def __truediv__(self, y, niter=100):
if self.explicit is True:
if sp.sparse.issparse(self.A):
# use scipy solver for sparse matrices
xest = spsolve(self.A, y)
elif isinstance(self.A, np.ndarray):
# use scipy solvers for dense matrices (used for backward
# compatibility, could be switched to numpy equivalents)
if self.A.shape[0] == self.A.shape[1]:
xest = solve(self.A, y)
else:
xest = lstsq(self.A, y)[0]
else:
# use numpy/cupy solvers for dense matrices
ncp = get_array_module(y)
if self.A.shape[0] == self.A.shape[1]:
xest = ncp.linalg.solve(self.A, y)
else:
xest = ncp.linalg.lstsq(self.A, y)[0]
else:
if isinstance(y, np.ndarray):
# numpy backend
xest = lsqr(self, y, iter_lim=niter)[0]
else:
# cupy backend
ncp = get_array_module(y)
xest = cgls(self,
y,
x0=ncp.zeros(int(self.shape[1]), dtype=self.dtype),
niter=niter)[0]
return xest
def _rmatvec_serial(self, x):
ncp = get_array_module(x)
y = ncp.zeros(self.mops, dtype=self.dtype)
for iop, oper in enumerate(self.ops):
y[self.mmops[iop]:self.mmops[iop + 1]] = \
oper.rmatvec(x[self.nnops[iop]:self.nnops[iop + 1]]).squeeze()
return y
def _rmatvec_centered(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[0:-2] -= (0.5 * x[1:-1]) / self.sampling
y[2:] += (0.5 * x[1:-1]) / self.sampling
if self.edge:
y[0] -= x[0] / self.sampling
y[1] += x[0] / self.sampling
y[-2] -= x[-1] / self.sampling
y[-1] += x[-1] / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0: # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[0:-2] -= (0.5 * x[1:-1]) / self.sampling
y[2:] += (0.5 * x[1:-1]) / self.sampling
if self.edge:
y[0] -= x[0] / self.sampling
y[1] += x[0] / self.sampling
y[-2] -= x[-1] / self.sampling
y[-1] += x[-1] / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y
def _sincinterp(M, iava, dims=None, dir=0, dtype='float64'):
"""Sinc interpolation.
"""
ncp = get_array_module(iava)
_checkunique(iava)
# create sinc interpolation matrix
nreg = M if dims is None else dims[dir]
ireg = ncp.arange(nreg)
sinc = ncp.tile(iava[:, np.newaxis], (1, nreg)) - \
ncp.tile(ireg, (len(iava), 1))
sinc = ncp.sinc(sinc)
# identify additional dimensions and create MatrixMult operator
otherdims = None
if dims is not None:
otherdims = ncp.array(dims)
otherdims = ncp.roll(otherdims, -dir)
otherdims = otherdims[1:]
Op = MatrixMult(sinc, dims=otherdims, dtype=dtype)
# create Transpose operator that brings dir to first dimension
if dir > 0:
axes = np.arange(len(dims), dtype=np.int)
axes = np.roll(axes, -dir)
dimsd = list(dims)
dimsd[dir] = len(iava)
Top = Transpose(dims, axes=axes, dtype=dtype)
T1op = Transpose(dimsd, axes=axes, dtype=dtype)
Op = T1op.H * Op * Top
return Op
def __init__(self, iava, dims, dtype="float64"):
ncp = get_array_module(iava)
# check non-unique pairs (works only with numpy arrays)
_checkunique(to_numpy(iava))
# define dimension of data
ndims = len(dims)
self.dims = dims
self.dimsd = [len(iava[1])] + list(dims[2:])
# find indices and weights
self.iava_t = ncp.floor(iava[0]).astype(int)
self.iava_b = self.iava_t + 1
self.weights_tb = iava[0] - self.iava_t
self.iava_l = ncp.floor(iava[1]).astype(int)
self.iava_r = self.iava_l + 1
self.weights_lr = iava[1] - self.iava_l
# expand dims to weights for nd-arrays
if ndims > 2:
for _ in range(ndims - 2):
self.weights_tb = ncp.expand_dims(self.weights_tb, axis=-1)
self.weights_lr = ncp.expand_dims(self.weights_lr, axis=-1)
self.shape = (np.prod(np.array(self.dimsd)), np.prod(np.array(self.dims)))
self.dtype = np.dtype(dtype)
self.explicit = False
def nonstationary_convmtx(H, n, hc=0, pad=(0, 0)):
r"""Convolution matrix from a bank of filters
Makes a dense convolution matrix :math:`\mathbf{C}`
such that the dot product ``np.dot(C, x)`` is the nonstationary
convolution of the bank of filters :math:`H=[h_1, h_2, h_n]`
and the input signal :math:`x`.
