Module `PDESolver.Sampling`

Expand source code

import tensorflow as tf
import numpy as np


class Region:
    def __init__(self):
        """
        Constructor for a region/domain.
        """

        pass

    def pick(self, n, sampler):
        """
        Samples n points from the region.

        Parameters
        -----------
        n: int
            Number of points to sample
        sampler: Sampler
            Sampler to use

        Returns
        -----------
        array: Array of shape (n, dimension)
        """
        
        raise NotImplementedError

    def get_bounds(self):
        """
        Returns the bounds of the region.

        Returns
        -----------
        array: Array of shape (2, dimension)
        """
        
        raise NotImplementedError

    def __and__(self, other):
        """
        Returns the union of two regions.

        Parameters
        -----------
        other: Region
            Other region

        Returns
        -----------
        Region
        """
        
        if isinstance(other, Union):
            return Union(*([self] + other.regions))
        elif isinstance(self, Union):
            return Union(*([other] + self.regions))
        else:
            return Union(self, other)


class Cuboid(Region):
    def __init__(self, corner1, corner2):
        """
        Constructor for a cuboid region which sides are parallel to all axis.

        Parameters
        -----------
        corner1: list
            Coordinates of the first corner
        corner2: list
            Coordinates of the second corner

        Examples
        -----------
        >>> Cuboid([0, 0], [1, 1]) # Unit square
        >>> Cuboid([0, 0, 0], [1, 1, 1]) # Unit cube
        >>> Cuboid([0, 0], [1, 0]) # Intersection of unit square with x-axis
        >>> Cuboid([0, 0], [0, 1]) # Intersection of unit square with y-axis
        """

        super().__init__()
        self.corner1 = tf.constant(corner1, dtype=tf.float32)
        self.corner2 = tf.constant(corner2, dtype=tf.float32)
        self.diff_idx = tf.reshape(tf.where(self.corner1 != self.corner2), [-1])
        self.dimension = len(self.diff_idx)

    def pick(self, n, sampler):
        if self.dimension == 0:
            result = tf.ones((n, len(self.corner1))) * self.corner1
        else:
            start = tf.gather(self.corner1, self.diff_idx)
            end = tf.gather(self.corner2, self.diff_idx)

            k = tf.cast(tf.math.ceil(n**(1 / self.dimension)), tf.int64)
            lines = []
            for i in range(self.dimension):
                ts = sampler.pick(k)
                lines += [(1 - ts) * start[i] + ts * end[i]]

            xy_matrix = tf.meshgrid(*lines)
            xy_vector = tf.concat([tf.reshape(variable, [-1, 1]) for variable in xy_matrix], axis=-1)

            n_approx = k**self.dimension
            result = tf.ones((n_approx, len(self.corner1))) * self.corner1

            row_idx = tf.range(n_approx, dtype=tf.int64)
            row_idx = tf.repeat(row_idx, self.dimension)

            col_idx = tf.transpose(tf.tile(self.diff_idx[:, tf.newaxis], [1, n_approx]))
            col_idx = tf.reshape(col_idx, [-1])

            indices = tf.transpose(tf.reshape(tf.concat([row_idx, col_idx], 0), [-1, n_approx * self.dimension]))
            result = tf.tensor_scatter_nd_update(result, indices, tf.reshape(xy_vector, [-1]))

        return result

    def get_bounds(self):
        return self.corner1.numpy(), self.corner2.numpy()


class Union(Region):
    def __init__(self, *regions):
        """
        Constructor for a union of regions.

        Parameters
        -----------
        regions: Region
            List of regions

        Examples
        -----------
        >>> Union(Cuboid([0, 0], [1, 0]), Cuboid([0, 0], [0, 1])) # Intersection of unit square with x- and y-axis
        >>> Cuboid([0, 0], [1, 0]) & Cuboid([0, 0], [0, 1]) # Shorthand notation of example above
        """

        super().__init__()

        samples = [r.pick(1, Random()) for r in regions]
        dimensions = [s.shape[1] for s in samples]
        assert len(set(dimensions)) == 1, "All regions must have the same (shape-)dimension" 

        self.regions = list(regions)

    def pick(self, n, sampler):
        num_regions = len(self.regions)
        ns = [n // num_regions] * num_regions
        ns[-1] += n % num_regions

        return tf.concat([r.pick(ns[i], sampler) for i, r in enumerate(self.regions)], axis=0)

    def get_bounds(self):
        upper = tf.maximum([np.max(region.get_bounds()) for region in self.regions], axis=0)
        lower = tf.minimum([np.min(region.get_bounds()) for region in self.regions], axis=0)

        return lower, upper


class Sampler:
    def __init__(self):
        """
        Constructor for a sampler.
        """

        pass

    def pick(self, n):
        """
        Samples n points from [0; 1] following a pattern determined by the used Sampler

