How to repeat elements of an array along two axes?

You could use the Kronecker product, see numpy.kron:

>>> a = np.arange(12).reshape(3,4)
>>> print(np.kron(a, np.ones((2,2), dtype=a.dtype)))
[[ 0  0  1  1  2  2  3  3]
 [ 0  0  1  1  2  2  3  3]
 [ 4  4  5  5  6  6  7  7]
 [ 4  4  5  5  6  6  7  7]
 [ 8  8  9  9 10 10 11 11]
 [ 8  8  9  9 10 10 11 11]]

Your original method is OK too, though!

You can make use of np.broadcast_to here:

def broadcast_tile(a, h, w):
    x, y = a.shape
    m, n = x * h, y * w
    return np.broadcast_to(
        a.reshape(x, 1, y, 1), (x, h, y, w)
    ).reshape(m, n)

broadcast_tile(a, 2, 2)

array([[ 0,  0,  1,  1,  2,  2,  3,  3],
       [ 0,  0,  1,  1,  2,  2,  3,  3],
       [ 4,  4,  5,  5,  6,  6,  7,  7],
       [ 4,  4,  5,  5,  6,  6,  7,  7],
       [ 8,  8,  9,  9, 10, 10, 11, 11],
       [ 8,  8,  9,  9, 10, 10, 11, 11]])

Performance

Functions

def chris(a, h, w):
    x, y = a.shape
    m, n = x * h, y * w
    return np.broadcast_to(
        a.reshape(x, 1, y, 1), (x, h, y, w)
    ).reshape(m, n)

def alex_riley(a, b0, b1):
    r, c = a.shape
    rs, cs = a.strides
    x = np.lib.stride_tricks.as_strided(a, (r, b0, c, b1), (rs, 0, cs, 0))
    return x.reshape(r*b0, c*b1)

def paul_panzer(a, b0, b1):
    r, c = a.shape
    out = np.empty((r, b0, c, b1), a.dtype)
    out[...] = a[:, None, :, None]
    return out.reshape(r*b0, c*b1)

def wim(a, h, w):
    return np.kron(a, np.ones((h,w), dtype=a.dtype))

Setup

import numpy as np
import pandas as pd
from timeit import timeit

res = pd.DataFrame(
       index=['chris', 'alex_riley', 'paul_panzer', 'wim'],
       columns=[5, 10, 20, 50, 100, 500, 1000],
       dtype=float
)

a = np.arange(100).reshape((10,10))

for f in res.index:
    for c in res.columns:
        h = w = c
        stmt = '{}(a, h, w)'.format(f)
        setp = 'from __main__ import h, w, a, {}'.format(f)
        res.at[f, c] = timeit(stmt, setp, number=50)

Output