pyproximal.optimization.primal.ADMML2¶

pyproximal.optimization.primal.ADMML2(proxg: ProxOperator, Op: LinearOperator, b: ndarray[tuple[Any, ...], dtype[_ScalarT]], A: LinearOperator, x0: ndarray[tuple[Any, ...], dtype[_ScalarT]], tau: float, niter: int = 10, z0: ndarray[tuple[Any, ...], dtype[_ScalarT]] | None = None, gfirst: bool = False, tol: float | None = None, rtol: float | None = None, callback: Callable[[ndarray[tuple[Any, ...], dtype[_ScalarT]]], None] | None = None, callbackz: bool = False, show: bool = False, itershow: tuple[int, int, int] = (10, 10, 10), **kwargs_solver: dict[str, Any]) → tuple[ndarray[tuple[Any, ...], dtype[_ScalarT]], ndarray[tuple[Any, ...], dtype[_ScalarT]]][source]¶

Alternating Direction Method of Multipliers for L2 misfit term

Solves the following minimization problem using Alternating Direction Method of Multipliers:

\[\begin{split}\mathbf{x},\mathbf{z} = \argmin_{\mathbf{x},\mathbf{z}} \frac{1}{2}||\mathbf{Op}\mathbf{x} - \mathbf{b}||_2^2 + g(\mathbf{z}) \\ s.t. \; \mathbf{Ax}=\mathbf{z}\end{split}\]

where \(g(\mathbf{z})\) is any convex function that has a known proximal operator.

Parameters:

proxgpyproximal.ProxOperator: Proximal operator of g function
Oppylops.LinearOperator: Linear operator of data misfit term
bnumpy.ndarray: Data
Apylops.LinearOperator: Linear operator of regularization term
x0numpy.ndarray: Initial vector
taufloat: Positive scalar weight, which should satisfy the following condition to guarantees convergence: \(\tau \in (0, 1/\lambda_{max}(\mathbf{A}^H\mathbf{A})]\).
niterint, optional: Number of iterations of iterative scheme
z0numpy.ndarray: Initial auxiliary vector. If None, initialized to A @ x0.
gfirstbool, optional: Apply Proximal of operator g first (True) or Proximal of operator f first (False)
tolfloat, optional: Tolerance on change of objective function (used as stopping criterion). If tol=None, run until niter is reached
rtolfloat, optional: Relative tolerance on objective function wrt initial value. Stops the solver when the ratio of the current objective function to the initial objective function is below this value. If rtol=None, run until niter is reached or the other tolerance criterion is met
callbackcallable, optional: Function with signature (callback(x)) to call after each iteration where x is the current model vector
showbool, optional: Display iterations log
itershowtuple, optional: Display set log for the first N1 steps, last N2 steps, and every N3 steps in between where N1, N2, N3 are the three element of the list.
**kwargs_solver: Arbitrary keyword arguments for scipy.sparse.linalg.lsqr used to solve the x-update

Returns:

xnumpy.ndarray: Inverted model
znumpy.ndarray: Inverted second model

Raises:

ValueError: If both x0 and z0 are set to None or x0 is set to None