil on `x` for improved numerical stability
s = s - x
x = np.zeros_like(x)
# get the powers for the added monomials
pwr = powers(order, dim)
# evaluate the RBF and monomials at each point in the stencil. This becomes
# the left-hand-side
A = phi(s, s, eps=eps)
P = mvmonos(s, pwr)
# Evaluate the RBF and monomials for each term in the differential operator.
# This becomes the right-hand-side.
a = np.zeros((1, size), dtype=float)
p = np.zeros((1, pwr.shape[0]), dtype=float)
for c, d in zip(coeffs, diffs):
a += c * phi(x[None, :], s, eps=eps, diff=d)
p += c * mvmonos(x[None, :], pwr, diff=d)
# squeeze `a` and `p` into 1d arrays. `a` is ran through as_array because it
# may be sparse.
a = as_array(a)[0]
p = p[0]
# attempt to compute the RBF-FD weights
try:
w = PartitionedSolver(A, P).solve(a, p)[0]
return w
except np.linalg.LinAlgError:
raise np.linalg.LinAlgError(
'An error was raised while computing the RBF-FD weights at point %s '
'with the RBF %s and the polynomial order %s. This may be due to a '
'stencil with duplicate or collinear points. The stencil contains the '
'following points:\n%s' % (x, phi, order, s))
