pylops_mpi.basicoperators.MPISecondDerivative

class pylops_mpi.basicoperators.MPISecondDerivative(dims, sampling=1.0, kind='centered', edge=False, base_comm=<mpi4py.MPI.Intracomm object>, dtype=<class 'numpy.float64'>)

MPI Second Derivative

Apply a second derivative using a three-point stencil finite-difference approximation with pylops_mpi.DistributedArray. The Second Derivative is calculated along axis=0.

Parameters:

dimsint or tuple

Number of samples for each dimension.

samplingfloat, optional

Sampling step \(\Delta x\).

kindstr, optional

Derivative kind (forward, centered, or backward).

edgebool, optional

Use reduced order derivative at edges (True) or ignore them (False). This is currently only available for centered derivative.

base_commmpi4py.MPI.Comm, optional

MPI Base Communicator. Defaults to mpi4py.MPI.COMM_WORLD.

dtypestr, optional

Type of elements in the input array.

Attributes:

shapetuple

Operator shape

Notes

The MPISecondDerivative operator applies a second derivative to a pylops_mpi.DistributedArray using either a second-order forward, backward or a centered stencil.

Similar to the concept of pylops_mpi.basicoperators.MPIFirstDerivative, ghosting is applied at each rank to include duplicated copies of border cells (ghost cells) from neighboring processes. These ghost cells enable each process to perform local computations independently, avoiding the necessity for direct communication with other processes. As a result, it becomes possible to calculate the Second Derivative of the entire distributed array efficiently.

Now, for a one-dimensional DistributedArray, the second-order forward stencil is:

\[y[i] = (x[i+2] - 2 * x[i+1] + x[i]) / \mathbf{\Delta x}^2\]

while the second-order backward stencil is:

\[y[i] = (x[i] - 2 * x[i-1] + x[i-2]) / \mathbf{\Delta x}^2\]

and the second-order centered stencil is:

\[y[i] = (x[i+1] - 2 * x[i] + x[i-1]) / \mathbf{\Delta x}^2\]

where \(y\) is a DistributedArray and \(x\) is a Ghosted array created by adding border cells of neighboring processes to the local array at each rank.

Methods

__init__(dims[, sampling, kind, edge, ...])
adjoint()	Adjoint MPI LinearOperator
conj()	Complex conjugate operator
dot(x)	Matrix Vector Multiplication
matvec(x)	Matrix-vector multiplication.
rmatvec(x)	Adjoint Matrix-vector multiplication.
transpose()	Transposition of MPI LinearOperator