pylops.Identity¶

class pylops.Identity(N, M=None, inplace=True, forceflat=None, dtype='float64', name='I')[source]¶

Identity operator.

Simply move model to data in forward model and viceversa in adjoint mode if \(M = N\). If \(M > N\) removes last \(M - N\) elements from model in forward and pads with \(0\) in adjoint. If \(N > M\) removes last \(N - M\) elements from data in adjoint and pads with \(0\) in forward.

Note that the identity operator can handle both 1d and nd arrays; in the case of nd arrays, all elements of N must be larger or equal than those of M (or all elements of M must be larger or equal than those of N).

Parameters:

Nint or tuple: Number of samples in data (and model, if M is not provided). If a tuple is provided, this is interpreted as the data (and model) are nd-arrays.
Mint or tuple, optional: Number of samples in model. If a tuple is provided, this is interpreted as the model is an nd-array. Note that when M is a tuple, N must be also a tuple with the same number of elements.
inplacebool, optional: Work inplace (True) or make a new copy (False). By default, data is a reference to the model (in forward) and model is a reference to the data (in adjoint).
forceflatbool, optional: Added in version 2.2.0.

Force an array to be flattened after matvec and rmatvec. Note that this is only required when N and M are tuples (input and output arrays are nd-arrays).
dtypestr, optional: Type of elements in input array.
namestr, optional: Added in version 2.0.0.

Name of operator (to be used by pylops.utils.describe.describe)

Attributes:

dimstuple

Shape of the array after the adjoint, but before flattening.

For example, x_reshaped = (Op.H * y.ravel()).reshape(Op.dims).

dimsdtuple

Shape of the array after the forward, but before flattening.

For example, y_reshaped = (Op * x.ravel()).reshape(Op.dimsd).

modestr

Operation mode: same for \(M = N\), model for \(N > M, and ``data`\) for \(M > N\).

shapetuple

Operator shape.

Raises:

ValueError

If M is a tuple with different number of elements of N
If N M are non-identical tuples and some values are largers in N and some in M

NotImplementedError

If N or M have type different from int or tuple/list

Notes

For \(M = N\), an Identity operator simply moves the model \(\mathbf{x}\) to the data \(\mathbf{y}\) in forward mode and viceversa in adjoint mode:

\[y_i = x_i \quad \forall i=1,2,\ldots,N\]

or in matrix form:

\[\mathbf{y} = \mathbf{I} \mathbf{x} = \mathbf{x}\]

and

\[\mathbf{x} = \mathbf{I} \mathbf{y} = \mathbf{y}\]

For \(M > N\), the Identity operator takes the first \(M\) elements of the model \(\mathbf{x}\) into the data \(\mathbf{y}\) in forward mode

\[y_i = x_i \quad \forall i=1,2,\ldots,N\]

and all the elements of the data \(\mathbf{y}\) into the first \(M\) elements of model in adjoint mode (other elements are O):

\[ \begin{align}\begin{aligned}x_i = y_i \quad \forall i=1,2,\ldots,M\\x_i = 0 \quad \forall i=M+1,\ldots,N\end{aligned}\end{align} \]

Methods

`__init__`(N[, M, inplace, forceflat, dtype, name])
`adjoint`()
`apply_columns`(cols)	Apply subset of columns of operator
`cond`([uselobpcg])	Condition number of linear operator.
`conj`()	Complex conjugate operator
`div`(y[, niter, densesolver])	Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
`dot`(x)	Matrix-matrix or matrix-vector multiplication.
`eigs`([neigs, symmetric, niter, uselobpcg])	Most significant eigenvalues of linear operator.
`matmat`(X)	Matrix-matrix multiplication.
`matvec`(x)	Matrix-vector multiplication.
`reset_count`()	Reset counters
`rmatmat`(X)	Matrix-matrix multiplication.
`rmatvec`(x)	Adjoint matrix-vector multiplication.
`todense`([backend])	Return dense matrix.
`toimag`([forw, adj])	Imag operator
`toreal`([forw, adj])	Real operator
`tosparse`()	Return sparse matrix.
`trace`([neval, method, backend])	Trace of linear operator.
`transpose`()