import numpy as np
from pylops import LinearOperator
[docs]class SecondDerivative(LinearOperator):
r"""Second derivative.
Apply second-order second derivative.
Parameters
----------
N : :obj:`int`
Number of samples in model.
dims : :obj:`tuple`, optional
Number of samples for each dimension
(``None`` if only one dimension is available)
dir : :obj:`int`, optional
Direction along which smoothing is applied.
sampling : :obj:`float`, optional
Sampling step ``dx``.
edge : :obj:`bool`, optional
Use reduced order derivative at edges (``True``) or
ignore them (``False``)
dtype : :obj:`str`, optional
Type of elements in input array.
Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly (``True``) or
not (``False``)
Notes
-----
The SecondDerivative operator applies a second derivative to any chosen
direction of a multi-dimensional array.
For simplicity, given a one dimensional array, the second-order centered
first derivative is:
.. math::
y[i] = (x[i+1] - 2x[i] + x[i-1]) / dx^2
"""
def __init__(self, N, dims=None, dir=0, sampling=1,
edge=False, dtype='float64'):
self.N = N
self.sampling = sampling
self.edge = edge
if dims is None:
self.dims = (self.N, )
self.reshape = False
else:
if np.prod(dims) != self.N:
raise ValueError('product of dims must equal N!')
else:
self.dims = dims
self.reshape = True
self.dir = dir if dir >= 0 else len(self.dims) + dir
self.shape = (self.N, self.N)
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
if not self.reshape:
x = x.squeeze()
y = np.zeros(self.N, self.dtype)
y[1:-1] = (x[2:] - 2*x[1:-1] + x[0:-2]) / self.sampling**2
if self.edge:
y[0] = (x[0] - 2*x[1] + x[2]) / self.sampling**2
y[-1] = (x[-3] - 2*x[-2] + x[-1]) / self.sampling**2
else:
x = np.reshape(x, self.dims)
if self.dir > 0: # need to bring the dim. to derive to first dim.
x = np.swapaxes(x, self.dir, 0)
y = np.zeros(x.shape, self.dtype)
y[1:-1] = (x[2:] - 2*x[1:-1] + x[0:-2])/self.sampling**2
if self.edge:
y[0] = (x[0] - 2*x[1] + x[2]) / self.sampling ** 2
y[-1] = (x[-3] - 2*x[-2] + x[-1]) / self.sampling ** 2
if self.dir > 0:
y = np.swapaxes(y, 0, self.dir)
y = y.ravel()
return y
def _rmatvec(self, x):
if not self.reshape:
x = x.squeeze()
y = np.zeros(self.N, self.dtype)
y[0:-2] += (x[1:-1]) / self.sampling**2
y[1:-1] -= (2*x[1:-1]) / self.sampling**2
y[2:] += (x[1:-1]) / self.sampling**2
if self.edge:
y[0] += x[0] / self.sampling**2
y[1] -= 2 * x[0] / self.sampling**2
y[2] += x[0] / self.sampling ** 2
y[-3] += x[-1] / self.sampling**2
y[-2] -= 2 * x[-1] / self.sampling**2
y[-1] += x[-1] / self.sampling**2
else:
x = np.reshape(x, self.dims)
if self.dir > 0: # need to bring the dim. to derive to first dim.
x = np.swapaxes(x, self.dir, 0)
y = np.zeros(x.shape, self.dtype)
y[0:-2] += (x[1:-1]) / self.sampling**2
y[1:-1] -= (2*x[1:-1]) / self.sampling**2
y[2:] += (x[1:-1]) / self.sampling**2
if self.edge:
y[0] += x[0] / self.sampling ** 2
y[1] -= 2 * x[0] / self.sampling ** 2
y[2] += x[0] / self.sampling ** 2
y[-3] += x[-1] / self.sampling ** 2
y[-2] -= 2 * x[-1] / self.sampling ** 2
y[-1] += x[-1] / self.sampling ** 2
if self.dir > 0:
y = np.swapaxes(y, 0, self.dir)
y = y.ravel()
return y