Parameters
----------
H : :obj:`np.ndarray`
Convolution filters (2D array of shape
:math:`[n_{filters} \times n_{h}]`
n : :obj:`int`
Number of columns of convolution matrix
hc : :obj:`np.ndarray`, optional
Index of center of first filter
pad : :obj:`np.ndarray`
Zero-padding to apply to the bank of filters before and after the
provided values (use it to avoid wrap-around or pass filters with
enough padding)
Returns
-------
C : :obj:`np.ndarray`
Convolution matrix
"""
ncp = get_array_module(H)
H = ncp.pad(H, ((0, 0), pad), mode='constant')
C = ncp.array([ncp.roll(h, ih) for ih, h in enumerate(H)])
C = C[:, pad[0] + hc:pad[0] + hc + n].T # take away edges
return C
def __init__(self, A, dims=None, dtype="float64"):
ncp = get_array_module(A)
self.A = A
if isinstance(A, ncp.ndarray):
self.complex = np.iscomplexobj(A)
else:
self.complex = np.iscomplexobj(A.data)
if dims is None:
self.reshape = False
self.shape = A.shape
self.explicit = True
else:
if isinstance(dims, int):
dims = (dims, )
self.reshape = True
self.dims = np.array(dims, dtype=int)
self.reshapedims = [
np.insert([np.prod(self.dims)], 0, self.A.shape[1]),
np.insert([np.prod(self.dims)], 0, self.A.shape[0]),
]
self.shape = (
A.shape[0] * np.prod(self.dims),
A.shape[1] * np.prod(self.dims),
)
self.explicit = False
self.dtype = np.dtype(dtype)
# Check dtype for correctness (upcast to complex when A is complex)
if np.iscomplexobj(A) and not np.iscomplexobj(
np.ones(1, dtype=self.dtype)):
self.dtype = A.dtype
logging.warning("Matrix A is a complex object, dtype cast to %s" %
self.dtype)
def convmtx(h, n):
r"""Convolution matrix
Equivalent of `MATLAB's convmtx function
<http://www.mathworks.com/help/signal/ref/convmtx.html>`_ .
Makes a dense convolution matrix :math:`\mathbf{C}`
such that the dot product ``np.dot(C, x)`` is the convolution of
the filter :math:`h` and the input signal :math:`x`.
Parameters
----------
h : :obj:`np.ndarray`
Convolution filter (1D array)
n : :obj:`int`
Number of columns (if :math:`len(h) < n`) or rows
(if :math:`len(h) \geq n`) of convolution matrix
Returns
-------
C : :obj:`np.ndarray`
Convolution matrix of size :math:`len(h)+n-1 \times n`
(if :math:`len(h) < n`) or :math:`n \times len(h)+n-1`
(if :math:`len(h) \geq n`)
"""
ncp = get_array_module(h)
if len(h) < n:
col_1 = ncp.r_[h[0], ncp.zeros(n - 1, dtype=h.dtype)]
row_1 = ncp.r_[h, ncp.zeros(n - 1, dtype=h.dtype)]
else:
row_1 = ncp.r_[h[0], ncp.zeros(n - 1, dtype=h.dtype)]
col_1 = ncp.r_[h, ncp.zeros(n - 1, dtype=h.dtype)]
C = get_toeplitz(h)(col_1, row_1)
return C
def _matvec(self, x):
ncp = get_array_module(x)
x = ncp.reshape(x, self.dims)
y = x[self.iava_t, self.iava_l] * (1 - self.weights_tb) * (1 - self.weights_lr) + \
x[self.iava_t, self.iava_r] * (1 - self.weights_tb) * self.weights_lr + \
x[self.iava_b, self.iava_l] * self.weights_tb * (1 - self.weights_lr) + \
x[self.iava_b, self.iava_r] * self.weights_tb * self.weights_lr
return y.ravel()
def _IRLS_model(Op, data, nouter, threshR=False, epsR=1e-10,
epsI=1e-10, x0=None, tolIRLS=1e-10,
returnhistory=False, **kwargs_solver):
r"""Iteratively reweighted least squares with L1 model term
"""
ncp = get_array_module(data)
if x0 is not None:
data = data - Op * x0
if returnhistory:
