`x_hat_b` and `x_hat_f` are the same in `convolutional_smoother`

[pavelkomarov]

Here is the current utility.convolutional_smoother code:

def convolutional_smoother(x, kernel, iterations=1):
    """Perform smoothing by convolving x with a kernel forward and backward and then combining

    :param np.array[float] x: 1D data
    :param np.array[float] kernel: kernel to use in convolution
    :param int iterations: number of iterations, >=1
    
    :return: **x_hat** (np.array[float]) -- smoothed x
    """
    x_hat = np.hstack((x[::-1], x, x[::-1])) # pad
    w = np.linspace(0, 1, len(x_hat)) # weights

    for _ in range(iterations):
        x_hat_f = np.convolve(x_hat, kernel, 'same')
        x_hat_b = np.convolve(x_hat[::-1], kernel, 'same')[::-1]
        
        x_hat = x_hat_f*w + x_hat_b*(1-w)

    return x_hat[len(x):len(x)*2]

But try this:

>>> import numpy as np
>>> x = np.arange(10)
>>> kernel = np.array([1/3, 1/3, 1/3])
>>> np.convolve(x, kernel, 'same')
array([0.33333333, 1.        , 2.        , 3.        , 4.        ,
       5.        , 6.        , 7.        , 8.        , 5.66666667])
>>> np.convolve(x[::-1], kernel, 'same')[::-1]
array([0.33333333, 1.        , 2.        , 3.        , 4.        ,
       5.        , 6.        , 7.        , 8.        , 5.66666667])
>>> x = np.random.randn(10)
>>> x
array([ 1.30231522,  1.28738889,  0.82759237, -0.41610496, -2.53010508,
        1.53911992,  0.1855655 ,  0.59859514, -0.48033393,  0.41225268])
>>> np.convolve(x, kernel, 'same')
array([ 0.8632347 ,  1.13909883,  0.5662921 , -0.70620589, -0.46903004,
       -0.26847322,  0.77442685,  0.10127557,  0.17683796, -0.02269375])
>>> np.convolve(x[::-1], kernel, 'same')[::-1]
array([ 0.8632347 ,  1.13909883,  0.5662921 , -0.70620589, -0.46903004,
       -0.26847322,  0.77442685,  0.10127557,  0.17683796, -0.02269375])
>>> from pynumdiff.utils.utility import friedrichs_kernel
>>> x = np.random.randn(100)
>>> a = np.convolve(x, kernel, 'same')
>>> b = np.convolve(x[::-1], kernel, 'same')[::-1]
>>> np.max(np.abs(a - b))
np.float64(3.1138286393783687e-16)

So essentially the code is doing more work than necessary. There is no need to do forward and backward and then weight them together.

[pavelkomarov]

The padding is also excessive, given we just chop off the ends. It also protects from edge effects for a lot of iterations, but not indefinitely.