Source code for discretize.utils.matutils

from __future__ import division
import numpy as np
import scipy.sparse as sp
from .codeutils import isScalar


[docs]def mkvc(x, numDims=1): """Creates a vector with the number of dimension specified e.g.:: a = np.array([1, 2, 3]) mkvc(a, 1).shape > (3, ) mkvc(a, 2).shape > (3, 1) mkvc(a, 3).shape > (3, 1, 1) """ if type(x) == np.matrix: x = np.array(x) if hasattr(x, 'tovec'): x = x.tovec() if isinstance(x, Zero): return x assert isinstance(x, np.ndarray), "Vector must be a numpy array" if numDims == 1: return x.flatten(order='F') elif numDims == 2: return x.flatten(order='F')[:, np.newaxis] elif numDims == 3: return x.flatten(order='F')[:, np.newaxis, np.newaxis]
[docs]def sdiag(h): """Sparse diagonal matrix""" if isinstance(h, Zero): return Zero() return sp.spdiags(mkvc(h), 0, h.size, h.size, format="csr")
[docs]def sdInv(M): "Inverse of a sparse diagonal matrix" return sdiag(1.0 / M.diagonal())
[docs]def speye(n): """Sparse identity""" return sp.identity(n, format="csr")
[docs]def kron3(A, B, C): """Three kron prods""" return sp.kron(sp.kron(A, B), C, format="csr")
[docs]def spzeros(n1, n2): """a sparse matrix of zeros""" return sp.dia_matrix((n1, n2))
[docs]def ddx(n): """Define 1D derivatives, inner, this means we go from n+1 to n""" return sp.spdiags( (np.ones((n+1, 1))*[-1, 1]).T, [0, 1], n, n+1, format="csr" )
[docs]def av(n): """Define 1D averaging operator from nodes to cell-centers.""" return sp.spdiags( (0.5*np.ones((n+1, 1))*[1, 1]).T, [0, 1], n, n+1, format="csr" )
[docs]def av_extrap(n): """Define 1D averaging operator from cell-centers to nodes.""" Av = ( sp.spdiags( (0.5 * np.ones((n, 1)) * [1, 1]).T, [-1, 0], n + 1, n, format="csr" ) + sp.csr_matrix(([0.5, 0.5], ([0, n], [0, n-1])), shape=(n+1, n)) ) return Av
[docs]def ndgrid(*args, **kwargs): """ Form tensorial grid for 1, 2, or 3 dimensions. Returns as column vectors by default. To return as matrix input: ndgrid(..., vector=False) The inputs can be a list or separate arguments. e.g.:: a = np.array([1, 2, 3]) b = np.array([1, 2]) XY = ndgrid(a, b) > [[1 1] [2 1] [3 1] [1 2] [2 2] [3 2]] X, Y = ndgrid(a, b, vector=False) > X = [[1 1] [2 2] [3 3]] > Y = [[1 2] [1 2] [1 2]] """ # Read the keyword arguments, and only accept a vector=True/False vector = kwargs.pop('vector', True) assert type(vector) == bool, "'vector' keyword must be a bool" assert len(kwargs) == 0, "Only 'vector' keyword accepted" # you can either pass a list [x1, x2, x3] or each seperately if type(args[0]) == list: xin = args[0] else: xin = args # Each vector needs to be a numpy array assert np.all( [isinstance(x, np.ndarray) for x in xin] ), "All vectors must be numpy arrays." if len(xin) == 1: return xin[0] elif len(xin) == 2: XY = np.broadcast_arrays(mkvc(xin[1], 1), mkvc(xin[0], 2)) if vector: X2, X1 = [mkvc(x) for x in XY] return np.c_[X1, X2] else: return XY[1], XY[0] elif len(xin) == 3: XYZ = np.broadcast_arrays( mkvc(xin[2], 1), mkvc(xin[1], 2), mkvc(xin[0], 3) ) if vector: X3, X2, X1 = [mkvc(x) for x in XYZ] return np.c_[X1, X2, X3] else: return XYZ[2], XYZ[1], XYZ[0]
[docs]def ind2sub(shape, inds): """From the given shape, returns the subscripts of the given index""" if type(inds) is not np.ndarray: inds = np.array(inds) assert len(inds.shape) == 1, ( 'Indexing must be done as a 1D row vector, e.g. [3,6,6,...]' ) return np.unravel_index(inds, shape, order='F')
[docs]def sub2ind(shape, subs): """From the given shape, returns the index of the given subscript""" if len(shape) == 1: return subs if type(subs) is not np.ndarray: subs = np.array(subs) if len(subs.shape) == 1: subs = subs[np.newaxis, :] assert subs.shape[1] == len(shape), ( 'Indexing must be done as a column vectors. e.g. [[3,6],[6,2],...]' ) inds = np.ravel_multi_index(subs.T, shape, order='F') return mkvc(inds)
[docs]def getSubArray(A, ind): """subArray""" assert type(ind) == list, "ind must be a list of vectors" assert len(A.shape) == len(ind), ( "ind must have the same length as the dimension of A" ) if len(A.shape) == 2: return A[ind[0], :][:, ind[1]] elif len(A.shape) == 3: return A[ind[0], :, :][:, ind[1], :][:, :, ind[2]] else: raise Exception("getSubArray does not support dimension asked.")
