stencil.py 16.1 KB
Newer Older
1
"""This submodule offers functions to work with stencils in expression an offset-list form."""
2
3
from typing import Sequence
import numpy as np
4
5
6
7
8
import sympy as sp
from collections import defaultdict


def inverse_direction(direction):
9
10
11
12
13
14
    """Returns inverse i.e. negative of given direction tuple

    Example:
        >>> inverse_direction((1, -1, 0))
        (-1, 1, 0)
    """
15
16
17
    return tuple([-i for i in direction])


18
def is_valid(stencil, max_neighborhood=None):
19
20
21
22
    """
    Tests if a nested sequence is a valid stencil i.e. all the inner sequences have the same length.
    If max_neighborhood is specified, it is also verified that the stencil does not contain any direction components
    with absolute value greater than the maximal neighborhood.
23
24
25
26
27
28
29
30

    Examples:
        >>> is_valid([(1, 0), (1, 0, 0)])  # stencil entries have different length
        False
        >>> is_valid([(2, 0), (1, 0)])
        True
        >>> is_valid([(2, 0), (1, 0)], max_neighborhood=1)
        False
31
32
33
34
35
36
37
38
39
40
41
42
    """
    expected_dim = len(stencil[0])
    for d in stencil:
        if len(d) != expected_dim:
            return False
        if max_neighborhood is not None:
            for d_i in d:
                if abs(d_i) > max_neighborhood:
                    return False
    return True


43
44
45
46
47
48
49
50
51
def is_symmetric(stencil):
    """Tests for every direction d, that -d is also in the stencil

    Examples:
        >>> is_symmetric([(1, 0), (0, 1)])
        False
        >>> is_symmetric([(1, 0), (-1, 0)])
        True
    """
52
53
54
55
56
57
    for d in stencil:
        if inverse_direction(d) not in stencil:
            return False
    return True


58
59
60
61
62
63
64
65
66
def have_same_entries(s1, s2):
    """Checks if two stencils are the same

    Examples:
        >>> stencil1 = [(1, 0), (-1, 0), (0, 1), (0, -1)]
        >>> stencil2 = [(-1, 0), (0, -1), (1, 0), (0, 1)]
        >>> have_same_entries(stencil1, stencil2)
        True
    """
67
68
69
70
71
72
73
74
    if len(s1) != len(s2):
        return False
    return len(set(s1) - set(s2)) == 0


# -------------------------------------Expression - Coefficient Form Conversion ----------------------------------------


75
def coefficient_dict(expr):
76
77
78
79
80
81
82
83
84
85
86
87
88
    """Extracts coefficients in front of field accesses in a expression.

    Expression may only access a single field at a single index.

    Returns:
        center, coefficient dict, nonlinear part
        where center is the single field that is accessed in expression accessed at center
        and coefficient dict maps offsets to coefficients. The nonlinear part is everything that is not in the form of
        coefficient times field access.

    Examples:
        >>> import pystencils as ps
        >>> f = ps.fields("f(3) : double[2D]")
89
        >>> field, coeffs, nonlinear_part = coefficient_dict(2 * f[0, 1](1) + 3 * f[-1, 0](1) + 123)
90
91
92
93
        >>> assert nonlinear_part == 123 and field == f(1)
        >>> sorted(coeffs.items())
        [((-1, 0), 3), ((0, 1), 2)]
    """
94
    from pystencils import Field
95
96
97
98
99
100
101
102
103
104
105
106
107
108
    expr = expr.expand()
    field_accesses = expr.atoms(Field.Access)
    fields = set(fa.field for fa in field_accesses)
    accessed_indices = set(fa.index for fa in field_accesses)

    if len(fields) != 1:
        raise ValueError("Could not extract stencil coefficients. "
                         "Expression has to be a linear function of exactly one field.")
    if len(accessed_indices) != 1:
        raise ValueError("Could not extract stencil coefficients. Field is accessed at multiple indices")

    field = fields.pop()
    idx = accessed_indices.pop()

109
110
    coeffs = defaultdict(lambda: 0)
    coeffs.update({fa.offsets: expr.coeff(fa) for fa in field_accesses})
111

112
    linear_part = sum(c * field[off](*idx) for off, c in coeffs.items())
113
    nonlinear_part = expr - linear_part
114
    return field(*idx), coeffs, nonlinear_part
115
116


117
def coefficients(expr):
118
119
    """Returns two lists - one with accessed offsets and one with their coefficients.

