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High-accuracy approximation of piecewise smooth functions using the Truncation and Encode approach

Rosa Donat
Rosa Donat
 and 
Sergio López-Ureña
Sergio López-Ureña
   | Sep 06, 2017
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Published Online: Sep 06, 2017
Page range: 367 - 384
Received: Apr 07, 2017
Accepted: Sep 06, 2017
DOI: https://doi.org/10.21042/AMNS.2017.2.00030
Keywords
© 2017 Rosa Donat, Sergio López-Ureña, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Sketch of recurrence (9). The value of v^2i+1k+1 $\hat v^{k+1}_{2i+1}$ , i ∈ {2j, 2j + 1}, depends on d^jk−1=v^2j+1k−(Pk−1kv^k−1)2j+1 $\hat d_j^{k-1}=\hat v_{2j+1}^{k} - (P_{k-1}^{k}\hat v^{k-1})_{2j+1}$ .
Sketch of recurrence (9). The value of v^2i+1k+1 $\hat v^{k+1}_{2i+1}$ , i ∈ {2j, 2j + 1}, depends on d^jk−1=v^2j+1k−(Pk−1kv^k−1)2j+1 $\hat d_j^{k-1}=\hat v_{2j+1}^{k} - (P_{k-1}^{k}\hat v^{k-1})_{2j+1}$ .

Fig. 2

Graphic representation of: left, f1; right, f2. Both functions are defined in (12).
Graphic representation of: left, f1; right, f2. Both functions are defined in (12).

Fig. 3

In continuous line, the graphics of ‖v15−v^15‖∞ $\|v^{15} - \hat v^{15}\|_\infty$ as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f1. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .
In continuous line, the graphics of ‖v15−v^15‖∞ $\|v^{15} - \hat v^{15}\|_\infty$ as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f1. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .

Fig. 4

In continuous line, the graphics of |E − Ê15| as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f1. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .
In continuous line, the graphics of |E − Ê15| as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f1. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .

Fig. 5

In continuous line, the graphics of ‖v15−v^15‖∞ $\|v^{15} - \hat v^{15}\|_\infty$ as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f2. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .
In continuous line, the graphics of ‖v15−v^15‖∞ $\|v^{15} - \hat v^{15}\|_\infty$ as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f2. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .

Fig. 6

In continuous line, the graphics of |E − Ê15| as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f2. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .
In continuous line, the graphics of |E − Ê15| as a function of neval for εk = ε varying from 10−1 to 10−10, applied to each local rule and f2. For comparison, in discontinuous line are shown the growth rates O(neval−2) $O(n_{eval}^{-2})$ and O(neval−4) $O(n_{eval}^{-4})$ .

Parameters as specified in Table 1 for f1, ε = εk = 10-1 and the cubic rule.

KnevalnK +1τ‖vK−v^K‖∞$||v^{k}- \hat v^K||_\infty $‖vK−v^K‖1$||v^{k}- \hat v^K||_{1}$ |E − ÊK|
31.3000e+011.7000e+011.3077e+005.6435e-037.8277e-044.4742e-03
41.3000e+013.3000e+012.5385e+001.1245e-021.7642e-031.1215e-03
51.3000e+016.5000e+015.0000e+001.1245e-021.9634e-032.7580e-04
61.3000e+011.2900e+029.9231e+001.2010e-022.0193e-036.3898e-05
71.3000e+012.5700e+021.9769e+011.2010e-022.0373e-031.0892e-05
81.3000e+015.1300e+023.9462e+011.2010e-022.0439e-032.3614e-06
91.3000e+011.0250e+037.8846e+011.2010e-022.0465e-035.6749e-06
101.3000e+012.0490e+031.5762e+021.2012e-022.0477e-036.5032e-06
111.3000e+014.0970e+033.1515e+021.2012e-022.0482e-036.7103e-06

Parameters as specified in Table 1 for f1, ε = εk = 10-1 and the PCHIP rule.

KnevalnK +1τ‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty $‖vK−v^K‖1 $||v^{k}- \hat v^K||_{1}$ |E − ÊK|
31.3000e+011.7000e+011.3077e+001.0606e-021.5196e-034.2125e-03
41.3000e+013.3000e+012.5385e+002.3126e-023.8381e-031.0814e-03
51.3000e+016.5000e+015.0000e+002.3126e-024.3212e-032.3874e-04
61.3000e+011.2900e+029.9231e+002.3126e-024.4675e-032.6119e-05
71.3000e+012.5700e+021.9769e+012.3126e-024.5131e-032.7146e-05
81.3000e+015.1300e+023.9462e+012.3144e-024.5297e-034.0470e-05
91.3000e+011.0250e+037.8846e+012.3177e-024.5360e-034.3801e-05
101.3000e+012.0490e+031.5762e+022.3177e-024.5387e-034.4634e-05
111.3000e+014.0970e+033.1515e+022.3177e-024.5399e-034.4842e-05

Parameters as specified in Table 1 for f1, K = 15 and the PCHIP rule. εk = ε varies between 10-1 and 10-10.

