def test_Sliding3D(par):
"""Dot-test and inverse for Sliding3D operator"""
Op = MatrixMult(
np.ones((par["nwiny"] * par["nwinx"] * par["nt"],
par["ny"] * par["nx"] * par["nt"])))
Slid = Sliding3D(
Op,
dims=(par["ny"] * par["winsy"], par["nx"] * par["winsx"], par["nt"]),
dimsd=(par["npy"], par["npx"], par["nt"]),
nwin=(par["nwiny"], par["nwinx"]),
nover=(par["novery"], par["noverx"]),
nop=(par["ny"], par["nx"]),
tapertype=par["tapertype"],
)
assert dottest(
Slid,
par["npy"] * par["npx"] * par["nt"],
par["ny"] * par["nx"] * par["nt"] * par["winsy"] * par["winsx"],
)
x = np.ones(
(par["ny"] * par["nx"] * par["winsy"] * par["winsx"], par["nt"]))
y = Slid * x.ravel()
xinv = LinearOperator(Slid) / y
assert_array_almost_equal(x.ravel(), xinv)
def FirstDirectionalDerivative(dims, v, sampling=1, edge=False,
dtype='float64'):
r"""First Directional derivative.
Apply directional derivative operator to a multi-dimensional
array (at least 2 dimensions are required) along either a single common
direction or different directions for each point of the array.
Parameters
----------
dims : :obj:`tuple`
Number of samples for each dimension.
v : :obj:`np.ndarray`, optional
Single direction (array of size :math:`n_{dims}`) or group of directions
(array of size :math:`[n_{dims} \times n_{d0} \times ... \times n_{n_{dims}}`)
sampling : :obj:`tuple`, optional
Sampling steps for each direction.
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.
Returns
-------
ddop : :obj:`pylops.LinearOperator`
First directional derivative linear operator
Notes
-----
The FirstDirectionalDerivative applies a first-order derivative
to a multi-dimensional array along the direction defined by the unitary
vector \mathbf{v}:
.. math::
df_\mathbf{v} =
\nabla f \mathbf{v}
or along the directions defined by the unitary vectors
:math:`\mathbf{v}(x, y)`:
.. math::
df_\mathbf{v}(x,y) =
\nabla f(x,y) \mathbf{v}(x,y)
where we have here considered the 2-dimensional case.
This operator can be easily implemented as the concatenation of the
:py:class:`pylops.Gradient` operator and the :py:class:`pylops.Diagonal`
operator with :math:\mathbf{v} along the main diagonal.
"""
Gop = Gradient(dims, sampling=sampling, edge=edge, dtype=dtype)
if v.ndim == 1:
Dop = Diagonal(v, dims=[len(dims)]+list(dims), dir=0, dtype=dtype)
else:
Dop = Diagonal(v.ravel(), dtype=dtype)
Sop = Sum(dims=[len(dims)]+list(dims), dir=0, dtype=dtype)
ddop = Sop * Dop * Gop
return LinearOperator(ddop)
def SecondDirectionalDerivative(dims,
v,
sampling=1,
edge=False,
dtype="float64"):
r"""Second Directional derivative.
Apply a second directional derivative operator to a multi-dimensional array
along either a single common direction or different directions for each
point of the array.
.. note:: At least 2 dimensions are required, consider using
:py:func:`pylops.SecondDerivative` for 1d arrays.
Parameters
----------
dims : :obj:`tuple`
Number of samples for each dimension.
v : :obj:`np.ndarray`, optional
Single direction (array of size :math:`n_\text{dims}`) or group of directions
(array of size :math:`[n_\text{dims} \times n_{d_0} \times ... \times n_{d_{n_\text{dims}}}]`)
sampling : :obj:`tuple`, optional
Sampling steps for each direction.
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.
Returns
-------
ddop : :obj:`pylops.LinearOperator`
Second directional derivative linear operator
Notes
-----
The SecondDirectionalDerivative applies a second-order derivative
to a multi-dimensional array along the direction defined by the unitary
vector :math:`\mathbf{v}`:
.. math::
d^2f_\mathbf{v} =
- D_\mathbf{v}^T [D_\mathbf{v} f]
where :math:`D_\mathbf{v}` is the first-order directional derivative
implemented by :func:`pylops.SecondDirectionalDerivative`.
This operator is sometimes also referred to as directional Laplacian
in the literature.
