A new regularization approach for numerical differentiation

Figures & data

Table 1. Relative errors in derivative approximation.

Methods	$m=100(h=0.01)$, $\sigma =0.01$	$m=100(h=0.01)$, $\sigma =0.1$	$m=10(h=0.1)$, $\sigma =0.01$
Degree-2 polynomial	0.0287	0.3190	0.2786
Tikhonov, k = 0	0.7393	0.8297	0.7062
Tikhonov, k = 1	0.1803	0.3038	0.6420
Tikhonov, k = 2	0.0186	0.0301	0.4432
Cubic Spline	0.1060	1.15	0.3004
Convolution smoothing	0.1059	0.8603	0.2098
Variational method	0.1669	0.7149	0.3419
Our method (k = 0)	0.0607	0.0839	0.1355

Table 2. Relative errors and the descent rates using different gradients.

$m=100(h=0.01)$, extreme error $\sigma =0.1$	Sobolev gradient (${\mathrm{\nabla }}_{{\mathcal{H}}^1}^{}G$)	${\mathcal{L}}^2$–${\mathcal{H}}^1$ conjugate gradient	${\mathcal{H}}^1$–${\mathcal{H}}^1$ conjugate gradient
Recovery errors (using SC-I Section 5.3, known δ)	0.0839	0.4364	0.1522
Recovery errors (using SC-II Section 8, unknown δ)	0.1299	0.1299	0.1299
# iterations (using SC-I Section 5.3, known δ)	39	4	6
# iterations (using SC-II Section 8, unknown δ)	2	2	2

Table 3. Relative errors and the descent rates using different gradients.

$m=100(h=0.01)$, moderate error $\sigma =0.01$	Sobolev gradient (${\mathrm{\nabla }}_{{\mathcal{H}}^1}^{}G$)	${\mathcal{L}}^2$–${\mathcal{H}}^1$ conjugate gradient	${\mathcal{H}}^1$–${\mathcal{H}}^1$ conjugate gradient
Recovery errors (using SC-I Section 5.3, known δ)	0.0607	0.0589	0.0613
Recovery errors (using SC-II Section 8, unknown δ)	0.1129	0.1129	0.1129
# iterations (using SC-I Section 5.3, known δ)	84	5	18
# iterations (using SC-II Section 8, unknown δ)	3	3	3

Figure 1. Noisy $\tilde{g}$.

Figure 2. Inverse recovery of the derivative $\tilde{\phi }$ and $T\tilde{\phi }$: (a) derivative $\tilde{\phi }$ vs. ϕ and (b) recovery of $\tilde{\phi }$.

Figure 3. Recoveries using out method: (a) numerical derivative $\tilde{\phi }$ and (b) smooth approximation $T\tilde{\phi }$.

Figure 4. (a) Noisy data for Example 7.4 and (b) total variation regularization from [1].

Figure 5. Integration smooths out the noise present in the data: (a) noisy $\tilde{g}$ vs exact g and (b) noisy $\tilde{u}$ vs exact u.

Figure 6. Fluctuations during the descent process. (a) $||u_{{\psi }_m}-u_{\delta }|{|}_{{\mathcal{L}}^2}$, Example 7.1 ($\sigma =0.1$) and (b) $||\tilde{g}-g_m|{|}_{{\mathcal{L}}^2}$, Example 7.2.

Figure 7. Relative errors in the recoveries during the descent process: (a) Example 7.2 and (b) Example 7.1 ($\sigma =0.01$).

Figure 8. Landweber iterations on Example 7.1 ($\sigma =0.01$): (a) $||T{\psi }_m-g_{\delta }|{|}_{{\mathcal{L}}^2}$-descent and (b) relative errors descent.

Knowles I, Renka RJ. 2014. Methods for numerical differentiation of noisy data. Proceedings of the variational and topological methods: theory, applications, numerical simulations, and open problems, Vol. 21. San Marcos (TX): Texas State University; 2014. p. 235–246.

 Google Scholar