ε2-4εneval‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty$ ‖vK−v^K‖1 $||v^{k}- \hat v^K||_{1}$ |E − ÊK|
1e-016.25e-031.3000e+012.3177e-024.5411e-034.4912e-05
1e-026.25e-042.7000e+011.8042e-021.1650e-035.0535e-04
1e-036.25e-056.9000e+011.4751e-041.3376e-051.3180e-06
1e-046.25e-061.2300e+022.7379e-051.4956e-061.4601e-07
1e-056.25e-072.3300e+022.8833e-062.0876e-071.4208e-08
1e-066.25e-084.2100e+027.0168e-071.1231e-081.6390e-09
1e-076.25e-097.5700e+026.8925e-081.3645e-096.3232e-11
1e-086.25e-101.3610e+031.5733e-091.1174e-102.6616e-12
1e-096.25e-112.4230e+031.1084e-101.1353e-111.4272e-11
1e-106.25e-124.3150e+032.2858e-111.1252e-121.5153e-11

Parameters as specified in Table 1 for f2, ε = εk = 10-1 and the cubic rule.

KnevalnK +1τ‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty$ ‖vK−v^K‖1 $||v^{k}- \hat v^K||_{1}$ |E − ÊK|
31.3000e+011.7000e+011.3077e+006.1742e-024.3700e-031.9318e-02
41.5000e+013.3000e+012.2000e+009.7336e-027.6400e-031.6578e-03
51.7000e+016.5000e+013.8235e+009.7336e-029.0579e-039.4329e-03
61.9000e+011.2900e+026.7895e+009.7601e-029.8133e-035.2198e-03
72.1000e+012.5700e+021.2238e+019.8885e-021.0169e-023.5400e-03
82.3000e+015.1300e+022.2304e+019.8885e-021.0359e-024.6595e-03
92.5000e+011.0250e+034.1000e+019.9210e-021.0448e-025.0960e-03
102.7000e+012.0490e+037.5889e+019.9210e-021.0496e-024.8203e-03
112.9000e+014.0970e+031.4128e+029.9272e-021.0518e-024.7122e-03

Parameters as specified in Table 1 for f1, K = 15 and the polygonal rule. εk = ε varies between 10-1 and 10-10.

ε2-4εneval‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty$ ‖vK−v^K‖1 $||v^{k}- \hat v^K||_{1}$ |E − ÊK|
1e-016.25e-031.3000e+011.2012e-022.0488e-036.7794e-06
1e-026.25e-042.7000e+014.4450e-049.2550e-054.9354e-05
1e-036.25e-054.3000e+012.7188e-042.2672e-057.3722e-06
1e-046.25e-068.1000e+017.0349e-069.1132e-075.9591e-08
1e-056.25e-071.3700e+026.5275e-071.1135e-074.9200e-09
1e-066.25e-082.5500e+026.3729e-081.0246e-082.4199e-09
1e-076.25e-094.0900e+021.1524e-081.4623e-097.9961e-10
1e-086.25e-107.1900e+026.2685e-101.1760e-108.0592e-11
1e-096.25e-111.3610e+036.9934e-119.8732e-121.9488e-11
1e-106.25e-122.5150e+037.3016e-129.4366e-131.5464e-11

Parameters as specified in Table 1 for f1, ε = εk = 10-1 and the polygonal rule.

KnevalnK +1τ‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty $ ‖vK−v^K‖1 $||v^{k}- \hat v^K||_{2}$ |E − ÊK|
31.3000e+011.7000e+011.3077e+004.6488e-025.5437e-034.2125e-03
41.7000e+013.3000e+011.9412e+004.6488e-029.2544e-031.0395e-03
51.7000e+016.5000e+013.8235e+004.6488e-021.1582e-021.0395e-03
61.7000e+011.2900e+027.5882e+004.7016e-021.2225e-021.0395e-03
71.7000e+012.5700e+021.5118e+014.7016e-021.2411e-021.0395e-03
81.7000e+015.1300e+023.0176e+014.7026e-021.2470e-021.0395e-03
91.7000e+011.0250e+036.0294e+014.7033e-021.2491e-021.0395e-03
101.7000e+012.0490e+031.2053e+024.7033e-021.2499e-021.0395e-03
111.7000e+014.0970e+032.4100e+024.7034e-021.2503e-021.0395e-03

Parameters as specified in Table 1 for f2, K = 15 and the polygonal rule. εk = ε varies between 10-1 and 10-10.

