Source code for probnum.linalg.solvers.belief_updates.solution_based._solution_based_proj_rhs_belief_update

"""Belief update in a solution-based inference view where the information is given by
projecting the current residual to a subspace."""
import numpy as np

import probnum  # pylint: disable="unused-import"
from probnum import randvars
from probnum.linalg.solvers.beliefs import LinearSystemBelief
from probnum.typing import FloatLike

from .._linear_solver_belief_update import LinearSolverBeliefUpdate

class SolutionBasedProjectedRHSBeliefUpdate(LinearSolverBeliefUpdate):
    r"""Gaussian belief update in a solution-based inference framework assuming projected right-hand-side information.

    Updates the belief over the quantities of interest of a linear system :math:`Ax=b` given a Gaussian belief over the solution :math:`x` and information of the form :math:`y = s\^top b=s^\top Ax`. The belief update computes the posterior belief about the solution, given by :math:`p(x \mid y) = \mathcal{N}(x; x_{i+1}, \Sigma_{i+1})`, [1]_ such that

    .. math ::
            x_{i+1} &= x_i + \Sigma_i A^\top s (s^\top A \Sigma_i A^\top s + \lambda)^\dagger s^\top (b - Ax_i),\\
            \Sigma_{i+1} &= \Sigma_i - \Sigma_i A^\top s (s^\top A \Sigma_i A s + \lambda)^\dagger s^\top A \Sigma_i,

    where :math:`\lambda` is the noise variance.

    noise_var :
        Variance of the scalar observation noise.

    .. [1] Cockayne, J. et al., A Bayesian Conjugate Gradient Method, *Bayesian
       Analysis*, 2019, 14, 937-1012

    def __init__(self, noise_var: FloatLike = 0.0) -> None:
        if noise_var < 0.0:
            raise ValueError(f"Noise variance {noise_var} must be non-negative.")
        self._noise_var = noise_var

[docs] def __call__( self, solver_state: "probnum.linalg.solvers.LinearSolverState" ) -> LinearSystemBelief: # Compute projected residual action_A = solver_state.action @ solver_state.problem.A pred = action_A @ solver_state.belief.x.mean proj_resid = solver_state.observation - pred # Compute gain and covariance update cov_xy = solver_state.belief.x.cov @ action_A.T gram = action_A @ cov_xy + self._noise_var gram_pinv = 1.0 / gram if gram > 0.0 else 0.0 gain = cov_xy * gram_pinv cov_update = np.outer(gain, cov_xy) x = randvars.Normal( mean=solver_state.belief.x.mean + gain * proj_resid, cov=solver_state.belief.x.cov - cov_update, ) if solver_state.belief.Ainv is None: Ainv = randvars.Constant(cov_update) else: Ainv = solver_state.belief.Ainv + cov_update return LinearSystemBelief( x=x, A=solver_state.belief.A, Ainv=Ainv, b=solver_state.belief.b )