Restoring a smooth function from its noisy integrals

Numerical (and experimental) data analysis often requires the restoration of a smooth function from a set of sampled integrals over finite bins. We present the bin hierarchy method that efficiently computes the maximally smooth function from the sampled integrals using essentially all the information contained in the data. We perform extensive tests with different classes of functions and levels of data quality, including Monte Carlo data suffering from a severe sign problem and physical data for the Green's function of the Fröhlich polaron.

Figure 1

Cubic polynomial test function. The BHM fit (red dashed line with error band) in comparison with the basis projection method (blue dotted line with error band). The left panel (a) shows the function and the fits, while the right panel (b) shows the difference between the fitted and the true function for both methods.

Figure 2

Quartic polynomial test function. The BHM fit (red dashed line with error band) in comparison with the basis projection method (blue dotted line with error band). The left panel (a) shows the function and the fits, while the right panel (b) shows the difference between the fitted and the true function for both methods.

Figure 3

BHM fit with error band of an exponential test function. The left panel (a) shows the function and the fit, while the right panel (b) shows the difference between the fitted and the true function.

Figure 4

Exponential test function. The BHM fit (red dashed line with error band) in comparison with the fit using only elementary bins (blue dotted line with error band). The latter does not properly resolve all sampled integrals of the test function.

Figure 5

Oscillating test function using ${10}^4$ sampled points [(a) and (b)] and ${10}^6$ sampled points [(c) and (d)]. The BHM fit (red dashed line) is shown with two error estimates on the spline coefficients: errors calculated using Eq. (12) (red band) and errors calculated using bootstrap (blue band). Bootstrap provides a tighter bound on the errors in this example. The left panels, (a) and (c), show the function and the fit, while the right panels, (b) and (d), show the difference between the fitted and the true function.

Figure 6

Oscillating test function. Difference between the fitted and the true function normalized by one standard error obtained with different methods.

Figure 7

Smoothed histogram of $\tilde{f}\left(x\right)$ values for the oscillating test function at $x=1,2,2.9,3.74$ (from left to right). The vertical lines are histogram mean (red solid line), true function value (blue dotted line), the particular $\tilde{f}\left(x\right)$ value from the simulation shown in the bottom panel of Fig. 5 (black dashed line). The latter is shown for reference. The horizontal lines are the $\pm 1\sigma$ interval around the histogram mean, where $\sigma$ is the median standard error from Eq. (12) (red solid line); the $\pm 1\sigma$ interval around the histogram mean, where $\sigma$ is the robust standard error (blue dotted line); the $\pm 1\sigma$ interval around the histogram mean, where $\sigma$ is the typical bootstrap error, taken from the example shown in the bottom panel of Fig. 5 (black dashed line).

Figure 8

Sampling a divergent function defined on a semi-infinite domain using the BHM with appropriate transforms. The left panel (a) shows the function and the fit, while the right panel (b) shows the difference between the fitted and the true function.

Figure 9

Divergent test function $f\left(x\right)\propto x^{-0.5}{(1+x)}^{-1}$ with an $x^{-0.5}$ divergence at $x=0$. Data is sampled with a compensation for the divergence that overestimates [left panel (a)] and underestimates [right panel (b)] the power of the divergence by using weighting factors of $x^{0.8}$ and $x^{0.2}$, respectively. The plots show the difference between the BHM fit and the true function.

Figure 10

Sampling with a severe sign problem using ${10}^5$ sampled points [left panel (a)] and ${10}^7$ sampled points [right panel (b)]. The sampled integrals used for the fit in the left panel are consistent with zero, which is correctly captured by the check presented in Sec. 3g. In this case, the displayed BHM spline would not be used for further data analysis.

Figure 11

Diagrammatic Monte Carlo sampling of the Fröhlich polaron Green's function. (a) shows a very precise numerical calculation of the function, while (b) shows the difference between the fitted and the reference function for the BHM and the basis projection method.