[docs]def inv3X3BlockDiagonal( a11, a12, a13, a21, a22, a23, a31, a32, a33, returnMatrix=True ): """ B = inv3X3BlockDiagonal(a11, a12, a13, a21, a22, a23, a31, a32, a33) inverts a stack of 3x3 matrices Input: A - a11, a12, a13, a21, a22, a23, a31, a32, a33 Output: B - inverse """ a11 = mkvc(a11) a12 = mkvc(a12) a13 = mkvc(a13) a21 = mkvc(a21) a22 = mkvc(a22) a23 = mkvc(a23) a31 = mkvc(a31) a32 = mkvc(a32) a33 = mkvc(a33) detA = ( a31*a12*a23 - a31*a13*a22 - a21*a12*a33 + a21*a13*a32 + a11*a22*a33 - a11*a23*a32 ) b11 = +(a22*a33 - a23*a32)/detA b12 = -(a12*a33 - a13*a32)/detA b13 = +(a12*a23 - a13*a22)/detA b21 = +(a31*a23 - a21*a33)/detA b22 = -(a31*a13 - a11*a33)/detA b23 = +(a21*a13 - a11*a23)/detA b31 = -(a31*a22 - a21*a32)/detA b32 = +(a31*a12 - a11*a32)/detA b33 = -(a21*a12 - a11*a22)/detA if not returnMatrix: return b11, b12, b13, b21, b22, b23, b31, b32, b33 return sp.vstack((sp.hstack((sdiag(b11), sdiag(b12), sdiag(b13))), sp.hstack((sdiag(b21), sdiag(b22), sdiag(b23))), sp.hstack((sdiag(b31), sdiag(b32), sdiag(b33)))))
[docs]def inv2X2BlockDiagonal(a11, a12, a21, a22, returnMatrix=True): """ B = inv2X2BlockDiagonal(a11, a12, a21, a22) Inverts a stack of 2x2 matrices by using the inversion formula inv(A) = (1/det(A)) * cof(A)^T Input: A - a11, a12, a21, a22 Output: B - inverse """ a11 = mkvc(a11) a12 = mkvc(a12) a21 = mkvc(a21) a22 = mkvc(a22) # compute inverse of the determinant. detAinv = 1./(a11*a22 - a21*a12) b11 = +detAinv*a22 b12 = -detAinv*a12 b21 = -detAinv*a21 b22 = +detAinv*a11 if not returnMatrix: return b11, b12, b21, b22 return sp.vstack((sp.hstack((sdiag(b11), sdiag(b12))), sp.hstack((sdiag(b21), sdiag(b22)))))
[docs]class TensorType(object): def __init__(self, M, tensor): if tensor is None: # default is ones self._tt = -1 self._tts = 'none' elif isScalar(tensor): self._tt = 0 self._tts = 'scalar' elif tensor.size == M.nC: self._tt = 1 self._tts = 'isotropic' elif ( (M.dim == 2 and tensor.size == M.nC*2) or (M.dim == 3 and tensor.size == M.nC*3) ): self._tt = 2 self._tts = 'anisotropic' elif ( (M.dim == 2 and tensor.size == M.nC*3) or (M.dim == 3 and tensor.size == M.nC*6) ): self._tt = 3 self._tts = 'tensor' else: raise Exception( 'Unexpected shape of tensor: {}'.format(tensor.shape) ) def __str__(self): return 'TensorType[{0:d}]: {1!s}'.format(self._tt, self._tts) def __eq__(self, v): return self._tt == v def __le__(self, v): return self._tt <= v def __ge__(self, v): return self._tt >= v def __lt__(self, v): return self._tt < v def __gt__(self, v): return self._tt > v
[docs]def makePropertyTensor(M, tensor): if tensor is None: # default is ones tensor = np.ones(M.nC) if isScalar(tensor): tensor = tensor * np.ones(M.nC) propType = TensorType(M, tensor) if propType == 1: # Isotropic! Sigma = sp.kron(sp.identity(M.dim), sdiag(mkvc(tensor))) elif propType == 2: # Diagonal tensor Sigma = sdiag(mkvc(tensor)) elif M.dim == 2 and tensor.size == M.nC*3: # Fully anisotropic, 2D tensor = tensor.reshape((M.nC, 3), order='F') row1 = sp.hstack((sdiag(tensor[:, 0]), sdiag(tensor[:, 2]))) row2 = sp.hstack((sdiag(tensor[:, 2]), sdiag(tensor[:, 1]))) Sigma = sp.vstack((row1, row2)) elif M.dim == 3 and tensor.size == M.nC*6: # Fully anisotropic, 3D tensor = tensor.reshape((M.nC, 6), order='F') row1 = sp.hstack( (sdiag(tensor[:, 0]), sdiag(tensor[:, 3]), sdiag(tensor[:, 4])) ) row2 = sp.hstack( (sdiag(tensor[:, 3]), sdiag(tensor[:, 1]), sdiag(tensor[:, 5])) ) row3 = sp.hstack( (sdiag(tensor[:, 4]), sdiag(tensor[:, 5]), sdiag(tensor[:, 2])) ) Sigma = sp.vstack((row1, row2, row3)) else: raise Exception('Unexpected shape of tensor') return Sigma
