Table 2
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Mean approximation of |$Y_{0}$|⁠, its mean relative MSE from DBDP, OSM and DLBDP schemes and their average runtimes in Example 2 for |$d=50$| and |$N \in \{2, 8, 32, 64\}$|⁠. The STD of the approximations of |$Y_{0}$| and its relative MSE values are given in the brackets

 |$N = 2$||$N = 8$||$N = 32$||$N = 64$|
 DBDPDBDPDBDPDBDP
 OSMOSMOSMOSM
MetricDLBDPDLBDPDLBDPDLBDP
|$Y_{0}$| (E et al., 2019)|$17.9743$|
|$\overline{Y}_{0}^{\varDelta , \hat{\theta }}$||$17.5602$|  |$({4.11\text{e}\!-\!01})$||$17.7981$|  |$({4.50\text{e}\!-\!01})$||$17.9276$|  |$({5.15\text{e}\!-\!01})$||$17.9112$|  |$({4.91\text{e}\!-\!01})$|
|$17.6537$|  |$({2.57\text{e}\!-\!01})$||$17.5056$|  |$({7.75\text{e}\!-\!01})$||$17.8351$|  |$({3.88\text{e}\!-\!01})$||$17.8865$|  |$({8.77\text{e}\!-\!02})$|
|$17.8329$|  |$({1.83\text{e}\!-\!01})$||$17.4669$|  |$({6.58\text{e}\!-\!01})$||$17.9714$|  |$({1.63\text{e}\!-\!01})$||$17.9117$|  |$({9.41\text{e}\!-\!02})$|
|$\overline{{\tilde{\varepsilon }}}^{y, r}_{0}$||${1.05\text{e}\!-\!03}$|  |$({1.48\text{e}\!-\!03})$||${7.24\text{e}\!-\!04}$|  |$({1.79\text{e}\!-\!03})$||${8.29\text{e}\!-\!04}$|  |$({1.40\text{e}\!-\!03})$||${7.58\text{e}\!-\!04}$|  |$({8.88\text{e}\!-\!04})$|
|${5.23\text{e}\!-\!04}$|  |$({5.25\text{e}\!-\!04})$||${2.54\text{e}\!-\!03}$|  |$({5.66\text{e}\!-\!03})$||${5.27\text{e}\!-\!04}$|  |$({1.08\text{e}\!-\!03})$||${4.77\text{e}\!-\!05}$|  |$({9.41\text{e}\!-\!05})$|
|${1.65\text{e}\!-\!04}$|  |$({2.77\text{e}\!-\!04})$||${2.14\text{e}\!-\!03}$|  |$({3.50\text{e}\!-\!03})$||${8.22\text{e}\!-\!05}$|  |$({7.96\text{e}\!-\!05})$||${3.95\text{e}\!-\!05}$|  |$({4.65\text{e}\!-\!05})$|
|$\overline{\tau }$||${5.54\text{e}\!+\!02}$||${2.82\text{e}\!+\!03}$||${2.87\text{e}\!+\!04}$||${1.12\text{e}\!+\!05}$|
|${2.60\text{e}\!+\!03}$||${9.74\text{e}\!+\!03}$||${7.30\text{e}\!+\!04}$||${2.55\text{e}\!+\!05}$|
|${2.36\text{e}\!+\!03}$||${7.67\text{e}\!+\!03}$||${4.67\text{e}\!+\!04}$||${1.47\text{e}\!+\!05}$|
 |$N = 2$||$N = 8$||$N = 32$||$N = 64$|
 DBDPDBDPDBDPDBDP
 OSMOSMOSMOSM
MetricDLBDPDLBDPDLBDPDLBDP
|$Y_{0}$| (E et al., 2019)|$17.9743$|
|$\overline{Y}_{0}^{\varDelta , \hat{\theta }}$||$17.5602$|  |$({4.11\text{e}\!-\!01})$||$17.7981$|  |$({4.50\text{e}\!-\!01})$||$17.9276$|  |$({5.15\text{e}\!-\!01})$||$17.9112$|  |$({4.91\text{e}\!-\!01})$|
|$17.6537$|  |$({2.57\text{e}\!-\!01})$||$17.5056$|  |$({7.75\text{e}\!-\!01})$||$17.8351$|  |$({3.88\text{e}\!-\!01})$||$17.8865$|  |$({8.77\text{e}\!-\!02})$|
