ConvexOptimization

class boulderopal.noise_reconstruction.ConvexOptimization(power_density_lower_bound, power_density_upper_bound, regularization_hyperparameter)

Configuration for noise reconstruction with the convex optimization (CVX) method.

Parameters

  • power_density_lower_bound (float) – The lower bound for the reconstructed power spectral densities. It must be greater than or equal to 0.
  • power_density_upper_bound (float) – The upper bound for the reconstructed power spectral densities. It must be greater than the power_density_lower_bound.
  • regularization_hyperparameter (float) – The regularization hyperparameter λ\lambda.

Notes

The CVX method finds the estimation of the power spectral density (PSD) matrix S{\mathbf S} by solving the optimization problem:

Sest=argminS(FSI22+λL1S22), {\mathbf S}_{\mathrm{est}} = \mathrm{argmin}_{\textbf S} (\| F'{\mathbf S} - {\mathbf I} \|_2^2 + \lambda \| L_1 {\mathbf S} \|_2^2) ,

where FF^\prime is the matrix of weighted filter functions and 2| \bullet |_2 denotes the Euclidean norm and L1L_1 is the first-order derivative operator defined as

L1=[1111](K1)×K. \begin{align} L_1 = \begin{bmatrix} -1 & 1 & & \\ & \ddots & \ddots & \\ & & -1 & 1 \\ \end{bmatrix}_{(K - 1) \times K} . \end{align}

λ\lambda is a positive regularization hyperparameter which determines the smoothness of Sest{\mathbf S}_{\mathrm{est}}. If you provide uncertainties in measurements, this method calculates the uncertainties in estimation using a Monte Carlo method.

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