## 2 Monte Carlo Algorithms Based on Modified Sobol' Sequences
\begin{array}{rcl}\mathbf{V}[f(\xi)&+&f(\xi^{\prime})]=\\ &=&\mathbf{V}\biggl{\{}\frac{1}{2}\Bigl{[}(\xi-\mathrm{s})^{T}[V^{2}f(\eta) ](\xi-\mathrm{s})+(\xi^{\prime}-\mathrm{s})^{T}[V^{2}f(\eta^{\prime})](\xi^{ \prime}-\mathrm{s})\Bigr{]}\biggr{\}}\\ &\leq&\mathbf{E}\Bigl{\{}\frac{1}{2}\Bigl{[}(\xi-\mathrm{s})^{T}[V^{2}f(\eta )](\xi-\mathrm{s})+(\xi^{\prime}-\mathrm{s})^{T}[V^{2}f(\eta^{\prime})](\xi^{ \prime}-\mathrm{s})\Bigr{]}\Bigr{\}}^{2}.\end{array} \]  
Given that \(f\in\mathbf{W}^{2}(L;U^{d})\), it is possible to enhance the last inequality by substituting the terms \([V^{2}f(\eta)]\) and \([V^{2}f(\eta^{\prime})]\) with the seminorm \(L\) (while removing the front bracket) and replacing the products \((\xi-\mathrm{s})^{T}\ (\xi-\mathrm{s})\) and \((\xi^{\prime}-\mathrm{s})^{T}\ (\xi^{\prime}-\mathrm{s})\) with the squared diameter of the sub-domain \(K_{j}\). Now, let us return to the notation with superscripts, bearing in mind that the preceding considerations apply to an arbitrary sub-domain \(K_{j}\). The variance can be estimated as follows:  
\[\mathbf{V}[f(\xi)+f(\xi^{\prime})] \leq L^{2}\sup_{x_{1}^{(j)},x_{2}^{(j)}}\left|x_{1}^{(j)}-x_{2}^{(j)} \right|^{4}\leq L^{2}(c_{2}^{(j)})^{4}n^{-4/d}.\]  
Therefore, the variance of \(\theta_{n}=\sum_{j=1}^{n}\theta^{(j)}\) can be estimated as follows:  
\[ \begin{array}{rcl}\mathbf{V}\theta_{n}=\sum_{j=1}^{n}p_{j}^{2}\mathbf{V}[f( \xi)+f(\xi^{\prime})]&\leq&\sum_{j=1}^{n}(c_{1}^{(j)})^{2}n^{-2}L^{2}(c_{2}^{(j )})^{4}n^{-4/d}\\ &\leq&\left(Lc_{1}^{(j)}c_{2}^{(j)2}\right)^{2}n^{-1-4/d}.\end{array}  \tag{15}\]  
Thus, \(r(f,d)\leq\widetilde{c}_{d}^{\prime\prime}\|f\|n^{-\frac{1}{2}-\frac{2}{d}}\). Following [PERSON]'s inequality to the variance (15) leads to the following estimation  
\[\varepsilon(f,d)\leq\widetilde{c}_{d}^{\prime}\|f\|n^{-\frac{1}{2}-\frac{2}{d}}\]  
for the probable error \(\varepsilon\), where \(\widetilde{c}_{d}^{\prime}=\sqrt{2d}\), which concludes the proof. It is noteworthy that the Monte Carlo algorithm MCA-MSS-S possesses an optimal rate of convergence for functions with continuous and bounded second derivatives [9]. This signifies that the rate of convergence (\(n^{-\frac{1}{2}-\frac{2}{d}}\)) cannot be enhanced for the functional class \(\mathbf{W}^{2}\) within the class of randomized algorithms \(\mathcal{A}^{\mathcal{R}}\). Note that both the MCA-MSS and MCA-MSS-S have a single control parameter, namely the radius \(\kappa\) of the _sphere of shaking_. However, effectively utilizing this control parameter increases the computational complexity. The challenge arises because, after _shaking_, the random point might leave the multidimensional sub-domain. Consequently, after each such operation, one must verify whether the random point remains within the same sub-domain. Verifying if a random point lies within a given domain is computationally expensive when dealing with a large number of points. A minor modification to the MCA-MSS-S can help to overcome this difficulty. Instead of shaking, one can simply generate a random point \(\xi^{(j)}\in K_{j}\) uniformly distributed inside \(K_{j}\), and then take the symmetric point \(\xi^{(j)^{\prime}}\) according to the central point \(s^{(j)}\). This fully randomized approach simulates the MCA-MSS-S, but the _shaking_ occurs with different radii \(\kappa\) in each sub-domain. We call this algorithm SS-MCA as it resembles the stratified symmetrized Monte Carlo method [11]. Clearly, the SS-MCA is less computationally expensive than the MCA-MSS-S, but it lacks a control parameter like the radius \(\kappa\), which can be considered as a randomly chosen parameter in each sub-domain \(K_{j}\). It is important to note that all three algorithms, the MCA-MSS, MCA-MSS-S, and SS-MCA, have optimal rates of convergence for their respective functional classes. Specifically, the MCA-MSS is optimal in \(\mathbf{W}^{1}(L;U^{d})\), while both the MCA-MSS-S and SS-MCA are optimal in \(\mathbf{W}^{2}(L;U^{d})\).