xinv_hist = ncp.zeros((nouter + 1, int(Op.shape[1])))
rw_hist = ncp.zeros((nouter + 1, int(Op.shape[0])))
Iop = Identity(data.size, dtype=data.dtype)
# first iteration (unweighted least-squares)
if ncp == np:
xinv = Op.H @ \
lsqr(Op @ Op.H + (epsI ** 2) * Iop, data, **kwargs_solver)[0]
else:
xinv = Op.H @ cgls(Op @ Op.H + (epsI ** 2) * Iop, data,
ncp.zeros(int(Op.shape[0]), dtype=Op.dtype),
**kwargs_solver)[0]
if returnhistory:
xinv_hist[0] = xinv
for iiter in range(nouter):
# other iterations (weighted least-squares)
xinvold = xinv.copy()
rw = np.abs(xinv)
rw = rw / rw.max()
R = Diagonal(rw, dtype=rw.dtype)
if ncp == np:
xinv = R @ Op.H @ lsqr(Op @ R @ Op.H + epsI ** 2 * Iop,
data, **kwargs_solver)[0]
else:
xinv = R @ Op.H @ cgls(Op @ R @ Op.H + epsI ** 2 * Iop,
data,
ncp.zeros(int(Op.shape[0]), dtype=Op.dtype),
**kwargs_solver)[0]
# save history
if returnhistory:
rw_hist[iiter] = rw
xinv_hist[iiter + 1] = xinv
# check tolerance
if np.linalg.norm(xinv - xinvold) < tolIRLS:
nouter = iiter
break
# adding initial guess
if x0 is not None:
xinv = x0 + xinv
if returnhistory:
xinv_hist = x0 + xinv_hist
if returnhistory:
return xinv, nouter, xinv_hist[:nouter + 1], rw_hist[:nouter + 1]
else:
return xinv, nouter
def _matvec(self, x):
ncp = get_array_module(x)
if self.explicit:
y = self.Opcol @ x
else:
y = ncp.zeros(int(self.Op.shape[1]), dtype=self.dtype)
y[self.cols] = x
y = self.Op._matvec(y)
return y
def _matvec(self, x):
ncp = get_array_module(x)
if self.reshape:
x = ncp.reshape(x, self.reshapedims[0])
y = self.A.dot(x)
if self.reshape:
return y.ravel()
else:
return y
def _rmatvec(self, x):
ncp = get_array_module(x)
y = self.Op._rmatvec(x)
if self.adj:
if self.real:
y = ncp.real(y)
else:
y = -ncp.imag(y)
return y
def __init__(self,
N,
h,
dims,
offset=None,
dirs=None,
method='fft',
dtype='float64'):
ncp = get_array_module(h)
self.h = h
self.nh = np.array(self.h.shape)
self.dirs = np.arange(len(dims)) if dirs is None else np.array(dirs)
# padding
if offset is None:
offset = np.zeros(self.h.ndim, dtype=np.int)
else:
offset = np.array(offset, dtype=np.int)
self.offset = 2 * (self.nh // 2 - offset)
pad = [(0, 0) for _ in range(self.h.ndim)]
dopad = False
for inh, nh in enumerate(self.nh):
if nh % 2 == 0:
self.offset[inh] -= 1
if self.offset[inh] != 0:
pad[inh] = [
self.offset[inh] if self.offset[inh] > 0 else 0,
-self.offset[inh] if self.offset[inh] < 0 else 0
]
dopad = True
if dopad:
self.h = ncp.pad(self.h, pad, mode='constant')
self.nh = self.h.shape
# find out which directions are used for convolution and define offsets
if len(dims) != len(self.nh):
dimsh = np.ones(len(dims), dtype=np.int)
for idir, dir in enumerate(self.dirs):
dimsh[dir] = self.nh[idir]
self.h = self.h.reshape(dimsh)
if np.prod(dims) != N:
raise ValueError('product of dims must equal N!')
else:
self.dims = np.array(dims)
self.reshape = True
# convolve and correate functions
self.convolve = get_convolve(h)
self.correlate = get_correlate(h)
self.method = method
self.shape = (np.prod(self.dims), np.prod(self.dims))
self.dtype = np.dtype(dtype)
self.explicit = False
def __init__(self, taxis, order, dtype='float64'):
ncp = get_array_module(taxis)
if not isinstance(taxis, ncp.ndarray):
logging.error('t must be numpy.ndarray...')
raise TypeError('t must be numpy.ndarray...')
else:
self.taxis = taxis
self.order = order
self.shape = (len(self.taxis), self.order+1)
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
ncp = get_array_module(x)
if not self.inplace:
x = x.copy()
if not self.reshape:
y = x[self.iava]
else:
x = ncp.reshape(x, self.dims)
y = ncp.take(x, self.iava, axis=self.dir)
y = y.ravel()
return y
def _rmatvec(self, x):
ncp = get_array_module(x)
if not self.inplace:
x = x.copy()
if self.shape[0] == self.shape[1]:
y = x
elif self.shape[0] < self.shape[1]:
y = ncp.zeros(self.shape[1], dtype=self.dtype)
y[:self.shape[0]] = x
else:
y = x[:self.shape[1]]
return y
def _IRLS_data(Op, data, nouter, threshR=False, epsR=1e-10,
epsI=1e-10, x0=None, tolIRLS=1e-10,
returnhistory=False, **kwargs_solver):
r"""Iteratively reweighted least squares with L1 data term
"""
ncp = get_array_module(data)
if x0 is not None:
data = data - Op * x0
if returnhistory:
xinv_hist = ncp.zeros((nouter + 1, int(Op.shape[1])))
rw_hist = ncp.zeros((nouter + 1, int(Op.shape[0])))
# first iteration (unweighted least-squares)
xinv = NormalEquationsInversion(Op, None, data, epsI=epsI,
returninfo=False,
**kwargs_solver)
r = data - Op * xinv
if returnhistory:
xinv_hist[0] = xinv
for iiter in range(nouter):
# other iterations (weighted least-squares)
xinvold = xinv.copy()
if threshR:
rw = 1. / ncp.maximum(ncp.abs(r), epsR)
else:
rw = 1. / (ncp.abs(r) + epsR)
rw = rw / rw.max()
R = Diagonal(rw)
xinv = NormalEquationsInversion(Op, [], data, Weight=R,
epsI=epsI,
returninfo=False,
**kwargs_solver)
r = data - Op * xinv
# save history
if returnhistory:
rw_hist[iiter] = rw
xinv_hist[iiter + 1] = xinv
# check tolerance
if ncp.linalg.norm(xinv - xinvold) < tolIRLS:
nouter = iiter
break
# adding initial guess
if x0 is not None:
xinv = x0 + xinv
if returnhistory:
xinv_hist = x0 + xinv_hist
if returnhistory:
return xinv, nouter, xinv_hist[:nouter + 1], rw_hist[:nouter + 1]
else:
return xinv, nouter
def matrix(self):
"""Return diagonal matrix as dense :obj:`numpy.ndarray`
Returns
-------
densemat : :obj:`numpy.ndarray`
Dense matrix.
"""
ncp = get_array_module(self.diag)
densemat = ncp.diag(self.diag.squeeze())
return densemat
def _matvec(self, x):
ncp = get_array_module(x)
x = ncp.squeeze(x.reshape(self.nsl, self.ny, self.nz))
if self.usematmul:
if self.nz == 1:
x = x[..., ncp.newaxis]
y = ncp.matmul(self.G, x)
else:
y = ncp.squeeze(
ncp.zeros((self.nsl, self.nx, self.nz), dtype=self.dtype))
for isl in range(self.nsl):
y[isl] = ncp.dot(self.G[isl], x[isl])
return y.ravel()
def _rmatvec(self, x):
ncp = get_array_module(x)
if self.reshape:
x = ncp.reshape(x, self.reshapedims[1])
if self.complex:
y = (self.A.T.dot(x.conj())).conj()
else:
y = self.A.T.dot(x)
if self.reshape:
return y.ravel()
else:
return y
def _rmatvec(self, x):
ncp = get_array_module(x)
if self.reshape:
x = ncp.reshape(x, self.dimsd)
if self.dir > 0: # bring the dimension to symmetrize to first
x = ncp.swapaxes(x, self.dir, 0)
y = x[self.nsym - 1:].copy()
y[1:] += x[self.nsym - 2::-1]
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
if self.reshape:
y = ncp.ndarray.flatten(y)
return y
def _rmatvec(self, x):
ncp = get_array_module(x)
ncp_add_at = get_add_at(x)
x = ncp.reshape(x, self.dimsd)
y = ncp.zeros(self.dims, dtype=self.dtype)
ncp_add_at(y, [self.iava_t, self.iava_l],
x * (1 - self.weights_tb) * (1 - self.weights_lr))
ncp_add_at(y, [self.iava_t, self.iava_r],
x * (1 - self.weights_tb) * self.weights_lr)
ncp_add_at(y, [self.iava_b, self.iava_l],
x * self.weights_tb * (1 - self.weights_lr))
ncp_add_at(y, [self.iava_b, self.iava_r],
x * self.weights_tb * self.weights_lr)
return y.ravel()
def inv(self):
r"""Return the inverse of :math:`\mathbf{A}`.
