1import itertools

2from collections import defaultdict

4import numpy as np

5import sympy as sp

7from pystencils.field import Field

8from pystencils.stencil import direction_string_to_offset

9from pystencils.sympyextensions import multidimensional_sum, prod

10from pystencils.utils import LinearEquationSystem, fully_contains

13class FiniteDifferenceStencilDerivation:

14 """Derives finite difference stencils.

16 Can derive standard finite difference stencils, as well as isotropic versions

17 (see Isotropic Finite Differences by A. Kumar)

19 Args:

20 derivative_coordinates: tuple indicating which derivative should be approximated,

21 (1, ) stands for first derivative in second direction (y),

22 (0, 1) would be a mixed second derivative in x and y

23 (0, 0, 0) would be a third derivative in x direction

24 stencil: list of offset tuples, defining the stencil

25 dx: spacing between grid points, one for all directions, i.e. dx=dy=dz

27 Examples:

28 Central differences

29 >>> fd_1d = FiniteDifferenceStencilDerivation((0,), stencil=[(-1,), (0,), (1,)])

30 >>> result = fd_1d.get_stencil()

31 >>> result

32 Finite difference stencil of accuracy 2, isotropic error: False

33 >>> result.weights

34 [-1/2, 0, 1/2]

36 Forward differences

37 >>> fd_1d = FiniteDifferenceStencilDerivation((0,), stencil=[(0,), (1,)])

38 >>> result = fd_1d.get_stencil()

39 >>> result

40 Finite difference stencil of accuracy 1, isotropic error: False

41 >>> result.weights

42 [-1, 1]

43 """

45 def __init__(self, derivative_coordinates, stencil, dx=1):

46 self.dim = len(stencil)

47 self.field = Field.create_generic('f', spatial_dimensions=self.dim)

48 self._derivative = tuple(sorted(derivative_coordinates))

49 self._stencil = stencil

50 self._dx = dx

51 self.weights = {tuple(d): self.symbolic_weight(*d) for d in self._stencil}

53 def assume_symmetric(self, dim, anti_symmetric=False):

54 """Adds restriction that weight in opposite directions of a dimension are equal (symmetric) or

55 the negative of each other (anti symmetric)

57 For example: dim=1, assumes that w(1, 1) == w(1, -1), if anti_symmetric=False or

58 w(1, 1) == -w(1, -1) if anti_symmetric=True

59 """

60 update = {}

61 for direction, value in self.weights.items():

62 inv_direction = tuple(-offset if i == dim else offset for i, offset in enumerate(direction))

63 if direction[dim] < 0:

64 inv_weight = self.weights[inv_direction]

65 update[direction] = -inv_weight if anti_symmetric else inv_weight

66 self.weights.update(update)

68 def set_weight(self, offset, value):

69 assert offset in self.weights

70 self.weights[offset] = value

72 def get_stencil(self, isotropic=False) -> 'FiniteDifferenceStencilDerivation.Result':

73 weights = [self.weights[d] for d in self._stencil]

74 system = LinearEquationSystem(sp.Matrix(weights).atoms(sp.Symbol))

76 order = 0

78 while True:

79 new_system = system.copy()

80 eq = self.error_term_equations(order)

82 sol_structure = new_system.solution_structure()

83 if sol_structure == 'single':

84 system = new_system

85 elif sol_structure == 'multiple':

86 system = new_system

87 elif sol_structure == 'none':

88 break

89 else:

90 assert False

91 order += 1

93 accuracy = order - len(self._derivative)

94 error_is_isotropic = False

95 if isotropic:

96 new_system = system.copy()

98 sol_structure = new_system.solution_structure()

99 error_is_isotropic = sol_structure != 'none'

100 if error_is_isotropic:

101 system = new_system

103 solve_res = system.solution()

104 weight_list = [self.weights[d].subs(solve_res) for d in self._stencil]

105 return self.Result(self._stencil, weight_list, accuracy, error_is_isotropic)

107 @staticmethod

108 def symbolic_weight(*args):

109 str_args = [str(e) for e in args]

110 return sp.Symbol(f"w_({','.join(str_args)})")

112 def error_term_dict(self, order):

113 error_terms = defaultdict(lambda: 0)

114 for direction in self._stencil:

115 weight = self.weights[tuple(direction)]

116 x = tuple(self._dx * d_i for d_i in direction)

