smbinterp/interp/baker/__init__.py

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from collections import namedtuple
from functools import wraps
import itertools
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import numpy as np
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import interp
AGGRESSIVE_ERROR_SOLVE = True
RAISE_PATHOLOGICAL_EXCEPTION = False
__version__ = interp.__version__
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Answer = namedtuple("Answer", ['qlin', 'final', 'error', 'abc'])
def get_phis(X, R):
"""
The get_phis function is used to get barycentric coordonites for a
point on a triangle or tetrahedron (Equation (*\ref{eq:qlinarea}*))
in 2D:
X - the destination point (2D)
X = [0,0]
R - the three points that make up the 2-D triangular simplex
R = [[-1, -1], [0, 2], [1, -1]]
this will return [0.333, 0.333, 0.333]
in 3D:
X - the destination point (3D)
X = [0,0,0]
R - the four points that make up the 3-D simplex (tetrahedron)
R = [
[ 0.0000, 0.0000, 1.0000],
[ 0.9428, 0.0000, -0.3333],
[-0.4714, 0.8165, -0.3333],
[-0.4714, -0.8165, -0.3333],
]
this will return [0.25, 0.25, 0.25, 0.25]
"""
# equations (*\ref{eq:lin3d}*) and (*\ref{eq:lin2d}*)
if len(X) == 2:
A = np.array([
[1, 1, 1],
[R[0][0], R[1][0], R[2][0]],
[R[0][1], R[1][1], R[2][1]],
])
b = np.array([1, X[0], X[1]])
elif len(X) == 3:
A = np.array([
[1, 1, 1, 1],
[R[0][0], R[1][0], R[2][0], R[3][0]],
[R[0][1], R[1][1], R[2][1], R[3][1]],
[R[0][2], R[1][2], R[2][2], R[3][2]],
])
b = np.array([1, X[0], X[1], X[2]])
else:
raise Exception("inapropriate demension on X")
phi = np.linalg.solve(A, b)
return phi
def qlinear(X, R, q):
"""
this calculates the linear portion of q from R to X
This is equation (*\ref{eq:qlinbasis}*)
X = destination point
R = a inter.grid object; must have R.points and R.q
"""
phis = get_phis(X, R)
qlin = np.sum([q_i * phi_i for q_i, phi_i in zip(q, phis)])
return phis, qlin
def get_error(phi, R, R_q, S, S_q, order=2):
"""
Calculate the error approximation terms, returning the unknowns
a,b, and c in equation (*\ref{eq:quadratic2d}*).
"""
B = [] # equation ((*\ref{eq:B2d}*)
w = [] # equation ((*\ref{eq:w}*)
cur_pattern = pattern(len(phi), order)
for (s, cur_q) in zip(S, S_q):
cur_phi, cur_qlin = qlinear(s, R, R_q)
l = []
for i in cur_pattern:
cur_sum = cur_phi[i[0]]
for j in i[1:]:
cur_sum *= cur_phi[j]
l.append(cur_sum)
B.append(l)
w.append(cur_q - cur_qlin)
B = np.array(B)
w = np.array(w)
A = np.dot(B.T, B)
b = np.dot(B.T, w)
try:
abc = np.linalg.solve(A, b)
except np.linalg.LinAlgError:
if not AGGRESSIVE_ERROR_SOLVE:
return None, None
abc = np.dot(np.linalg.pinv(A), b)
error_term = 0.0
for (a, i) in zip(abc, cur_pattern):
cur_sum = a
for j in i:
cur_sum *= phi[j]
error_term += cur_sum
return error_term, abc
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def interpolate(X, R, R_q, S=None, S_q=None, order=2):
"""
This is the main function to call to get an interpolation to X from the
input meshes
X -- the destination point
R = Simplex
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R_q = q values at R
S = extra points
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S_q = q values at S
order - order of interpolation - 1
"""
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qlin=None
error_term=None
final=None
abc={}
# calculate values only for the simplex triangle
phi, qlin = qlinear(X, R, R_q)
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if order in xrange(2, 11) and S:
error_term, abc = get_error(phi, R, R_q, S, S_q, order)
# if a pathological vertex configuration was encountered and
# AGGRESSIVE_ERROR_SOLVE is False, get_error will return (None, None)
# indicating that only linear interpolation should be performed
if (error_term is None) and (abc is None):
if RAISE_PATHOLOGICAL_EXCEPTION:
raise np.linalg.LinAlgError("Pathological Vertex Config")
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else:
final = qlin + error_term
elif order not in xrange(2,11):
raise Exception('unsupported order "%d" for baker method' % order)
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return Answer(qlin=qlin, error=error_term, final=final, abc=abc)
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def memoize(f):
"""
for more information on what I'm doing here, please read:
http://en.wikipedia.org/wiki/Memoize
"""
cache = {}
@wraps(f)
def memf(simplex_size, nu):
x = (simplex_size, nu)
if x not in cache:
cache[x] = f(simplex_size, nu)
return cache[x]
return memf
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@memoize
def pattern(simplex_size, nu):
"""
This function returns the pattern requisite to compose the error
approximation function, and the matrix B.
"""
r = []
for i in itertools.product(xrange(simplex_size), repeat=nu):
if len(set(i)) != 1:
r.append(tuple(sorted(i)))
unique_r = list(set(r))
return unique_r