"""
Dop = FirstDirectionalDerivative(dims,
v,
sampling=sampling,
edge=edge,
dtype=dtype)
ddop = -Dop.H * Dop
return LinearOperator(ddop)
def test_Sliding1D(par):
"""Dot-test and inverse for Sliding1D operator
"""
Op = MatrixMult(np.ones((par['nwiny'], par['ny'])))
Slid = Sliding1D(Op,
dim=par['ny'] * par['winsy'],
dimd=par['npy'],
nwin=par['nwiny'],
nover=par['novery'],
tapertype=par['tapertype'])
assert dottest(Slid, par['npy'], par['ny'] * par['winsy'])
x = np.ones(par['ny'] * par['winsy'])
y = Slid * x.flatten()
xinv = LinearOperator(Slid) / y
assert_array_almost_equal(x.flatten(), xinv)
def test_Patch2D(par):
"""Dot-test and inverse for Patch2D operator
"""
Op = MatrixMult(
np.ones((par['nwiny'] * par['nwint'], par['ny'] * par['nt'])))
Pop = Patch2D(Op,
dims=(par['ny'] * par['winsy'], par['nt'] * par['winst']),
dimsd=(par['npy'], par['npt']),
nwin=(par['nwiny'], par['nwint']),
nover=(par['novery'], par['novert']),
nop=(par['ny'], par['nt']),
tapertype=par['tapertype'])
assert dottest(Pop, par['npy'] * par['nt'],
par['ny'] * par['nt'] * par['winsy'] * par['winst'])
x = np.ones((par['ny'] * par['winsy'], par['nt'] * par['winst']))
y = Pop * x.flatten()
xinv = LinearOperator(Pop) / y
assert_array_almost_equal(x.flatten(), xinv)
def test_Patch2D(par):
"""Dot-test and inverse for Patch2D operator"""
Op = MatrixMult(
np.ones((par["nwiny"] * par["nwint"], par["ny"] * par["nt"])))
Pop = Patch2D(
Op,
dims=(par["ny"] * par["winsy"], par["nt"] * par["winst"]),
dimsd=(par["npy"], par["npt"]),
nwin=(par["nwiny"], par["nwint"]),
nover=(par["novery"], par["novert"]),
nop=(par["ny"], par["nt"]),
tapertype=par["tapertype"],
)
assert dottest(Pop, par["npy"] * par["nt"],
par["ny"] * par["nt"] * par["winsy"] * par["winst"])
x = np.ones((par["ny"] * par["winsy"], par["nt"] * par["winst"]))
y = Pop * x.ravel()
xinv = LinearOperator(Pop) / y
assert_array_almost_equal(x.ravel(), xinv)
def FISTA(Op,
data,
niter,
eps=0.1,
alpha=None,
eigsiter=None,
eigstol=0,
tol=1e-10,
returninfo=False,
show=False,
threshkind='soft',
perc=None,
callback=None):
r"""Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
Solve an optimization problem with :math:`L1` regularization function given
the operator ``Op`` and data ``y``. The operator can be real or complex,
and should ideally be either square :math:`N=M` or underdetermined
:math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, the maximum eigenvalue is
estimated and the optimal step size is chosen. If provided, the
condition will not be checked internally).
eigsiter : :obj:`int`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
returninfo : :obj:`bool`, optional
Return info of FISTA solver
show : :obj:`bool`, optional
Display iterations log
threshkind : :obj:`str`, optional
Kind of thresholding ('hard', 'soft', 'half', 'soft-percentile', or
'half-percentile' - 'soft' used as default)
perc : :obj:`float`, optional
Percentile, as percentage of values to be kept by thresholding (to be
provided when thresholding is soft-percentile or half-percentile)
callback : :obj:`callable`, optional
Function with signature (``callback(x)``) to call after each iteration
where ``x`` is the current model vector
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
Raises
------
NotImplementedError
If ``threshkind`` is different from hard, soft, half, soft-percentile,
or half-percentile
ValueError
If ``perc=None`` when ``threshkind`` is soft-percentile or
half-percentile
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
ISTA: Iterative Shrinkage-Thresholding Algorithm (ISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_p
using the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) [1]_,
where :math:`p=0, 1, 1/2`. This is a modified version of ISTA solver with
improved convergence properties and limited additional computational cost.
Similarly to the ISTA solver, the choice of the thresholding algorithm to
apply at every iteration is based on the choice of :math:`p`.
.. [1] Beck, A., and Teboulle, M., “A Fast Iterative Shrinkage-Thresholding
Algorithm for Linear Inverse Problems”, SIAM Journal on
Imaging Sciences, vol. 2, pp. 183-202. 2009.
