Robust determination of relaxation times spectra of long-time multirelaxation processes

Long-time relaxation processes occur in numerous physical systems. They are often regarded as multirelaxation processes, which are a superposition of exponential decays with a certain distribution of relaxation times. The relaxation times spectra often convey information about the underlying physics. Extracting the spectrum of relaxation times from experimental data is, however, difficult. This is partly due to the mathematical properties of the problem and partly due to experimental limitations. In this paper, we perform the inversion of time-series relaxation data into a relaxation spectrum using the singular value decomposition accompanied by the Akaike information criterion estimator. We show that this approach does not need any a priori information on the spectral shape and that it delivers a solution that consistently approximates the best one achievable for given experimental dataset. On the contrary, we show that the solution obtained imposing an optimal fit of experimental data is often far from reconstructing well the distribution of relaxation times.

Figure 1

Left column: Noiseless $y\left(t\right)$ functions data used for the analysis shown in the logarithmic time. Right column: Corresponding relaxation distributions $G\left(\tau \right)$. Details of the three distributions are reported in Table 1.

Figure 2

Schematic representation of the different phases of the inversion procedure. Here SSR (quality of the temporal fit) and $p$ (quality of the correlation of reconstructed and theoretical distributions) are shown as a function of the threshold $s_0$. Circles represent the threshold values to obtain optimal fit of the data ($s_F$) and the best correlation of reconstructed and theoretical distributions ($s_C$). The threshold $s_A$ shown in the plot corresponds to the estimated threshold found with our approach, whose definition will be given later.

Figure 3

Dependence of the reconstructed $\widehat{y}$ (left column) and $\tau$ spectrum ($\widehat{G}$) on the threshold level $s_0$. Data for case A without added noise ($n_0=0$) have been inverted.

Figure 4

Estimated optimal threshold using the different approaches described in the text for different noise levels $n_0$. (a) SSR vs. threshold. (b) $p$ vs. threshold. Symbols in subplots (a) and (b) indicate the estimated optimal threshold from the three approaches. (c) Akaike indicator vs. threshold, the minimum being corresponding to the Akaike threshold $s_A$. (d) $p$ vs. noise when using thresholds obtained in the three approaches.

Figure 5

Reconstructed distribution of relaxation times obtained using different thresholds (case A).

Figure 6

Reconstructed distribution of relaxation times obtained for the three cases discussed in Table 1. Two noise levels have been considered in the two columns.

Figure 7

Accuracy of the approach in determining the most relevant relaxation time [$\tau$ value corresponding to the maximum in the distribution function for the Weibull distribution (a) and corresponding to the maximum relaxation time for the rectangular distribution (b)]. We report the error in the estimation of ${\tau }_m$ [see Eq. (15)] as a function of noise level. The typical intervals of the experimental noise level are shown as dashed vertical lines.

Figure 8

Effects of noise on the procedure of deriving the time relaxation distribution function. (a) Fraction of elements set to zero in the diagonal SVD matrix; (b) cross-correlation between the reconstructed and expected distribution function ($p=1-r$); (c) residuals of the reconstructed and used dataset (SSR). On the $x$ axes we always have the noise level $n_0$.

Figure 9

Effects of incompleteness of the experimental dataset on the reconstruction of the time relaxation distribution function. (a) Effects of missing data for early time relaxation. $t_1$ indicates the time corresponding to the first data acquisition. (b) Effects of a wrong estimate of the onset time of relaxation ($t_0$). (c) Effects of a wrong estimate of the asymptotic value of the variable (offset $y_0$ in the data). In all plots, data for exact (cyan symbols) and noisy (red symbols) data are reported. Noise was $n_0=0.005$.

Figure 10

Resolution accuracy: (a) Examples of double Weibull distributions used for the analysis for three values of resolution parameter (distance between peaks): Red solid lines are the theoretical distributions while dashed blue and solid green lines represent the reconstructed distributions in the absence and in the presence of noise ($n_0=0.005$), respectively. (b) Minimum peak distance detectable with the proposed approach as a function of noise level.

Figure 11

Inversion using different approaches. Case A with small noise ($n_0=0.001$). First column: $y$ vs. $t$. Second column: distribution of relaxation times vs. $\tau$. [(a) and (d)] SVD $+$ AIC approach; [(b) and (e)] SVD with zeros elimination; [(c) and (f)] inversion fitting with a discrete number of exponentials.

Figure 12

Inversion of experimental data. (a) Experimental data and fitting functions; (b) distribution of relaxation times.

Figure 13

Robustness of the approach. Left column: Rank of the solution as a function of noise for the three choices of the threshold. Right column: Correlation between predicted and theoretical distributions defined by $p=1-r$ vs. noise. [(a) and (b)] Results are obtained averaging solutions obtained inverting the same set of data but changing the range of $\tau$ values over which the solution is searched. [(c) and (d)] Results are obtained averaging solutions obtained inverting the same set of data with different noise realizations. [(e) and (f)] Results are obtained averaging solutions obtained inverting the same set of data with different multiplicative noise realizations (dynamic noise).