Source code for pylops.basicoperators.kronecker

__all__ = ["Kronecker"]

import numpy as np

from pylops import LinearOperator
from pylops.utils.typing import DTypeLike, NDArray

[docs]class Kronecker(LinearOperator): r"""Kronecker operator. Perform Kronecker product of two operators. Note that the combined operator is never created explicitly, rather the product of this operator with the model vector is performed in forward mode, or the product of the adjoint of this operator and the data vector in adjoint mode. Parameters ---------- Op1 : :obj:`pylops.LinearOperator` First operator Op2 : :obj:`pylops.LinearOperator` Second operator dtype : :obj:`str`, optional Type of elements in input array. name : :obj:`str`, optional .. versionadded:: 2.0.0 Name of operator (to be used by :func:`pylops.utils.describe.describe`) Attributes ---------- shape : :obj:`tuple` Operator shape explicit : :obj:`bool` Operator contains a matrix that can be solved explicitly (``True``) or not (``False``) Notes ----- The Kronecker product (denoted with :math:`\otimes`) is an operation on two operators :math:`\mathbf{Op}_1` and :math:`\mathbf{Op}_2` of sizes :math:`\lbrack n_1 \times m_1 \rbrack` and :math:`\lbrack n_2 \times m_2 \rbrack` respectively, resulting in a block matrix of size :math:`\lbrack n_1 n_2 \times m_1 m_2 \rbrack`. .. math:: \mathbf{Op}_1 \otimes \mathbf{Op}_2 = \begin{bmatrix} \text{Op}_1^{1,1} \mathbf{Op}_2 & \ldots & \text{Op}_1^{1,m_1} \mathbf{Op}_2 \\ \vdots & \ddots & \vdots \\ \text{Op}_1^{n_1,1} \mathbf{Op}_2 & \ldots & \text{Op}_1^{n_1,m_1} \mathbf{Op}_2 \end{bmatrix} The application of the resulting matrix to a vector :math:`\mathbf{x}` of size :math:`\lbrack m_1 m_2 \times 1 \rbrack` is equivalent to the application of the second operator :math:`\mathbf{Op}_2` to the rows of a matrix of size :math:`\lbrack m_2 \times m_1 \rbrack` obtained by reshaping the input vector :math:`\mathbf{x}`, followed by the application of the first operator to the transposed matrix produced by the first operator. In adjoint mode the same procedure is followed but the adjoint of each operator is used. """ def __init__( self, Op1: LinearOperator, Op2: LinearOperator, dtype: DTypeLike = "float64", name: str = "K", ) -> None: self.Op1 = Op1 self.Op2 = Op2 self.Op1H = self.Op1.H self.Op2H = self.Op2.H shape = ( self.Op1.shape[0] * self.Op2.shape[0], self.Op1.shape[1] * self.Op2.shape[1], ) super().__init__(dtype=np.dtype(dtype), shape=shape, name=name) def _matvec(self, x: NDArray) -> NDArray: x = x.reshape(self.Op1.shape[1], self.Op2.shape[1]) y = self.Op2.matmat(x.T).T y = self.Op1.matmat(y).ravel() return y def _rmatvec(self, x: NDArray) -> NDArray: x = x.reshape(self.Op1.shape[0], self.Op2.shape[0]) y = self.Op2H.matmat(x.T).T y = self.Op1H.matmat(y).ravel() return y