"""
if not threshkind in [
'hard', 'soft', 'half', 'hard-percentile', 'soft-percentile',
'half-percentile'
]:
raise NotImplementedError('threshkind should be hard, soft, half,'
'hard-percentile, soft-percentile, '
'or half-percentile')
if threshkind in ['hard-percentile', 'soft-percentile', 'half-percentile'
] and perc is None:
raise ValueError('Provide a percentile when choosing hard-percentile,'
'soft-percentile, or half-percentile thresholding')
# choose thresholding function
if threshkind == 'soft':
threshf = _softthreshold
elif threshkind == 'hard':
threshf = _hardthreshold
elif threshkind == 'half':
threshf = _halfthreshold
elif threshkind == 'hard-percentile':
threshf = _hardthreshold_percentile
elif threshkind == 'soft-percentile':
threshf = _softthreshold_percentile
else:
threshf = _halfthreshold_percentile
if show:
tstart = time.time()
print('FISTA optimization (%s thresholding)\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' %
(threshkind, Op.shape[0], Op.shape[1], eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
maxeig = np.abs(
Op1.eigs(neigs=1,
symmetric=True,
niter=eigsiter,
**dict(tol=eigstol, which='LM')))[0]
alpha = 1. / maxeig
# define threshold
thresh = eps * alpha * 0.5
if show:
if perc is None:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
else:
print('alpha = %10e\tperc = %.1f' % (alpha, perc))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
zinv = xinv.copy()
t = 1
if returninfo:
cost = np.zeros(niter + 1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
resz = data - Op.matvec(zinv)
# compute gradient
grad = alpha * Op.rmatvec(resz)
# update inverted model
xinv_unthesh = zinv + grad
if perc is None:
xinv = threshf(xinv_unthesh, thresh)
else:
xinv = threshf(xinv_unthesh, 100 - perc)
# update auxiliary coefficients
told = t
t = (1. + np.sqrt(1. + 4. * t**2)) / 2.
zinv = xinv + ((told - 1.) / t) * (xinv - xinvold)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5 * np.linalg.norm(data - Op.matvec(xinv))**2
costreg = eps * np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
# run callback
if callback is not None:
callback(xinv)
if show:
if iiter < 10 or niter - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter+1, xinv[0], costdata, costdata+costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
# xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f' %
(niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter
def ISTA(Op,
data,
niter,
eps=0.1,
alpha=None,
eigsiter=None,
eigstol=0,
tol=1e-10,
monitorres=False,
returninfo=False,
show=False,
threshkind='soft',
perc=None,
callback=None):
r"""Iterative Shrinkage-Thresholding Algorithm (ISTA).
Solve an optimization problem with :math:`Lp, \quad p=0, 1/2, 1`
regularization, given the operator ``Op`` and data ``y``. The operator
can be real or complex, and should ideally be either square :math:`N=M`
or underdetermined :math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, the maximum eigenvalue is
estimated and the optimal step size is chosen. If provided, the
condition will not be checked internally).
eigsiter : :obj:`float`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
monitorres : :obj:`bool`, optional
Monitor that residual is decreasing
returninfo : :obj:`bool`, optional
Return info of CG solver
show : :obj:`bool`, optional
Display iterations log
threshkind : :obj:`str`, optional
Kind of thresholding ('hard', 'soft', 'half', 'hard-percentile',
'soft-percentile', or 'half-percentile' - 'soft' used as default)
perc : :obj:`float`, optional
Percentile, as percentage of values to be kept by thresholding (to be
provided when thresholding is soft-percentile or half-percentile)
callback : :obj:`callable`, optional
Function with signature (``callback(x)``) to call after each iteration
where ``x`` is the current model vector
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
Raises
------
NotImplementedError
If ``threshkind`` is different from hard, soft, half, soft-percentile,
or half-percentile
ValueError
If ``perc=None`` when ``threshkind`` is soft-percentile or
half-percentile
ValueError
If ``monitorres=True`` and residual increases
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
FISTA: Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_p
using the Iterative Shrinkage-Thresholding Algorithms (ISTA) [1]_, where
:math:`p=0, 1, 1/2`. This is a very simple iterative algorithm which
applies the following step:
.. math::
\mathbf{x}^{(i+1)} = T_{(\epsilon \alpha /2, p)} (\mathbf{x}^{(i)} +
\alpha \mathbf{Op}^H (\mathbf{d} - \mathbf{Op} \mathbf{x}^{(i)}))
where :math:`\epsilon \alpha /2` is the threshold and :math:`T_{(\tau, p)}`
is the thresholding rule. The most common variant of ISTA uses the
so-called soft-thresholding rule :math:`T(\tau, p=1)`. Alternatively an
hard-thresholding rule is used in the case of `p=0` or a half-thresholding
rule is used in the case of `p=1/2`. Finally, percentile bases thresholds
are also implemented: the damping factor is not used anymore an the
threshold changes at every iteration based on the computed percentile.
.. [1] Daubechies, I., Defrise, M., and De Mol, C., “An iterative
thresholding algorithm for linear inverse problems with a sparsity
constraint”, Communications on pure and applied mathematics, vol. 57,
pp. 1413-1457. 2004.