117 for offset in multidimensional_sum(order, dim=self.field.spatial_dimensions):

118 fac = sp.factorial(order)

119 error_terms[tuple(sorted(offset))] += weight / fac * prod(x[off] for off in offset)

120 if self._derivative in error_terms:

121 error_terms[self._derivative] -= 1

122 return error_terms

124 def error_term_equations(self, order):

125 return list(self.error_term_dict(order).values())

127 def isotropy_equations(self, order):

128 def cycle_int_sequence(sequence, modulus):

129 result = []

130 arr = np.array(sequence, dtype=int)

131 while True:

132 if tuple(arr) in result:

133 break

134 result.append(tuple(arr))

135 arr = (arr + 1) % modulus

136 return tuple(set(tuple(sorted(t)) for t in result))

138 error_dict = self.error_term_dict(order)

139 eqs = []

140 for derivative_tuple in list(error_dict.keys()):

141 if fully_contains(self._derivative, derivative_tuple):

142 remaining = list(derivative_tuple)

143 for e in self._derivative:

144 del remaining[remaining.index(e)]

145 permutations = cycle_int_sequence(remaining, self.dim)

146 if len(permutations) == 1:

147 eqs.append(error_dict[derivative_tuple])

148 else:

149 for i in range(1, len(permutations)):

150 new_eq = (error_dict[tuple(sorted(permutations[i] + self._derivative))]

151 - error_dict[tuple(sorted(permutations[i - 1] + self._derivative))])

152 if new_eq:

153 eqs.append(new_eq)

154 else:

155 eqs.append(error_dict[derivative_tuple])

156 return eqs

158 class Result:

159 def __init__(self, stencil, weights, accuracy, is_isotropic):

160 self.stencil = stencil

161 self.weights = weights

162 self.accuracy = accuracy

163 self.is_isotropic = is_isotropic

165 def visualize(self):

166 from pystencils.stencil import plot

167 plot(self.stencil, data=self.weights)

169 def apply(self, field_access: Field.Access):

170 f = field_access

171 return sum(f.get_shifted(*offset) * weight for offset, weight in zip(self.stencil, self.weights))

173 def __array__(self):

174 return np.array(self.as_array().tolist())

176 def as_array(self):

177 dim = len(self.stencil)

178 assert (dim == 2 or dim == 3), "Only 2D or 3D matrix representations are available"

179 max_offset = max(max(abs(e) for e in direction) for direction in self.stencil)

180 shape_list = []

181 for i in range(dim):

182 shape_list.append(2 * max_offset + 1)

184 number_of_elements = np.prod(shape_list)

185 shape = tuple(shape_list)

186 result = sp.MutableDenseNDimArray( * number_of_elements, shape)

188 if dim == 2:

189 for direction, weight in zip(self.stencil, self.weights):

190 result[max_offset - direction, max_offset + direction] = weight

191 if dim == 3:

192 for direction, weight in zip(self.stencil, self.weights):

193 result[max_offset - direction, max_offset + direction, max_offset + direction] = weight

195 return result

197 def rotate_weights_and_apply(self, field_access: Field.Access, axes):

198 """derive gradient weights of other direction with already calculated weights of one direction

199 via rotation and apply them to a field."""

200 dim = len(self.stencil)

201 assert (dim == 2 or dim == 3), "This function is only for 2D or 3D stencils available"

202 rotated_weights = np.rot90(np.array(self.__array__()), 1, axes)

204 result = []

205 max_offset = max(max(abs(e) for e in direction) for direction in self.stencil)

206 if dim == 2:

207 for direction in self.stencil:

208 result.append(rotated_weights[max_offset - direction,

209 max_offset + direction])

210 if dim == 3:

211 for direction in self.stencil:

212 result.append(rotated_weights[max_offset - direction,

213 max_offset + direction,

214 max_offset + direction])

216 f = field_access

217 return sum(f.get_shifted(*offset) * weight for offset, weight in zip(self.stencil, result))

219 def __repr__(self):

220 return "Finite difference stencil of accuracy {}, isotropic error: {}".format(self.accuracy,

221 self.is_isotropic)

224class FiniteDifferenceStaggeredStencilDerivation:

225 """Derives a finite difference stencil for application at a staggered position

227 Args:

228 neighbor: the neighbor direction string or vector at whose staggered position to calculate the derivative

