Source code for gemseo_umdo.statistics.multilevel.mlmc.pilots.variance

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
# Lesser General Public License for more details.
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"""The variance-based pilot for the MLMC algorithm."""
from __future__ import annotations

from typing import Any
from typing import Iterable
from typing import Sequence

from numpy import array
from numpy import nanmean
from numpy import nansum
from numpy import nanvar
from numpy.typing import NDArray

from gemseo_umdo.statistics.multilevel.mlmc.pilots.pilot import MLMCPilot


[docs]class Variance(MLMCPilot): """The variance-based pilot for the MLMC algorithm.""" __delta: list[NDArray[float]] r"""The samples of $Y_0-Y_{-1},Y_2-Y_1,\ldots,Y_L-Y_{L-1}$. Namely, $(Y_\ell^{(\ell,n_\ell)}-Y_{\ell-1}^{(\ell,n_\ell)})_{0\leq \ell \leq L}$. """ __sigma: list[NDArray[float]] r"""The samples of $Y_0+Y_{-1},Y_2+Y_1,\ldots,Y_L+Y_{L-1}$. Namely, $(Y_\ell^{(\ell,n_\ell)}+Y_{\ell-1}^{(\ell,n_\ell)})_{0\leq \ell \leq L}$. """ def __init__( # noqa: D107 self, sampling_ratios: NDArray[float], costs: NDArray[float] ) -> None: super().__init__(sampling_ratios, costs) n_levels = len(sampling_ratios) self.__delta = [array([]) for _ in range(n_levels)] self.__sigma = [array([]) for _ in range(n_levels)] def _compute_statistic(self) -> float: # noqa: D102 # El Amri et al., Eq. 29, Multilevel Surrogate-based Control Variates, 2023. # Var_MLMC[Y] = sum_l Var^l[Y_l] - Var^l[Y_{l-1}] # = sum_l Var^l[(D_l+S_l)/2] - Var^l[(S_l+D_l)/2] # where D_l = Y_l-Y_{l-1} and S_l = Y_l+Y_{l-1} return nansum( [ nanvar((delta + sigma) / 2) - nanvar((sigma - delta) / 2) for delta, sigma in zip(self.__delta, self.__sigma) ] ) def _compute_V_l( # noqa: D102 N802 self, levels: Iterable[int], samples: Sequence[NDArray[float]], *pilot_parameters: Any, ) -> NDArray[float]: # Mycek and De Lozzo, Table 1, # Multilevel Monte Carlo covariance estimation # for the computation of Sobol' indices, 2019. # V_l = (M4[D_l]M4[S_l])**0.5 for level in levels: _samples = samples[level] self.__delta[level] = _samples[:, 0] - _samples[:, 1] self.__sigma[level] = _samples.sum(1) return array( [ ( nanmean((delta - delta.mean()) ** 4) * nanmean((sigma - sigma.mean()) ** 4) ) ** 0.5 for delta, sigma in zip(self.__delta, self.__sigma) ] )