"""
if not threshkind in [
'hard', 'soft', 'half', 'hard-percentile', 'soft-percentile',
'half-percentile'
]:
raise NotImplementedError('threshkind should be hard, soft, half,'
'hard-percentile, soft-percentile, '
'or half-percentile')
if threshkind in ['hard-percentile', 'soft-percentile', 'half-percentile'
] and perc is None:
raise ValueError('Provide a percentile when choosing hard-percentile,'
'soft-percentile, or half-percentile thresholding')
# choose thresholding function
if threshkind == 'soft':
threshf = _softthreshold
elif threshkind == 'hard':
threshf = _hardthreshold
elif threshkind == 'half':
threshf = _halfthreshold
elif threshkind == 'hard-percentile':
threshf = _hardthreshold_percentile
elif threshkind == 'soft-percentile':
threshf = _softthreshold_percentile
else:
threshf = _halfthreshold_percentile
if show:
tstart = time.time()
print('ISTA optimization (%s thresholding)\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' %
(threshkind, Op.shape[0], Op.shape[1], eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
maxeig = np.abs(
Op1.eigs(neigs=1,
symmetric=True,
niter=eigsiter,
**dict(tol=eigstol, which='LM')))[0]
alpha = 1. / maxeig
# define threshold
thresh = eps * alpha * 0.5
if show:
if perc is None:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
else:
print('alpha = %10e\tperc = %.1f' % (alpha, perc))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
if monitorres:
normresold = np.inf
if returninfo:
cost = np.zeros(niter + 1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
res = data - Op.matvec(xinv)
if monitorres:
normres = np.linalg.norm(res)
if normres > normresold:
raise ValueError('ISTA stopped at iteration %d due to '
'residual increasing, consider modifying '
'eps and/or alpha...' % iiter)
else:
normresold = normres
# compute gradient
grad = alpha * Op.rmatvec(res)
# update inverted model
xinv_unthesh = xinv + grad
if perc is None:
xinv = threshf(xinv_unthesh, thresh)
else:
xinv = threshf(xinv_unthesh, 100 - perc)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5 * np.linalg.norm(res)**2
costreg = eps * np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
# run callback
if callback is not None:
callback(xinv)
if show:
if iiter < 10 or niter - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter+1, xinv[0], costdata, costdata+costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
logging.warning('update smaller that tolerance for '
'iteration %d' % iiter)
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
# xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f' %
(niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter
def OMP(Op,
data,
niter_outer=10,
niter_inner=40,
sigma=1e-4,
normalizecols=False,
show=False):
r"""Orthogonal Matching Pursuit (OMP).
Solve an optimization problem with :math:`L0` regularization function given
the operator ``Op`` and data ``y``. The operator can be real or complex,
and should ideally be either square :math:`N=M` or underdetermined
:math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter_outer : :obj:`int`
Number of iterations of outer loop
niter_inner : :obj:`int`
Number of iterations of inner loop
sigma : :obj:`list`
Maximum L2 norm of residual. When smaller stop iterations.
normalizecols : :obj:`list`
Normalize columns (``True``) or not (``False``). Note that this can be
expensive as it requires applying the forward operator
:math:`n_{cols}` times to unit vectors (i.e., containing 1 at
position j and zero otherwise); use only when the columns of the
operator are expected to have highly varying norms.
show : :obj:`bool`, optional
Display iterations log
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
iiter : :obj:`int`
Number of effective outer iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
See Also
--------
ISTA: Iterative Shrinkage-Thresholding Algorithm (ISTA).