        Parameters
        -----------
        n: int
            Number of points to sample

        Returns
        -----------
        array: Array of shape (n,)
        """
        
        raise NotImplementedError


class Random(Sampler):
    def __init__(self):
        """
        Constructor for a sampler sampling from a uniform distibution on [0; 1]. 

        Examples
        -----------
        >>> sampler = Random()
        >>> sampler.pick(5) 
        <tf.Tensor: shape=(5,), dtype=float32, numpy=
        array([0.41073012, 0.6588371 , 0.03179121, 0.96656334, 0.14362192],
            dtype=float32)>
        """

        super().__init__()

    def pick(self, n):
        return tf.random.uniform(shape=(n,))


class Equidistant(Sampler):
    def __init__(self):
        """
        Constructor for a sampler sampling equidistantly from [0; 1].

        Examples
        -----------
        >>> sampler = Equidistant() 
        >>> sampler.pick(5)
        <tf.Tensor: shape=(5,), dtype=float32, numpy=array([0.  , 0.25, 0.5 , 0.75, 1.  ], dtype=float32)>
        """

        super().__init__()

    def pick(self, n):
        return tf.linspace(0., 1., n)


class EquidistantRandom(Sampler):
    def __init__(self, n):
        """
        Constructor for a sampler sampling a random subset of a equidistant grid in [0; 1].

        Parameters
        -----------
        n: int
            Number of points in the equidistant grid

        Examples
        ------------
        >>> sampler = EquidistantRandom(5)
        >>> sampler.pick(3)
        <tf.Tensor: shape=(1, 3), dtype=float32, numpy=array([[0.5 , 0.75, 0.  ]], dtype=float32)>
        """

        super().__init__()
        self.n = n
        self.original = tf.linspace(0., 1., n)

    def pick(self, n):
        idx = tf.random.uniform(shape=(int(n),), minval=0, maxval=self.n - 1, dtype=tf.int64)
        return tf.gather(self.original, idx)

Classes

class Cuboid (corner1, corner2)

Constructor for a cuboid region which sides are parallel to all axis.

Parameters

corner1 : list: Coordinates of the first corner
corner2 : list: Coordinates of the second corner

Examples

>>> Cuboid([0, 0], [1, 1]) # Unit square
>>> Cuboid([0, 0, 0], [1, 1, 1]) # Unit cube
>>> Cuboid([0, 0], [1, 0]) # Intersection of unit square with x-axis
>>> Cuboid([0, 0], [0, 1]) # Intersection of unit square with y-axis

Expand source code

class Cuboid(Region):
    def __init__(self, corner1, corner2):
        """
        Constructor for a cuboid region which sides are parallel to all axis.

        Parameters
        -----------
        corner1: list
            Coordinates of the first corner
        corner2: list
            Coordinates of the second corner

        Examples
        -----------
        >>> Cuboid([0, 0], [1, 1]) # Unit square
        >>> Cuboid([0, 0, 0], [1, 1, 1]) # Unit cube
        >>> Cuboid([0, 0], [1, 0]) # Intersection of unit square with x-axis
        >>> Cuboid([0, 0], [0, 1]) # Intersection of unit square with y-axis
        """

        super().__init__()
        self.corner1 = tf.constant(corner1, dtype=tf.float32)
        self.corner2 = tf.constant(corner2, dtype=tf.float32)
        self.diff_idx = tf.reshape(tf.where(self.corner1 != self.corner2), [-1])
        self.dimension = len(self.diff_idx)

    def pick(self, n, sampler):
        if self.dimension == 0:
            result = tf.ones((n, len(self.corner1))) * self.corner1
        else:
            start = tf.gather(self.corner1, self.diff_idx)
            end = tf.gather(self.corner2, self.diff_idx)