|$17.8329$|  |$({1.83\text{e}\!-\!01})$||$17.4669$|  |$({6.58\text{e}\!-\!01})$||$17.9714$|  |$({1.63\text{e}\!-\!01})$||$17.9117$|  |$({9.41\text{e}\!-\!02})$|
|$\overline{{\tilde{\varepsilon }}}^{y, r}_{0}$||${1.05\text{e}\!-\!03}$|  |$({1.48\text{e}\!-\!03})$||${7.24\text{e}\!-\!04}$|  |$({1.79\text{e}\!-\!03})$||${8.29\text{e}\!-\!04}$|  |$({1.40\text{e}\!-\!03})$||${7.58\text{e}\!-\!04}$|  |$({8.88\text{e}\!-\!04})$|
|${5.23\text{e}\!-\!04}$|  |$({5.25\text{e}\!-\!04})$||${2.54\text{e}\!-\!03}$|  |$({5.66\text{e}\!-\!03})$||${5.27\text{e}\!-\!04}$|  |$({1.08\text{e}\!-\!03})$||${4.77\text{e}\!-\!05}$|  |$({9.41\text{e}\!-\!05})$|
|${1.65\text{e}\!-\!04}$|  |$({2.77\text{e}\!-\!04})$||${2.14\text{e}\!-\!03}$|  |$({3.50\text{e}\!-\!03})$||${8.22\text{e}\!-\!05}$|  |$({7.96\text{e}\!-\!05})$||${3.95\text{e}\!-\!05}$|  |$({4.65\text{e}\!-\!05})$|
|$\overline{\tau }$||${5.54\text{e}\!+\!02}$||${2.82\text{e}\!+\!03}$||${2.87\text{e}\!+\!04}$||${1.12\text{e}\!+\!05}$|
|${2.60\text{e}\!+\!03}$||${9.74\text{e}\!+\!03}$||${7.30\text{e}\!+\!04}$||${2.55\text{e}\!+\!05}$|
|${2.36\text{e}\!+\!03}$||${7.67\text{e}\!+\!03}$||${4.67\text{e}\!+\!04}$||${1.47\text{e}\!+\!05}$|
Table 2
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Mean approximation of |$Y_{0}$|⁠, its mean relative MSE from DBDP, OSM and DLBDP schemes and their average runtimes in Example 2 for |$d=50$| and |$N \in \{2, 8, 32, 64\}$|⁠. The STD of the approximations of |$Y_{0}$| and its relative MSE values are given in the brackets

 |$N = 2$||$N = 8$||$N = 32$||$N = 64$|
 DBDPDBDPDBDPDBDP
 OSMOSMOSMOSM
MetricDLBDPDLBDPDLBDPDLBDP
|$Y_{0}$| (E et al., 2019)|$17.9743$|
|$\overline{Y}_{0}^{\varDelta , \hat{\theta }}$||$17.5602$|  |$({4.11\text{e}\!-\!01})$||$17.7981$|  |$({4.50\text{e}\!-\!01})$||$17.9276$|  |$({5.15\text{e}\!-\!01})$||$17.9112$|  |$({4.91\text{e}\!-\!01})$|
|$17.6537$|  |$({2.57\text{e}\!-\!01})$||$17.5056$|  |$({7.75\text{e}\!-\!01})$||$17.8351$|  |$({3.88\text{e}\!-\!01})$||$17.8865$|  |$({8.77\text{e}\!-\!02})$|
|$17.8329$|  |$({1.83\text{e}\!-\!01})$||$17.4669$|  |$({6.58\text{e}\!-\!01})$||$17.9714$|  |$({1.63\text{e}\!-\!01})$||$17.9117$|  |$({9.41\text{e}\!-\!02})$|
|$\overline{{\tilde{\varepsilon }}}^{y, r}_{0}$||${1.05\text{e}\!-\!03}$|  |$({1.48\text{e}\!-\!03})$||${7.24\text{e}\!-\!04}$|  |$({1.79\text{e}\!-\!03})$||${8.29\text{e}\!-\!04}$|  |$({1.40\text{e}\!-\!03})$||${7.58\text{e}\!-\!04}$|  |$({8.88\text{e}\!-\!04})$|