FISTA: Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
||\mathbf{x}||_0 \quad subj. to \quad
||\mathbf{Op}\mathbf{x}-\mathbf{b}||_2 <= \sigma,
using Orthogonal Matching Pursuit (OMP). This is a very
simple iterative algorithm which applies the following step:
.. math::
\Lambda_k = \Lambda_{k-1} \cup \{ arg max_j
|\mathbf{Op}_j^H \mathbf{r}_k| \} \\
\mathbf{x}_k = \{ arg min_{\mathbf{x}}
||\mathbf{Op}_{\Lambda_k} \mathbf{x} - \mathbf{b}||_2
"""
Op = LinearOperator(Op)
if show:
tstart = time.time()
print(
'OMP optimization\n'
'-----------------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'sigma = %.2e\tniter_outer = %d\tniter_inner = %d\n'
'normalization=%s' % (Op.shape[0], Op.shape[1], sigma, niter_outer,
niter_inner, normalizecols))
# find normalization factor for each column
if normalizecols:
ncols = Op.shape[1]
norms = np.zeros(ncols)
for icol in range(ncols):
unit = np.zeros(ncols)
unit[icol] = 1
norms[icol] = np.linalg.norm(Op.matvec(unit))
if show:
print(
'-----------------------------------------------------------------'
)
head1 = ' Itn r2norm'
print(head1)
cols = []
res = data.copy()
cost = np.zeros(niter_outer + 1)
cost[0] = np.linalg.norm(data)
iiter = 0
while iiter < niter_outer and cost[iiter] > sigma:
cres = np.abs(Op.rmatvec(res))
if normalizecols:
cres = cres / norms
# exclude columns already chosen by putting them negative
if iiter > 0:
cres[cols] = -1
# choose column with max cres
imax = np.argwhere(cres == np.max(cres)).ravel()
nimax = len(imax)
if nimax > 0:
imax = imax[np.random.permutation(nimax)[0]]
else:
imax = imax[0]
cols.append(imax)
# estimate model for current set of columns
Opcol = Op.apply_columns(cols)
x = lsqr(Opcol, data, iter_lim=niter_inner)[0]
res = data - Opcol.matvec(x)
iiter += 1
cost[iiter] = np.linalg.norm(res)
if show:
if iiter < 10 or niter_outer - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e' % (iiter + 1, cost[iiter])
print(msg)
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
xinv[cols] = x
if show:
print('\nIterations = %d Total time (s) = %.2f' %
(iiter, time.time() - tstart))
print(
'-----------------------------------------------------------------\n'
)
return xinv, iiter, cost
def PhaseShift(vel, dz, nt, freq, kx, ky=None, dtype="float64"):
r"""Phase shift operator
Apply positive (forward) phase shift with constant velocity in
forward mode, and negative (backward) phase shift with constant velocity in
adjoint mode. Input model and data should be 2- or 3-dimensional arrays
in time-space domain of size :math:`[n_t \times n_x \;(\times n_y)]`.
Parameters
----------
vel : :obj:`float`, optional
Constant propagation velocity
dz : :obj:`float`, optional
Depth step
nt : :obj:`int`, optional
Number of time samples of model and data
freq : :obj:`numpy.ndarray`
Positive frequency axis
kx : :obj:`int`, optional
Horizontal wavenumber axis (centered around 0) of size
:math:`[n_x \times 1]`.
ky : :obj:`int`, optional
Second horizontal wavenumber axis for 3d phase shift
(centered around 0) of size :math:`[n_y \times 1]`.
dtype : :obj:`str`, optional
Type of elements in input array
Returns
-------
Pop : :obj:`pylops.LinearOperator`
Phase shift operator
Notes
-----
The phase shift operator implements a one-way wave equation forward
propagation in frequency-wavenumber domain by applying the following
transformation to the input model:
.. math::
d(f, k_x, k_y) = m(f, k_x, k_y)
e^{-j \Delta z \sqrt{\omega^2/v^2 - k_x^2 - k_y^2}}
where :math:`v` is the constant propagation velocity and
:math:`\Delta z` is the propagation depth. In adjoint mode, the data is
propagated backward using the following transformation:
.. math::
m(f, k_x, k_y) = d(f, k_x, k_y)
e^{j \Delta z \sqrt{\omega^2/v^2 - k_x^2 - k_y^2}}
Effectively, the input model and data are assumed to be in time-space
domain and forward Fourier transform is applied to both dimensions, leading
to the following operator:
.. math::
\mathbf{d} = \mathbf{F}^H_t \mathbf{F}^H_x \mathbf{P}
\mathbf{F}_x \mathbf{F}_t \mathbf{m}
where :math:`\mathbf{P}` perfoms the phase-shift as discussed above.
"""
dtypefft = (np.ones(1, dtype=dtype) + 1j * np.ones(1, dtype=dtype)).dtype
if ky is None:
dims = (nt, kx.size)
dimsfft = (freq.size, kx.size)
else:
dims = (nt, kx.size, ky.size)
dimsfft = (freq.size, kx.size, ky.size)
Fop = FFT(dims, dir=0, nfft=nt, real=True, dtype=dtype)
Kxop = FFT(
dimsfft, dir=1, nfft=kx.size, real=False, fftshift_after=True, dtype=dtypefft
)
if ky is not None:
Kyop = FFT(
dimsfft,
dir=2,
nfft=ky.size,
real=False,
fftshift_after=True,
dtype=dtypefft,
)
Pop = _PhaseShift(vel, dz, freq, kx, ky, dtypefft)
if ky is None:
Pop = Fop.H * Kxop * Pop * Kxop.H * Fop
else:
Pop = Fop.H * Kxop * Kyop * Pop * Kyop.H * Kxop.H * Fop
# Recasting of type is required to avoid FFT operators to cast to complex.
# We know this is correct because forward and inverse FFTs are applied at
# the beginning and end of this combined operator
Pop.dtype = dtype
return LinearOperator(Pop)
def Interp(M, iava, dims=None, dir=0, kind='linear', dtype='float64'):
r"""Interpolation operator.
