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