[docs]def invPropertyTensor(M, tensor, returnMatrix=False): propType = TensorType(M, tensor) if isScalar(tensor): T = 1./tensor elif propType < 3: # Isotropic or Diagonal T = 1./mkvc(tensor) # ensure it is a vector. elif M.dim == 2 and tensor.size == M.nC*3: # Fully anisotropic, 2D tensor = tensor.reshape((M.nC, 3), order='F') B = inv2X2BlockDiagonal(tensor[:, 0], tensor[:, 2], tensor[:, 2], tensor[:, 1], returnMatrix=False) b11, b12, b21, b22 = B T = np.r_[b11, b22, b12] elif M.dim == 3 and tensor.size == M.nC*6: # Fully anisotropic, 3D tensor = tensor.reshape((M.nC, 6), order='F') B = inv3X3BlockDiagonal(tensor[:, 0], tensor[:, 3], tensor[:, 4], tensor[:, 3], tensor[:, 1], tensor[:, 5], tensor[:, 4], tensor[:, 5], tensor[:, 2], returnMatrix=False) b11, b12, b13, b21, b22, b23, b31, b32, b33 = B T = np.r_[b11, b22, b33, b12, b13, b23] else: raise Exception('Unexpected shape of tensor') if returnMatrix: return makePropertyTensor(M, T) return T
[docs]class Zero(object): __numpy_ufunc__ = True __array_ufunc__ = None def __add__(self, v): return v def __radd__(self, v): return v def __iadd__(self, v): return v def __sub__(self, v): return -v def __rsub__(self, v): return v def __isub__(self, v): return v def __mul__(self, v): return self def __rmul__(self, v): return self def __div__(self, v): return self def __truediv__(self, v): return self def __rdiv__(self, v): raise ZeroDivisionError('Cannot divide by zero.') def __rtruediv__(self, v): raise ZeroDivisionError('Cannot divide by zero.') def __rfloordiv__(self, v): raise ZeroDivisionError('Cannot divide by zero.') def __pos__(self): return self def __neg__(self): return self def __lt__(self, v): return 0 < v def __le__(self, v): return 0 <= v def __eq__(self, v): return v == 0 def __ne__(self, v): return not (0 == v) def __ge__(self, v): return 0 >= v def __gt__(self, v): return 0 > v
[docs] def transpose(self): return self
@property def T(self): return self
[docs]class Identity(object): __numpy_ufunc__ = True __array_ufunc__ = None _positive = True def __init__(self, positive=True): self._positive = positive is True def __pos__(self): return self def __neg__(self): return Identity(not self._positive) def __add__(self, v): if sp.issparse(v): return ( v + speye(v.shape[0]) if self._positive else v - speye(v.shape[0]) ) return v + 1 if self._positive else v - 1 def __radd__(self, v): return self.__add__(v) def __sub__(self, v): return self+-v def __rsub__(self, v): return -self+v def __mul__(self, v): return v if self._positive else -v def __rmul__(self, v): return v if self._positive else -v def __div__(self, v): if sp.issparse(v): raise NotImplementedError('Sparse arrays not divisibile.') return 1/v if self._positive else -1/v def __truediv__(self, v): if sp.issparse(v): raise NotImplementedError('Sparse arrays not divisibile.') return 1.0/v if self._positive else -1.0/v def __rdiv__(self, v): return v if self._positive else -v def __rtruediv__(self, v): return v if self._positive else -v def __floordiv__(self, v): return 1//v if self._positive else -1//v def __rfloordiv__(self, v): return 1//v if self._positivie else -1//v def __lt__(self, v): return 1 < v if self._positive else -1 < v def __le__(self, v): return 1 <= v if self._positive else -1 <= v def __eq__(self, v): return v == 1 if self._positive else v == -1 def __ne__(self, v): return (not (1 == v))if self._positive else (not (-1 == v)) def __ge__(self, v): return 1 >= v if self._positive else -1 >= v def __gt__(self, v): return 1 > v if self._positive else -1 > v @property def T(self): return self
[docs] def transpose(self): return self