120
    Same restrictions as `coefficient_dict` apply. Expression must not have any nonlinear part
121
122
123

    >>> import pystencils as ps
    >>> f = ps.fields("f(3) : double[2D]")
124
    >>> coff = coefficients(2 * f[0, 1](1) + 3 * f[-1, 0](1))
125
    """
126
    field_center, coeffs, nonlinear_part = coefficient_dict(expr)
127
    assert nonlinear_part == 0
128
129
    stencil = list(coeffs.keys())
    entries = [coeffs[c] for c in stencil]
130
131
132
    return stencil, entries


133
def coefficient_list(expr, matrix_form=False):
134
135
    """Returns stencil coefficients in the form of nested lists

136
    Same restrictions as `coefficient_dict` apply. Expression must not have any nonlinear part
137
138
139
140

    Examples:
        >>> import pystencils as ps
        >>> f = ps.fields("f: double[2D]")
141
        >>> coefficient_list(2 * f[0, 1] + 3 * f[-1, 0])
142
        [[0, 0, 0], [3, 0, 0], [0, 2, 0]]
143
        >>> coefficient_list(2 * f[0, 1] + 3 * f[-1, 0], matrix_form=True)
144
145
146
147
148
        Matrix([
        [0, 2, 0],
        [3, 0, 0],
        [0, 0, 0]])
    """
149
    field_center, coeffs, nonlinear_part = coefficient_dict(expr)
150
151
152
153
154
    assert nonlinear_part == 0
    field = field_center.field

    dim = field.spatial_dimensions
    max_offsets = defaultdict(lambda: 0)
155
    for offset in coeffs.keys():
156
157
158
159
        for d, off in enumerate(offset):
            max_offsets[d] = max(max_offsets[d], abs(off))

    if dim == 1:
160
        result = [coeffs[(i,)] for i in range(-max_offsets[0], max_offsets[0] + 1)]
161
162
163
164
165
166
        return sp.Matrix(result) if matrix_form else result
    else:
        y_range = list(range(-max_offsets[1], max_offsets[1] + 1))
        if matrix_form:
            y_range.reverse()
        if dim == 2:
167
            result = [[coeffs[(i, j)]
168
169
170
171
                       for i in range(-max_offsets[0], max_offsets[0] + 1)]
                      for j in y_range]
            return sp.Matrix(result) if matrix_form else result
        elif dim == 3:
172
            result = [[[coeffs[(i, j, k)]
173
174
175
176
177
178
179
180
                        for i in range(-max_offsets[0], max_offsets[0] + 1)]
                       for j in y_range]
                      for k in range(-max_offsets[2], max_offsets[2] + 1)]
            return [sp.Matrix(l) for l in result] if matrix_form else result
        else:
            raise ValueError("Can only handle fields with 1,2 or 3 spatial dimensions")


181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
# ------------------------------------- Point-on-compass notation ------------------------------------------------------


def offset_component_to_direction_string(coordinate_id: int, value: int) -> str:
    """Translates numerical offset to string notation.

    x offsets are labeled with east 'E' and 'W',
    y offsets with north 'N' and 'S' and
    z offsets with top 'T' and bottom 'B'
    If the absolute value of the offset is bigger than 1, this number is prefixed.

    Args:
        coordinate_id: integer 0, 1 or 2 standing for x,y and z
        value: integer offset

    Examples:
        >>> offset_component_to_direction_string(0, 1)
        'E'
        >>> offset_component_to_direction_string(1, 2)
        '2N'
    """
    assert 0 <= coordinate_id < 3, "Works only for at most 3D arrays"
    name_components = (('W', 'E'),  # west, east
                       ('S', 'N'),  # south, north
                       ('B', 'T'))  # bottom, top
    if value == 0:
        result = ""
    elif value < 0:
        result = name_components[coordinate_id][0]
    else:
        result = name_components[coordinate_id][1]
    if abs(value) > 1:
        result = "%d%s" % (abs(value), result)
    return result


def offset_to_direction_string(offsets: Sequence[int]) -> str:
    """
    Translates numerical offset to string notation.
    For details see :func:`offset_component_to_direction_string`
    Args:
        offsets: 3-tuple with x,y,z offset

    Examples:
        >>> offset_to_direction_string([1, -1, 0])
        'SE'
        >>> offset_to_direction_string(([-3, 0, -2]))
        '2B3W'
    """
    if len(offsets) > 3:
        return str(offsets)
    names = ["", "", ""]
    for i in range(len(offsets)):
        names[i] = offset_component_to_direction_string(i, offsets[i])
    name = "".join(reversed(names))
    if name == "":
        name = "C"
    return name


def direction_string_to_offset(direction: str, dim: int = 3):
    """
    Reverse mapping of :func:`offset_to_direction_string`