Apply interpolation along direction ``dir``
from regularly sampled input vector into fractionary positions ``iava``
using one of the following algorithms:
- *Nearest neighbour* interpolation
is a thin wrapper around :obj:`pylops.Restriction` at ``np.round(iava)``
locations.
- *Linear interpolation* extracts values from input vector
at locations ``np.floor(iava)`` and ``np.floor(iava)+1`` and linearly
combines them in forward mode, places weighted versions of the
interpolated values at locations ``np.floor(iava)`` and
``np.floor(iava)+1`` in an otherwise zero vector in adjoint mode.
- *Sinc interpolation* performs sinc interpolation at locations ``iava``.
Note that this is the most accurate method but it has higher computational
cost as it involves multiplying the input data by a matrix of size
:math:`N \times M`.
.. note:: The vector ``iava`` should contain unique values. If the same
index is repeated twice an error will be raised. This also applies when
values beyond the last element of the input array for
*linear interpolation* as those values are forced to be just before this
element.
Parameters
----------
M : :obj:`int`
Number of samples in model.
iava : :obj:`list` or :obj:`numpy.ndarray`
Floating indices of locations of available samples for interpolation.
dims : :obj:`list`, optional
Number of samples for each dimension
(``None`` if only one dimension is available)
dir : :obj:`int`, optional
Direction along which restriction is applied.
kind : :obj:`str`, optional
Kind of interpolation (``nearest``, ``linear``, and ``sinc`` are
currently supported)
dtype : :obj:`str`, optional
Type of elements in input array.
Returns
-------
op : :obj:`pylops.LinearOperator`
Linear intepolation operator
iava : :obj:`list` or :obj:`numpy.ndarray`
Corrected indices of locations of available samples
(samples at ``M-1`` or beyond are forced to be at ``M-1-eps``)
Raises
------
ValueError
If the vector ``iava`` contains repeated values.
NotImplementedError
If ``kind`` is not ``nearest``, ``linear`` or ``sinc``
See Also
--------
pylops.Restriction : Restriction operator
Notes
-----
*Linear interpolation* of a subset of :math:`N` values at locations
``iava`` from an input (or model) vector :math:`\mathbf{x}` of size
:math:`M` can be expressed as:
.. math::
y_i = (1-w_i) x_{l^{l}_i} + w_i x_{l^{r}_i}
\quad \forall i=1,2,...,N
where :math:`\mathbf{l^l}=[\lfloor l_1 \rfloor, \lfloor l_2 \rfloor,...,
\lfloor l_N \rfloor]` and :math:`\mathbf{l^r}=[\lfloor l_1 \rfloor +1,
\lfloor l_2 \rfloor +1,...,
\lfloor l_N \rfloor +1]` are vectors containing the indeces
of the original array at which samples are taken, and
:math:`\mathbf{w}=[l_1 - \lfloor l_1 \rfloor, l_2 - \lfloor l_2 \rfloor,
..., l_N - \lfloor l_N \rfloor]` are the linear interpolation weights.
This operator can be implemented by simply summing two
:class:`pylops.Restriction` operators which are weighted
using :class:`pylops.basicoperators.Diagonal` operators.
*Sinc interpolation* of a subset of :math:`N` values at locations
``iava`` from an input (or model) vector :math:`\mathbf{x}` of size
:math:`M` can be expressed as:
.. math::
y_i = \sum_{j=0}^{M} x_j sinc(i-j) \quad \forall i=1,2,...,N
This operator can be implemented using the :class:`pylops.MatrixMult`
operator with a matrix containing the values of the sinc function at all
:math:`i,j` possible combinations.
"""
if kind == 'nearest':
interpop, iava = _nearestinterp(M, iava, dims=dims, dir=dir, dtype=dtype)
elif kind == 'linear':
interpop, iava = _linearinterp(M, iava, dims=dims, dir=dir, dtype=dtype)
elif kind == 'sinc':
interpop = _sincinterp(M, iava, dims=dims, dir=dir, dtype=dtype)
else:
raise NotImplementedError('kind is not correct...')
return LinearOperator(interpop), iava
def OMP(Op, data, niter_outer=10, niter_inner=40, sigma=1e-4,
normalizecols=False, show=False):
r"""Orthogonal Matching Pursuit (OMP).
Solve an optimization problem with :math:`L0` regularization function given
the operator ``Op`` and data ``y``. The operator can be real or complex,
and should ideally be either square :math:`N=M` or underdetermined
:math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter_outer : :obj:`int`, optional
Number of iterations of outer loop
niter_inner : :obj:`int`, optional
Number of iterations of inner loop. By choosing ``niter_inner=0``, the
Matching Pursuit (MP) algorithm is implemented.
sigma : :obj:`list`
Maximum L2 norm of residual. When smaller stop iterations.
normalizecols : :obj:`list`, optional
Normalize columns (``True``) or not (``False``). Note that this can be
expensive as it requires applying the forward operator
:math:`n_{cols}` times to unit vectors (i.e., containing 1 at
position j and zero otherwise); use only when the columns of the
operator are expected to have highly varying norms.
show : :obj:`bool`, optional
Display iterations log
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
iiter : :obj:`int`
Number of effective outer iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
See Also
--------
ISTA: Iterative Shrinkage-Thresholding Algorithm (ISTA).