229 dim: how many dimensions (2 or 3)

230 derivative: a tuple of directions over which to perform derivatives

231 """

233 def __init__(self, neighbor, dim, derivative=tuple()):

234 if type(neighbor) is str:

235 neighbor = direction_string_to_offset(neighbor)

236 if dim == 2:

237 assert neighbor[dim:] == 0

238 assert derivative is tuple() or max(derivative) < dim

239 neighbor = sp.Matrix(neighbor[:dim])

240 pos = neighbor / 2

242 def unitvec(i):

243 """return the `i`-th unit vector in three dimensions"""

244 a = np.zeros(dim, dtype=int)

245 a[i] = 1

246 return a

248 def flipped(a, i):

249 """return `a` with its `i`-th element's sign flipped"""

250 a = a.copy()

251 a[i] *= -1

252 return a

254 # determine the points to use, coordinates are relative to position

255 points = []

256 if np.linalg.norm(neighbor, 1) == 1:

257 main_points = [neighbor / 2, neighbor / -2]

258 elif np.linalg.norm(neighbor, 1) == 2:

259 nonzero_indices = [i for i, v in enumerate(neighbor) if v != 0 and i < dim]

260 main_points = [neighbor / 2, neighbor / -2, flipped(neighbor / 2, nonzero_indices),

261 flipped(neighbor / -2, nonzero_indices)]

262 else:

263 main_points = [neighbor.multiply_elementwise(sp.Matrix(c) / 2)

264 for c in itertools.product([-1, 1], repeat=3)]

265 points += main_points

266 zero_indices = [i for i, v in enumerate(neighbor) if v == 0 and i < dim]

267 for i in zero_indices:

268 points += [point + sp.Matrix(unitvec(i)) for point in main_points]

269 points += [point - sp.Matrix(unitvec(i)) for point in main_points]

270 points_tuple = tuple([tuple(p) for p in points])

271 self._stencil = points_tuple

273 # determine the stencil weights

274 if len(derivative) == 0:

275 weights = None

276 else:

277 derivation = FiniteDifferenceStencilDerivation(derivative, points_tuple).get_stencil()

278 if not derivation.accuracy:

279 raise Exception('the requested derivative cannot be performed with the available neighbors')

280 weights = derivation.weights

282 # if the weights are underdefined, we can choose the free symbols to find the sparsest stencil

283 free_weights = set(itertools.chain(*[w.free_symbols for w in weights]))

284 if len(free_weights) > 0:

285 zero_counts = defaultdict(list)

286 for values in itertools.product([-1, -sp.Rational(1, 2), 0, 1, sp.Rational(1, 2)],

287 repeat=len(free_weights)):

289 weights = [w.subs(subs) for w in derivation.weights]

290 if not all(a == 0 for a in weights):

291 zero_count = sum([1 for w in weights if w == 0])

292 zero_counts[zero_count].append(weights)

293 best = zero_counts[max(zero_counts.keys())]

294 if len(best) > 1: # if there are multiple, pick the one that contains a nonzero center weight

295 center = [tuple(p + pos) for p in points].index((0, 0, 0)[:dim])

296 best = [b for b in best if b[center] != 0]

297 if len(best) > 1: # if there are still multiple, they are equivalent, so we average

298 weights = [sum([b[i] for b in best]) / len(best) for i in range(len(weights))]

299 else:

300 weights = best

301 assert weights

303 points_tuple = tuple([tuple(p + pos) for p in points])

304 self._points = points_tuple

305 self._weights = weights

307 @property

308 def points(self):

309 """return the points of the stencil"""

310 return self._points

312 @property

313 def stencil(self):

314 """return the points of the stencil relative to the staggered position specified by neighbor"""

315 return self._stencil

317 @property

318 def weights(self):

319 """return the weights of the stencil"""

320 assert self._weights is not None

321 return self._weights

323 def visualize(self):

324 if self._weights is None:

325 ws = None

326 else:

327 ws = np.array([w for w in self.weights if w != 0], dtype=float)

328 pts = np.array([p for i, p in enumerate(self.points) if self.weights[i] != 0], dtype=int)

329 from pystencils.stencil import plot

330 plot(pts, data=ws)

332 def apply(self, access: Field.Access):

333 return sum([access.get_shifted(*point) * weight for point, weight in zip(self.points, self.weights)])