            k = tf.cast(tf.math.ceil(n**(1 / self.dimension)), tf.int64)
            lines = []
            for i in range(self.dimension):
                ts = sampler.pick(k)
                lines += [(1 - ts) * start[i] + ts * end[i]]

            xy_matrix = tf.meshgrid(*lines)
            xy_vector = tf.concat([tf.reshape(variable, [-1, 1]) for variable in xy_matrix], axis=-1)

            n_approx = k**self.dimension
            result = tf.ones((n_approx, len(self.corner1))) * self.corner1

            row_idx = tf.range(n_approx, dtype=tf.int64)
            row_idx = tf.repeat(row_idx, self.dimension)

            col_idx = tf.transpose(tf.tile(self.diff_idx[:, tf.newaxis], [1, n_approx]))
            col_idx = tf.reshape(col_idx, [-1])

            indices = tf.transpose(tf.reshape(tf.concat([row_idx, col_idx], 0), [-1, n_approx * self.dimension]))
            result = tf.tensor_scatter_nd_update(result, indices, tf.reshape(xy_vector, [-1]))

        return result

    def get_bounds(self):
        return self.corner1.numpy(), self.corner2.numpy()

Ancestors

Region

Inherited members

Region:
- get_bounds
- pick

class Equidistant

Constructor for a sampler sampling equidistantly from [0; 1].

Examples

>>> sampler = Equidistant() 
>>> sampler.pick(5)
<tf.Tensor: shape=(5,), dtype=float32, numpy=array([0.  , 0.25, 0.5 , 0.75, 1.  ], dtype=float32)>

Expand source code

class Equidistant(Sampler):
    def __init__(self):
        """
        Constructor for a sampler sampling equidistantly from [0; 1].

        Examples
        -----------
        >>> sampler = Equidistant() 
        >>> sampler.pick(5)
        <tf.Tensor: shape=(5,), dtype=float32, numpy=array([0.  , 0.25, 0.5 , 0.75, 1.  ], dtype=float32)>
        """

        super().__init__()

    def pick(self, n):
        return tf.linspace(0., 1., n)

Ancestors

Sampler

Inherited members

Sampler:
- pick

class EquidistantRandom (n)

Constructor for a sampler sampling a random subset of a equidistant grid in [0; 1].

Parameters

n : int: Number of points in the equidistant grid

Examples

>>> sampler = EquidistantRandom(5)
>>> sampler.pick(3)
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=array([[0.5 , 0.75, 0.  ]], dtype=float32)>

Expand source code

class EquidistantRandom(Sampler):
    def __init__(self, n):
        """
        Constructor for a sampler sampling a random subset of a equidistant grid in [0; 1].

        Parameters
        -----------
        n: int
            Number of points in the equidistant grid

        Examples
        ------------
        >>> sampler = EquidistantRandom(5)
        >>> sampler.pick(3)
        <tf.Tensor: shape=(1, 3), dtype=float32, numpy=array([[0.5 , 0.75, 0.  ]], dtype=float32)>
        """

        super().__init__()
        self.n = n
        self.original = tf.linspace(0., 1., n)

    def pick(self, n):
        idx = tf.random.uniform(shape=(int(n),), minval=0, maxval=self.n - 1, dtype=tf.int64)
        return tf.gather(self.original, idx)

Ancestors

Sampler

Inherited members

Sampler:
- pick

class Random

Constructor for a sampler sampling from a uniform distibution on [0; 1].

Examples

>>> sampler = Random()
>>> sampler.pick(5) 
<tf.Tensor: shape=(5,), dtype=float32, numpy=
array([0.41073012, 0.6588371 , 0.03179121, 0.96656334, 0.14362192],
    dtype=float32)>

Expand source code

class Random(Sampler):
    def __init__(self):
        """
        Constructor for a sampler sampling from a uniform distibution on [0; 1]. 

        Examples
        -----------
        >>> sampler = Random()
        >>> sampler.pick(5) 
        <tf.Tensor: shape=(5,), dtype=float32, numpy=
        array([0.41073012, 0.6588371 , 0.03179121, 0.96656334, 0.14362192],
            dtype=float32)>
        """

        super().__init__()

    def pick(self, n):
        return tf.random.uniform(shape=(n,))

Ancestors

Sampler

Inherited members

Sampler:
- pick

class Region

Constructor for a region/domain.