FISTA: Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
||\mathbf{x}||_0 \quad subj. to \quad
||\mathbf{Op}\mathbf{x}-\mathbf{b}||_2^2 <= \sigma,
using Orthogonal Matching Pursuit (OMP). This is a very
simple iterative algorithm which applies the following step:
.. math::
\Lambda_k = \Lambda_{k-1} \cup \{ arg max_j
|\mathbf{Op}_j^H \mathbf{r}_k| \} \\
\mathbf{x}_k = \{ arg min_{\mathbf{x}}
||\mathbf{Op}_{\Lambda_k} \mathbf{x} - \mathbf{b}||_2^2
Note that by choosing ``niter_inner=0`` the basic Matching Pursuit (MP)
algorithm is implemented instead. In other words, instead of solving an
optimization at each iteration to find the best :math:`\mathbf{x}` for the
currently selected basis functions, the vector :math:`\mathbf{x}` is just
updated at the new basis function by taking directly the value from
the inner product :math:`\mathbf{Op}_j^H \mathbf{r}_k`.
In this case it is highly reccomended to provide a normalized basis
function. If different basis have different norms, the solver is likely
to diverge. Similar observations apply to OMP, even though mild unbalancing
between the basis is generally properly handled.
"""
ncp = get_array_module(data)
Op = LinearOperator(Op)
if show:
tstart = time.time()
algname = 'OMP optimization\n' if niter_inner > 0 else 'MP optimization\n'
print(algname +
'-----------------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'sigma = %.2e\tniter_outer = %d\tniter_inner = %d\n'
'normalization=%s' %
(Op.shape[0], Op.shape[1], sigma, niter_outer,
niter_inner, normalizecols))
# find normalization factor for each column
if normalizecols:
ncols = Op.shape[1]
norms = ncp.zeros(ncols)
for icol in range(ncols):
unit = ncp.zeros(ncols, dtype=Op.dtype)
unit[icol] = 1
norms[icol] = np.linalg.norm(Op.matvec(unit))
if show:
print('-----------------------------------------------------------------')
head1 = ' Itn r2norm'
print(head1)
if niter_inner == 0:
x = []
cols = []
res = data.copy()
cost = ncp.zeros(niter_outer + 1)
cost[0] = np.linalg.norm(data)
iiter = 0
while iiter < niter_outer and cost[iiter] > sigma:
# compute inner products
cres = Op.rmatvec(res)
cres_abs = np.abs(cres)
if normalizecols:
cres_abs = cres_abs / norms
# choose column with max cres
cres_max = np.max(cres_abs)
imax = np.argwhere(cres_abs == cres_max).ravel()
nimax = len(imax)
if nimax > 0:
imax = imax[np.random.permutation(nimax)[0]]
else:
imax = imax[0]
# update active set
if imax not in cols:
addnew = True
cols.append(int(imax))
else:
addnew = False
imax_in_cols = cols.index(imax)
# estimate model for current set of columns
if niter_inner == 0:
# MP update
Opcol = Op.apply_columns([int(imax), ])
res -= Opcol.matvec(cres[imax] * ncp.ones(1))
if addnew:
x.append(cres[imax])
else:
x[imax_in_cols] += cres[imax]
else:
# OMP update
Opcol = Op.apply_columns(cols)
if ncp == np:
x = lsqr(Opcol, data, iter_lim=niter_inner)[0]
else:
x = cgls(Opcol, data, ncp.zeros(int(Opcol.shape[1]),
dtype=Opcol.dtype),
niter=niter_inner)[0]
res = data - Opcol.matvec(x)
iiter += 1
cost[iiter] = np.linalg.norm(res)
if show:
if iiter < 10 or niter_outer - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e' % (iiter + 1, cost[iiter])
print(msg)
xinv = ncp.zeros(int(Op.shape[1]), dtype=Op.dtype)
xinv[cols] = ncp.array(x)
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (iiter, time.time() - tstart))
print(
'-----------------------------------------------------------------\n')
return xinv, iiter, cost
def FISTA(Op,
data,
niter,
eps=0.1,
alpha=None,
eigsiter=None,
eigstol=0,
tol=1e-10,
returninfo=False,
show=False):
r"""Fast Iterative Soft Thresholding Algorithm (FISTA).