Returns
----------
Ainv : :obj:`numpy.ndarray`
Inverse matrix.
"""
if sp.sparse.issparse(self.A):
Ainv = inv(self.A)
else:
ncp = get_array_module(self.A)
Ainv = ncp.linalg.inv(self.A)
return Ainv
def _matvec(self, x):
ncp = get_array_module(x)
y = ncp.zeros(self.dimsd, dtype=self.dtype)
if self.reshape:
x = ncp.reshape(x, self.dims)
if self.dir > 0: # bring the dimension to symmetrize to first
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.swapaxes(y, self.dir, 0)
y[self.nsym - 1:] = x
y[:self.nsym - 1] = x[-1:0:-1]
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
if self.reshape:
y = ncp.ndarray.flatten(y)
return y
def _matvec_forward(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[:-1] = (x[1:] - x[:-1]) / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0: # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[:-1] = (x[1:] - x[:-1]) / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y
def __init__(
self,
theta,
vsvp=0.5,
nt0=1,
spatdims=None,
linearization="akirich",
dtype="float64",
):
self.ncp = get_array_module(theta)
self.nt0 = nt0 if not isinstance(vsvp, self.ncp.ndarray) else len(vsvp)
self.ntheta = len(theta)
if spatdims is None:
self.spatdims = ()
nspatdims = 1
else:
self.spatdims = spatdims if isinstance(spatdims,
tuple) else (spatdims, )
nspatdims = np.prod(spatdims)
# Compute AVO coefficients
if linearization == "akirich":
Gs = akirichards(theta, vsvp, n=self.nt0)
elif linearization == "fatti":
Gs = fatti(theta, vsvp, n=self.nt0)
elif linearization == "ps":
Gs = ps(theta, vsvp, n=self.nt0)
else:
logging.error("%s not an available "
"linearization...", linearization)
raise NotImplementedError("%s not an available linearization..." %
linearization)
self.G = self.ncp.concatenate([gs.T[:, self.ncp.newaxis] for gs in Gs],
axis=1)
# add dimensions to G to account for horizonal axes
for _ in range(len(self.spatdims)):
self.G = self.G[..., np.newaxis]
self.npars = len(Gs)
self.shape = (
self.nt0 * self.ntheta * nspatdims,
self.nt0 * self.npars * nspatdims,
)
self.dtype = np.dtype(dtype)
self.explicit = False
def __init__(self, diag, dims=None, dir=0, dtype='float64'):
ncp = get_array_module(diag)
self.diag = diag.flatten()
self.complex = True if ncp.iscomplexobj(self.diag) else False
if dims is None:
self.shape = (len(self.diag), len(self.diag))
self.dims = None
self.reshape = False
else:
diagdims = [1] * len(dims)
diagdims[dir] = dims[dir]
self.diag = self.diag.reshape(diagdims)
self.shape = (np.prod(dims), np.prod(dims))
self.dims = dims
self.reshape = True
self.dtype = np.dtype(dtype)
self.explicit = False
def _rmatvec(self, x):
ncp = get_array_module(x)
x = ncp.squeeze(x.reshape(self.nsl, self.nx, self.nz))
if self.usematmul:
if self.nz == 1:
x = x[..., ncp.newaxis]
if hasattr(self, 'GT'):
y = ncp.matmul(self.GT, x)
else:
y = ncp.matmul(self.G.transpose((0, 2, 1)).conj(), x)
else:
y = ncp.squeeze(ncp.zeros((self.nsl, self.ny, self.nz),
dtype=self.dtype))
if hasattr(self, 'GT'):
for isl in range(self.nsl):
y[isl] = ncp.dot(self.GT[isl], x[isl])
else:
for isl in range(self.nsl):
y[isl] = ncp.dot(self.G[isl].conj().T, x[isl])
return y.ravel()
def _matvec(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.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 = ncp.reshape(x, self.dims)
if self.dir > 0: # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.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 = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y