Expand source code

class Region:
    def __init__(self):
        """
        Constructor for a region/domain.
        """

        pass

    def pick(self, n, sampler):
        """
        Samples n points from the region.

        Parameters
        -----------
        n: int
            Number of points to sample
        sampler: Sampler
            Sampler to use

        Returns
        -----------
        array: Array of shape (n, dimension)
        """
        
        raise NotImplementedError

    def get_bounds(self):
        """
        Returns the bounds of the region.

        Returns
        -----------
        array: Array of shape (2, dimension)
        """
        
        raise NotImplementedError

    def __and__(self, other):
        """
        Returns the union of two regions.

        Parameters
        -----------
        other: Region
            Other region

        Returns
        -----------
        Region
        """
        
        if isinstance(other, Union):
            return Union(*([self] + other.regions))
        elif isinstance(self, Union):
            return Union(*([other] + self.regions))
        else:
            return Union(self, other)

Methods

def get_bounds(self)

Returns the bounds of the region.

Returns

array : Array of shape (2, dimension)

Expand source code

def get_bounds(self):
    """
    Returns the bounds of the region.

    Returns
    -----------
    array: Array of shape (2, dimension)
    """
    
    raise NotImplementedError

def pick(self, n, sampler)

Samples n points from the region.

Parameters

n : int: Number of points to sample
sampler : Sampler: Sampler to use

Returns

array : Array of shape (n, dimension)

Expand source code

def pick(self, n, sampler):
    """
    Samples n points from the region.

    Parameters
    -----------
    n: int
        Number of points to sample
    sampler: Sampler
        Sampler to use

    Returns
    -----------
    array: Array of shape (n, dimension)
    """
    
    raise NotImplementedError

class Sampler

Constructor for a sampler.

Expand source code

class Sampler:
    def __init__(self):
        """
        Constructor for a sampler.
        """

        pass

    def pick(self, n):
        """
        Samples n points from [0; 1] following a pattern determined by the used Sampler

        Parameters
        -----------
        n: int
            Number of points to sample

        Returns
        -----------
        array: Array of shape (n,)
        """
        
        raise NotImplementedError

Subclasses

Equidistant
EquidistantRandom
Random

Methods

def pick(self, n)

Samples n points from [0; 1] following a pattern determined by the used Sampler

Parameters

n : int: Number of points to sample

Returns

array : Array of shape (n,)

Expand source code

def pick(self, n):
    """
    Samples n points from [0; 1] following a pattern determined by the used Sampler

    Parameters
    -----------
    n: int
        Number of points to sample

    Returns
    -----------
    array: Array of shape (n,)
    """
    
    raise NotImplementedError

class Union (*regions)

Constructor for a union of regions.

Parameters

regions : Region: List of regions

Examples

>>> Union(Cuboid([0, 0], [1, 0]), Cuboid([0, 0], [0, 1])) # Intersection of unit square with x- and y-axis
>>> Cuboid([0, 0], [1, 0]) & Cuboid([0, 0], [0, 1]) # Shorthand notation of example above

Expand source code

class Union(Region):
    def __init__(self, *regions):
        """
        Constructor for a union of regions.

        Parameters
        -----------
        regions: Region
            List of regions

        Examples
        -----------
        >>> Union(Cuboid([0, 0], [1, 0]), Cuboid([0, 0], [0, 1])) # Intersection of unit square with x- and y-axis
        >>> Cuboid([0, 0], [1, 0]) & Cuboid([0, 0], [0, 1]) # Shorthand notation of example above
        """

        super().__init__()

        samples = [r.pick(1, Random()) for r in regions]
        dimensions = [s.shape[1] for s in samples]
        assert len(set(dimensions)) == 1, "All regions must have the same (shape-)dimension" 

        self.regions = list(regions)

    def pick(self, n, sampler):
        num_regions = len(self.regions)
        ns = [n // num_regions] * num_regions
        ns[-1] += n % num_regions

        return tf.concat([r.pick(ns[i], sampler) for i, r in enumerate(self.regions)], axis=0)

    def get_bounds(self):
        upper = tf.maximum([np.max(region.get_bounds()) for region in self.regions], axis=0)
        lower = tf.minimum([np.min(region.get_bounds()) for region in self.regions], axis=0)

        return lower, upper

Ancestors

Region

Inherited members

Region:
- get_bounds
- pick