Solve an optimization problem with :math:`L1` regularization function given
the operator ``Op`` and data ``y``. The operator can be real or complex,
and should ideally be either square :math:`N=M` or underdetermined
:math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, estimated to satisfy the
condition, otherwise the condition will not be checked)
eigsiter : :obj:`int`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
returninfo : :obj:`bool`, optional
Return info of FISTA solver
show : :obj:`bool`, optional
Display iterations log
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
ISTA: Iterative Soft Thresholding Algorithm (FISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_1
using the Fast Iterative Soft Thresholding Algorithm (FISTA) [1]_. This is
a modified version of ISTA solver with improved convergence properties and
limitied additional computational cost.
.. [1] Beck, A., and Teboulle, M., “A Fast Iterative Shrinkage-Thresholding
Algorithm for Linear Inverse Problems”, SIAM Journal on
Imaging Sciences, vol. 2, pp. 183-202. 2009.
"""
if show:
tstart = time.time()
print('FISTA optimization\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' %
(Op.shape[0], Op.shape[1], eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
maxeig = np.abs(
Op1.eigs(neigs=1,
symmetric=True,
niter=eigsiter,
**dict(tol=eigstol, which='LM')))[0]
alpha = 1. / maxeig
# define threshold
thresh = eps * alpha * 0.5
if show:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
zinv = xinv.copy()
t = 1
if returninfo:
cost = np.zeros(niter + 1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
resz = data - Op.matvec(zinv)
# compute gradient
grad = alpha * Op.rmatvec(resz)
# update inverted model
xinv_unthesh = zinv + grad
xinv = _softthreshold(xinv_unthesh, thresh)
# update auxiliary coefficients
told = t
t = (1. + np.sqrt(1. + 4. * t**2)) / 2.
zinv = xinv + ((told - 1.) / t) * (xinv - xinvold)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5 * np.linalg.norm(data - Op.matvec(xinv))**2
costreg = eps * np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
if show:
if iiter < 10 or niter - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter+1, xinv[0], costdata, costdata+costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
#xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f' %
(niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter
def ISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0,
tol=1e-10, monitorres=False, returninfo=False, show=False):
r"""Iterative Soft Thresholding Algorithm (ISTA).
Solve an optimization problem with :math:`L1` regularization function given
the operator ``Op`` and data ``y``. The operator can be real or complex,
and should ideally be either square :math:`N=M` or underdetermined
:math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, estimated to satisfy the
condition, otherwise the condition will not be checked)
eigsiter : :obj:`float`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
monitorres : :obj:`bool`, optional
Monitor that residual is decreasing
returninfo : :obj:`bool`, optional
Return info of CG solver
show : :obj:`bool`, optional
Display iterations log
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
Raises
------
ValueError
If ``monitorres=True`` and residual increases
See Also
--------
FISTA: Fast Iterative Soft Thresholding Algorithm (FISTA).
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_1
using the Iterative Soft Thresholding Algorithm (ISTA) [1]_. This is a very
simple iterative algorithm which applies the following step:
.. math::
\mathbf{x}^{(i+1)} = soft (\mathbf{x}^{(i)} + \alpha \mathbf{Op}^H
(\mathbf{d} - \mathbf{Op} \mathbf{x}^{(i)})), \epsilon \alpha /2)
where :math:`\epsilon \alpha /2` is the
threshold and :math:`soft()` is the so-called soft-thresholding rule.
.. [1] Beck, A., and Teboulle, M., “A Fast Iterative Shrinkage-Thresholding
Algorithm for Linear Inverse Problems”, SIAM Journal on
Imaging Sciences, vol. 2, pp. 183-202. 2009.
"""
if show:
tstart = time.time()
print('ISTA optimization\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' % (Op.shape[0],
Op.shape[1],
eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
maxeig = np.abs(Op1.eigs(neigs=1, symmetric=True, niter=eigsiter,
**dict(tol=eigstol, which='LM')))[0]
alpha = 1./maxeig
# define threshold
thresh = eps*alpha*0.5
if show:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
if monitorres:
normresold = np.inf
if returninfo:
cost = np.zeros(niter+1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
res = data - Op.matvec(xinv)
if monitorres:
normres = np.linalg.norm(res)
if normres > normresold:
raise ValueError('ISTA stopped at iteration %d due to '
'residual increasing, consider modyfing '
'eps and/or alpha...' % iiter)
else:
normresold = normres
# compute gradient
grad = alpha*Op.rmatvec(res)
# update inverted model
xinv_unthesh = xinv + grad
xinv = _softthreshold(xinv_unthesh, thresh)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5 * np.linalg.norm(res) ** 2
costreg = eps * np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
if show:
if iiter < 10 or niter - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter+1, xinv[0], costdata, costdata+costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
#xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter