The procedure

Our method consists in a set of rules that aim to provide an unbiased estimate for a smeared spectral density. The lsdensities library provides functions for the implementation of these rules.

1. Establish a parametric map between the input data \(C(t)\) and a smeared spectral density \(\rho_\sigma(\omega)\). Given a set of parameters \(\mathbb{p} = \lambda, \alpha\).

\[C(t) \underset{\mathbb{p}}{\mapsto} \rho^{\mathbb{p}}_\sigma(E)\]

This is implemented through the equation

\[\begin{split}\vec{g}(\sigma;\omega) = \argmin_{\vec{g} \in \mathbb{R}^{\tau_{\rm max}}} \int_0^\infty dE \, e^{\alpha E} | \sum_{r=1}^{\tau_{\rm max}} g_\tau(\omega) b_T(a \tau, E) - \mathcal{S}_\sigma(E,\omega) |^2 \\ + \lambda \; \sum_{\tau_1, \tau_2=1}^{\tau_{\rm max}} g_{\tau_1}(\omega) \, B_{\tau_1 \tau_2}\, g_{\tau_2}(\omega) \, , \;\;\;\;\; \lambda \in (0,\infty)\end{split}\]

For a generic set of parameters \(\mathbb{p}\), in particular for \(\lambda\), this will not be the solution we are after. However, for a subset of values of \(\mathbb{p}\), the map provides a solution that is compatible, within the statistical error of the input data, with the spectral density associated with \(C(t)\) and smeared with the desired kernel.

The second operation that must be performed is then:

  1. Identification of the value of \(\lambda\) (and \(\alpha\) if desired) such that the solution is compatible with the input within errors.

    This task can be implemented manually by the user, but this library manages it automatically via the InverseProblemWrapper class in two different ways. Two optimal values of \(\lambda\) will be suggested based on two different interpretations.

    One relies on the identification of a plateau in the values of \(\rho_\sigma\) as a function of \(\lambda\), as the latter variable is reduced. The results are collected in InverseProblemWrapper.rhoResultHLT.

    The other suggested value is based on Bayesian inference and is the result of the maximisation of a likelihood function. The results are collected in InverseProblemWrapper.rhoResultBayes.

    If everything works correctly, the two values should be compatible.

    The search for the optimal value of \(\lambda\) requires input values to be passed through the AlgorithmParameters class.