smbinterp/interp/baker/__init__.py

import sys

import numpy as np

from functools import wraps
import itertools

import interp
import logging
log = logging.getLogger('interp')

def get_phis(X, R):
  """
    The get_phis function is used to get barycentric coordonites for a point on
    a triangle or tetrahedron:


    in 2D:

    X -- the destination point (2D)
         X = [0,0]
    r -- the three points that make up the containing triangular simplex (2D)
         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 containing simplex, tetrahedron (3D)
         R = [
               [0.0, 0.0, 1.0],
               [0.94280904333606508, 0.0, -0.3333333283722672],
               [-0.47140452166803232, 0.81649658244673617, -0.3333333283722672],
               [-0.47140452166803298, -0.81649658244673584, -0.3333333283722672],
             ]

    this will return [0.25, 0.25, 0.25, 0.25]
  """

  # baker: eq 7
  # TODO: perhaps also test len(R[0])  .. ?
  if len(X) == 2:
    log.debug("running 2D")
    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:
    log.debug("running 3D")
    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")

  try:
    phi = np.linalg.solve(A,b)
  except np.linalg.LinAlgError as e:
    msg = "calculation of phis yielded a linearly dependant system (%s)" % e
    log.error(msg)
    # raise Exception(msg)
    phi = np.dot(np.linalg.pinv(A), b)

  log.debug("phi: %s", phi)

  return phi

def qlinear(X, R):
  """
    this calculates the linear portion of q from R to X

    also, this is baker eq 3

    X = destination point
    R = a inter.grid object; must have R.points and R.q
  """

  phis = get_phis(X, R.verts)
  qlin = np.sum([q_i * phi_i for q_i, phi_i in zip(R.q, phis)])

  log.debug("phis: %s", phis)
  log.debug("qlin: %s", qlin)

  return phis, qlin

def get_error(phi, R, S, order = 2):
  #TODO: change the equation names in the comments
  B = [] # baker eq 9
  w = [] # baker eq 11

  cur_pattern  = pattern(len(phi), order)
  log.info("pattern: %s" % cur_pattern)

  for (s,q) in zip(S.verts, S.q):
    cur_phi, cur_qlin = qlinear(s, R)
    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(q - cur_qlin)

  log.info("B: %s" % B)
  log.info("w: %s" % w)


  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 as e:
    log.error("linear calculation went bad, resorting to np.linalg.pinv: %s" % e)
    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

  log.debug("error_term: %s" % error_term)
  return error_term, abc

def run_baker(X, R, S, order=2):
  """
    This is the main function to call to get an interpolation to X from the
    input meshes

    X -- the destination point (2D)
         X = [0,0]

    R = Simplex
    S = extra points
  """
  log.debug("order = %d" % order)
  log.debug("extra points = %d" % len(S.verts))

  answer = {
             'qlin': None,
             'error': None,
             'final': None,
            }
  # calculate values only for the simplex triangle
  phi, qlin = qlinear(X, R)

  if order == 1:
    answer['qlin'] = qlin
    answer['final'] = qlin
    return answer
  elif order in xrange(2,11):
    error_term, abc = get_error(phi, R, S, order)
  else:
    raise Exception('unsupported order "%d" for baker method' % order)

  q_final = qlin + error_term

  answer['qlin' ] =  qlin
  answer['error'] =  error_term
  answer['final'] =  q_final
  answer['abc'  ] =  abc

  log.debug(answer)

  return answer


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:
      log.debug("adding to cache: %s", x)
      cache[x] = f(simplex_size, nu)
    return cache[x]
  return memf


@memoize
def pattern(simplex_size, nu):
  """
    This function returns the pattern requisite to compose the error
    approximation function, and the matrix B.
  """
  log.debug("pattern: simplex: %d, order: %d" % (simplex_size, nu))

  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