ε2-4εneval‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty$ ‖vK−v^K‖1 $||v^{k}- \hat v^K||_{1}$ |E − ÊK|
1e-016.25e-034.1000e+015.7922e-028.2502e-033.1449e-03
1e-026.25e-045.9000e+011.8042e-021.1149e-035.0165e-04
1e-036.25e-051.0700e+022.7833e-041.2775e-053.1659e-06
1e-046.25e-061.6900e+023.7144e-051.4538e-061.3326e-06
1e-056.25e-072.8500e+024.2689e-062.0742e-071.1600e-06
1e-066.25e-084.7500e+027.0168e-071.1186e-081.1461e-06
1e-076.25e-098.0500e+026.8925e-081.3631e-091.1445e-06
1e-086.25e-101.4090e+031.5733e-091.1040e-101.1444e-06
1e-096.25e-112.4650e+031.1084e-101.1313e-111.1444e-06
1e-106.25e-124.3510e+032.2858e-111.1239e-121.1444e-06

Parameters as specified in Table 1 for f2, ε = εk = 10-1 and the polygonal rule.

KnevalnK +1τ‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty$ ‖vK−v^K‖1 $||v^{k}- \hat v^K||_{1}$ |E − ÊK|
31.3000e+011.7000e+011.3077e+004.6488e-025.5437e-032.2962e-02
41.9000e+013.3000e+011.7368e+004.6488e-027.8089e-033.5761e-03
52.1000e+016.5000e+013.0952e+004.6488e-029.9419e-034.0614e-03
62.3000e+011.2900e+025.6087e+004.7016e-021.0549e-021.3192e-04
72.5000e+012.5700e+021.0280e+014.7016e-021.0726e-021.8242e-03
82.7000e+015.1300e+021.9000e+014.7026e-021.0781e-028.4798e-04
92.9000e+011.0250e+033.5345e+014.7033e-021.0800e-023.5975e-04
103.1000e+012.0490e+036.6097e+014.7033e-021.0808e-026.0389e-04
113.3000e+014.0970e+031.2415e+024.7034e-021.0811e-027.2597e-04

Parameters as specified in Table 1 for f2, ε = εk = 10-1and the PCHIP rule

KnevalnK +1τ‖vK−v^K‖∞ $||v^{k}- \hat v^K||_\infty$ ‖vK−v^K‖1 $||v^{k}- \hat v^K||_{1}$ |E − ÊK|
31.3000e+011.7000e+011.3077e+005.2400e-023.9780e-032.0350e-02
41.5000e+013.3000e+012.2000e+005.3922e-027.1545e-038.9677e-04
51.7000e+016.5000e+013.8235e+005.7556e-027.8826e-037.5834e-03
61.9000e+011.2900e+026.7895e+005.7616e-028.1229e-033.8474e-03
72.1000e+012.5700e+021.2238e+015.7920e-028.2001e-031.9526e-03
82.3000e+015.1300e+022.2304e+015.7920e-028.2293e-032.9445e-03
92.5000e+011.0250e+034.1000e+015.7920e-028.2408e-033.4357e-03
102.7000e+012.0490e+037.5889e+015.7922e-028.2458e-033.1923e-03
112.9000e+014.0970e+031.4128e+025.7922e-028.2481e-033.0704e-03

Parameters for the numerical experiments.

neval: Number of function evaluations done by TE to compute v^K $\hat v^K$ .
nK+1 = 2K+1: Number of function evaluations needed to compute vK.
τ=nK+1neval $\tau = \frac{n_K+1}{n_{eval}}$ .
eK=‖vK−v^K‖∞=supi|viK−v^iK| $e_K=\| v^K - \hat v^K\|_\infty = \sup_i |v^K_i - \hat v^K_i|$ .
‖vK−v^K‖1=nK−1∑i=0nK|viK−v^iK| $\| v^K - \hat v^K\|_1 = n_K^{-1}\sum_{i=0}^{n_K} |v^K_i - \hat v^K_i|$ .
E=∫01f(x)dx $E=\int_0^1 f(x) dx$ .
ÊK is obtained applying the trapezoidal rule to v^K $\hat v^K$ .
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