    Args:
        direction: string representation of offset
        dim: dimension of offset, i.e the length of the returned list

    Examples:
        >>> direction_string_to_offset('NW', dim=3)
        array([-1,  1,  0])
        >>> direction_string_to_offset('NW', dim=2)
        array([-1,  1])
        >>> direction_string_to_offset(offset_to_direction_string((3,-2,1)))
        array([ 3, -2,  1])
    """
    offset_dict = {
        'C': np.array([0, 0, 0]),

        'W': np.array([-1, 0, 0]),
        'E': np.array([1, 0, 0]),

        'S': np.array([0, -1, 0]),
        'N': np.array([0, 1, 0]),

        'B': np.array([0, 0, -1]),
        'T': np.array([0, 0, 1]),
    }
    offset = np.array([0, 0, 0])

    while len(direction) > 0:
        factor = 1
        first_non_digit = 0
        while direction[first_non_digit].isdigit():
            first_non_digit += 1
        if first_non_digit > 0:
            factor = int(direction[:first_non_digit])
            direction = direction[first_non_digit:]
        cur_offset = offset_dict[direction[0]]
        offset += factor * cur_offset
        direction = direction[1:]
    return offset[:dim]


285
286
287
# -------------------------------------- Visualization -----------------------------------------------------------------


288
def plot(stencil, **kwargs):
289
290
    dim = len(stencil[0])
    if dim == 2:
291
        plot_2d(stencil, **kwargs)
292
293
294
295
296
297
298
    else:
        slicing = False
        if 'slice' in kwargs:
            slicing = kwargs['slice']
            del kwargs['slice']

        if slicing:
299
            plot_3d_slicing(stencil, **kwargs)
300
        else:
301
            plot_3d(stencil, **kwargs)
302
303


304
def plot_2d(stencil, axes=None, figure=None, data=None, textsize='12', **kwargs):
305
306
307
    """
    Creates a matplotlib 2D plot of the stencil

308
309
310
311
312
    Args:
        stencil: sequence of directions
        axes: optional matplotlib axes
        data: data to annotate the directions with, if none given, the indices are used
        textsize: size of annotation text
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
    """
    from matplotlib.patches import BoxStyle
    import matplotlib.pyplot as plt

    if axes is None:
        if figure is None:
            figure = plt.gcf()
        axes = figure.gca()

    text_box_style = BoxStyle("Round", pad=0.3)
    head_length = 0.1
    max_offsets = [max(abs(d[c]) for d in stencil) for c in (0, 1)]

    if data is None:
        data = list(range(len(stencil)))

    for direction, annotation in zip(stencil, data):
        assert len(direction) == 2, "Works only for 2D stencils"

        if not(direction[0] == 0 and direction[1] == 0):
            axes.arrow(0, 0, direction[0], direction[1], head_width=0.08, head_length=head_length, color='k')

        if isinstance(annotation, sp.Basic):
            annotation = "$" + sp.latex(annotation) + "$"
        else:
            annotation = str(annotation)

        def position_correction(d, magnitude=0.18):
            if d < 0:
                return -magnitude
            elif d > 0:
                return +magnitude
            else:
                return 0
        text_position = [direction[c] + position_correction(direction[c]) for c in (0, 1)]
        axes.text(*text_position, annotation, verticalalignment='center',
                  zorder=30, horizontalalignment='center', size=textsize,
                  bbox=dict(boxstyle=text_box_style, facecolor='#00b6eb', alpha=0.85, linewidth=0))

    axes.set_axis_off()
    axes.set_aspect('equal')
    max_offsets = [m if m > 0 else 0.1 for m in max_offsets]
    border = 0.1
    axes.set_xlim([-border - max_offsets[0], border + max_offsets[0]])
    axes.set_ylim([-border - max_offsets[1], border + max_offsets[1]])


360
def plot_3d_slicing(stencil, slice_axis=2, figure=None, data=None, **kwargs):
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
    """Visualizes a 3D, first-neighborhood stencil by plotting 3 slices along a given axis.