|${5.23\text{e}\!-\!04}$|  |$({5.25\text{e}\!-\!04})$||${2.54\text{e}\!-\!03}$|  |$({5.66\text{e}\!-\!03})$||${5.27\text{e}\!-\!04}$|  |$({1.08\text{e}\!-\!03})$||${4.77\text{e}\!-\!05}$|  |$({9.41\text{e}\!-\!05})$|
|${1.65\text{e}\!-\!04}$|  |$({2.77\text{e}\!-\!04})$||${2.14\text{e}\!-\!03}$|  |$({3.50\text{e}\!-\!03})$||${8.22\text{e}\!-\!05}$|  |$({7.96\text{e}\!-\!05})$||${3.95\text{e}\!-\!05}$|  |$({4.65\text{e}\!-\!05})$|
|$\overline{\tau }$||${5.54\text{e}\!+\!02}$||${2.82\text{e}\!+\!03}$||${2.87\text{e}\!+\!04}$||${1.12\text{e}\!+\!05}$|
|${2.60\text{e}\!+\!03}$||${9.74\text{e}\!+\!03}$||${7.30\text{e}\!+\!04}$||${2.55\text{e}\!+\!05}$|
|${2.36\text{e}\!+\!03}$||${7.67\text{e}\!+\!03}$||${4.67\text{e}\!+\!04}$||${1.47\text{e}\!+\!05}$|
 |$N = 2$||$N = 8$||$N = 32$||$N = 64$|
 DBDPDBDPDBDPDBDP
 OSMOSMOSMOSM
MetricDLBDPDLBDPDLBDPDLBDP
|$Y_{0}$| (E et al., 2019)|$17.9743$|
|$\overline{Y}_{0}^{\varDelta , \hat{\theta }}$||$17.5602$|  |$({4.11\text{e}\!-\!01})$||$17.7981$|  |$({4.50\text{e}\!-\!01})$||$17.9276$|  |$({5.15\text{e}\!-\!01})$||$17.9112$|  |$({4.91\text{e}\!-\!01})$|
|$17.6537$|  |$({2.57\text{e}\!-\!01})$||$17.5056$|  |$({7.75\text{e}\!-\!01})$||$17.8351$|  |$({3.88\text{e}\!-\!01})$||$17.8865$|  |$({8.77\text{e}\!-\!02})$|
|$17.8329$|  |$({1.83\text{e}\!-\!01})$||$17.4669$|  |$({6.58\text{e}\!-\!01})$||$17.9714$|  |$({1.63\text{e}\!-\!01})$||$17.9117$|  |$({9.41\text{e}\!-\!02})$|
|$\overline{{\tilde{\varepsilon }}}^{y, r}_{0}$||${1.05\text{e}\!-\!03}$|  |$({1.48\text{e}\!-\!03})$||${7.24\text{e}\!-\!04}$|  |$({1.79\text{e}\!-\!03})$||${8.29\text{e}\!-\!04}$|  |$({1.40\text{e}\!-\!03})$||${7.58\text{e}\!-\!04}$|  |$({8.88\text{e}\!-\!04})$|
|${5.23\text{e}\!-\!04}$|  |$({5.25\text{e}\!-\!04})$||${2.54\text{e}\!-\!03}$|  |$({5.66\text{e}\!-\!03})$||${5.27\text{e}\!-\!04}$|  |$({1.08\text{e}\!-\!03})$||${4.77\text{e}\!-\!05}$|  |$({9.41\text{e}\!-\!05})$|
|${1.65\text{e}\!-\!04}$|  |$({2.77\text{e}\!-\!04})$||${2.14\text{e}\!-\!03}$|  |$({3.50\text{e}\!-\!03})$||${8.22\text{e}\!-\!05}$|  |$({7.96\text{e}\!-\!05})$||${3.95\text{e}\!-\!05}$|  |$({4.65\text{e}\!-\!05})$|
|$\overline{\tau }$||${5.54\text{e}\!+\!02}$||${2.82\text{e}\!+\!03}$||${2.87\text{e}\!+\!04}$||${1.12\text{e}\!+\!05}$|
|${2.60\text{e}\!+\!03}$||${9.74\text{e}\!+\!03}$||${7.30\text{e}\!+\!04}$||${2.55\text{e}\!+\!05}$|
|${2.36\text{e}\!+\!03}$||${7.67\text{e}\!+\!03}$||${4.67\text{e}\!+\!04}$||${1.47\text{e}\!+\!05}$|
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