    Args:
        stencil: stencil as sequence of directions
        slice_axis: 0, 1, or 2 indicating the axis to slice through
        data: optional data to print as text besides the arrows
    """
    import matplotlib.pyplot as plt

    for d in stencil:
        for element in d:
            assert element == -1 or element == 0 or element == 1, "This function can only first neighborhood stencils"

    if figure is None:
        figure = plt.gcf()

    axes = [figure.add_subplot(1, 3, i + 1) for i in range(3)]
    splitted_directions = [[], [], []]
    splitted_data = [[], [], []]
    axes_names = ['x', 'y', 'z']

    for i, d in enumerate(stencil):
        split_idx = d[slice_axis] + 1
        reduced_dir = tuple([element for j, element in enumerate(d) if j != slice_axis])
        splitted_directions[split_idx].append(reduced_dir)
        splitted_data[split_idx].append(i if data is None else data[i])

    for i in range(3):
389
        plot_2d(splitted_directions[i], axes=axes[i], data=splitted_data[i], **kwargs)
390
    for i in [-1, 0, 1]:
391
        axes[i + 1].set_title("Cut at %s=%d" % (axes_names[slice_axis], i), y=1.08)
392
393


394
def plot_3d(stencil, figure=None, axes=None, data=None, textsize='8'):
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
    """
    Draws 3D stencil into a 3D coordinate system, parameters are similar to :func:`visualize_stencil_2d`
    If data is None, no labels are drawn. To draw the labels as in the 2D case, use ``data=list(range(len(stencil)))``
    """
    from matplotlib.patches import FancyArrowPatch
    from mpl_toolkits.mplot3d import proj3d
    import matplotlib.pyplot as plt
    from matplotlib.patches import BoxStyle
    from itertools import product, combinations
    import numpy as np

    class Arrow3D(FancyArrowPatch):
        def __init__(self, xs, ys, zs, *args, **kwargs):
            FancyArrowPatch.__init__(self, (0, 0), (0, 0), *args, **kwargs)
            self._verts3d = xs, ys, zs

        def draw(self, renderer):
            xs3d, ys3d, zs3d = self._verts3d
            xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)
            self.set_positions((xs[0], ys[0]), (xs[1], ys[1]))
            FancyArrowPatch.draw(self, renderer)

    if axes is None:
        if figure is None:
            figure = plt.figure()
        axes = figure.gca(projection='3d')
        axes.set_aspect("equal")

    if data is None:
        data = [None] * len(stencil)

    text_offset = 1.25
    text_box_style = BoxStyle("Round", pad=0.3)

    # Draw cell (cube)
    r = [-1, 1]
    for s, e in combinations(np.array(list(product(r, r, r))), 2):
        if np.sum(np.abs(s - e)) == r[1] - r[0]:
            axes.plot3D(*zip(s, e), color="k", alpha=0.5)

    for d, annotation in zip(stencil, data):
        assert len(d) == 3, "Works only for 3D stencils"
        if not (d[0] == 0 and d[1] == 0 and d[2] == 0):
            if d[0] == 0:
                color = '#348abd'
            elif d[1] == 0:
                color = '#fac364'
            elif sum([abs(d) for d in d]) == 2:
                color = '#95bd50'
            else:
                color = '#808080'

            a = Arrow3D([0, d[0]], [0, d[1]], [0, d[2]], mutation_scale=20, lw=2, arrowstyle="-|>", color=color)
            axes.add_artist(a)

        if annotation:
            if isinstance(annotation, sp.Basic):
                annotation = "$" + sp.latex(annotation) + "$"
            else:
                annotation = str(annotation)

            axes.text(d[0] * text_offset, d[1] * text_offset, d[2] * text_offset,
                      annotation, verticalalignment='center', zorder=30,
                      size=textsize, bbox=dict(boxstyle=text_box_style, facecolor='#777777', alpha=0.6, linewidth=0))

    axes.set_xlim([-text_offset * 1.1, text_offset * 1.1])
    axes.set_ylim([-text_offset * 1.1, text_offset * 1.1])
    axes.set_zlim([-text_offset * 1.1, text_offset * 1.1])
    axes.set_axis_off()


466
def plot_expression(expr, **kwargs):
467
    """Displays coefficients of a linear update expression of a single field as matplotlib arrow drawing."""
468
    stencil, coeffs = coefficients(expr)
469
470
471
    dim = len(stencil[0])
    assert 0 < dim <= 3
    if dim == 1:
472
        return coefficient_list(expr, matrix_form=True)
473
    elif dim == 2:
474
        return plot_2d(stencil, data=coeffs, **kwargs)
475
    elif dim == 3:
476
        return plot_3d_slicing(stencil, data=coeffs, **kwargs)