## 2 Monte Carlo Algorithms Based on Modified Sobol' Sequences
The point \(\xi^{(j)}\) is selected during our MCA-MSS procedure, while \(\xi^{(j)^{\prime}}\) is chosen to be symmetrical to \(\xi^{(j)}\) with respect to the central point \(\mathrm{s}^{(j)}\) in each cube \(K_{j}\). This approach results in a total of \(2m^{d}\) random points. We can compute all function values \(f(\xi^{(j)})\) and \(f(\xi^{(j)^{\prime}})\) for \(j=1,\ldots,m^{d}\) and approximate the value of the integral as follows:  
\[I(f)\approx\frac{1}{2m^{d}}\sum_{j=1}^{2n}\Bigl{[}f(\xi^{(j)})+f(\xi^{(j)^{ \prime}})\Bigr{]}. \tag{11}\]  
This estimate corresponds to the MCA-MSS-S. We will demonstrate later that this algorithm exhibits an optimal rate of convergence for functions with bounded second derivatives, i.e., for functions \(f\in\mathbf{W}^{2}(L;U^{d})\), whereas the MCA-MSS has an optimal rate of convergence for functions with bounded first derivatives: \(f\in\mathbf{W}^{1}(L;U^{d})\). One can prove the following:  
Theorem 3.1: _The quadrature Formula (11) constructed above for integrands \(f\) from \(\mathbf{W}^{2}(L;U^{d})\) satisfies_  
\[err(f,d)\leq\tilde{c}_{d}^{\prime}\|f\|n^{-\frac{1}{2}-\frac{2}{d}}\]_and_  
\[r(f,d)\leq\widetilde{c}_{d}^{\prime\prime}\|f\|n^{-\frac{1}{2}-\frac{2}{d}},\]  
_where the constants \(\widetilde{c}_{d}^{\prime}\) and \(\widetilde{c}_{d}^{\prime\prime}\) do not depend on \(n\)._  
Proof of Theorem 1.: One can see that  
\[\mathbf{E}\left\{\frac{1}{2m^{d}}\sum_{j=1}^{2n}\Bigl{[}f(\xi^{(j)})+f(\xi^{(j ^{\prime})})\Bigr{]}\right\}=\int_{U^{d}}f(\mathrm{x})d\mathrm{x}.\]  
For the fixed \(\Lambda\Pi_{\tau}\) point \(\mathrm{x}^{(j)}\in K_{j}\), one can use the \(d\)-dimensional Taylor formula to present the function \(f(\mathrm{x}^{(j)})\) in \(K_{j}\) around the central point \(\mathrm{s}^{(j)}\). Since \(f\in\mathbf{W}^{2}(L;U^{d})\), there exists a \(d\)-dimensional point \(\eta^{(j)}\in K_{j}\) lying between \(\mathrm{x}^{(j)}\) and \(\mathrm{s}^{(j)}\) such that  
\[ \begin{array}{rcl}f(\mathrm{x}^{(j)})=f(\mathrm{s}^{(j)})&+&\
abla f( \mathrm{s}^{(j)})\ (\mathrm{x}^{(j)}-\mathrm{s}^{(j)})\\ &+&\frac{1}{2}(\mathrm{x}^{(j)}-\mathrm{s}^{(j)})^{T}\ [V^{2}f(\eta^{(j)})]( \mathrm{x}^{(j)}-\mathrm{s}^{(j)}),\end{array}  \tag{12}\]  
where \(\
abla f(\mathrm{x})=\left[\frac{\partial f(\mathrm{x})}{\partial\mathrm{x}_ {1}},\ldots,\frac{\partial f(\mathrm{x})}{\partial\mathrm{x}_{d}}\right]\) and \([V^{2}f(\mathrm{x})]=\left[\frac{\partial^{2}f(\mathrm{x})}{\partial \mathrm{x}_{i}\partial\mathrm{x}_{k}}\right]_{i,k=1}^{d}.\) To simplify matters, the superscript \((j)\) of the argument in the last two formulas is omitted, assuming that the formulas apply to the \(j^{th}\) cube \(K_{j}\). Now, we can rewrite Formula (12) at the previously defined random points \(\xi\) and \(\xi^{\prime}\), both located within \(K_{j}\). This yields  
\[f(\xi)=f(\mathrm{s})+\
abla f(\mathrm{s})\ (\xi-\mathrm{s})+\frac{1}{2!}(\xi- \mathrm{s})^{T}[V^{2}f(\eta)](\xi-\mathrm{s}), \tag{13}\]  
\[f(\xi^{\prime})=f(\mathrm{s})+\
abla f(\mathrm{s})\ (\xi^{\prime}-\mathrm{s})+ \frac{1}{2!}(\xi^{\prime}-\mathrm{s})^{T}[V^{2}f(\eta^{\prime})](\xi^{\prime} -\mathrm{s}), \tag{14}\]  
where \(\eta^{\prime}\) is another \(d\)-dimensional point lying between \(\xi^{\prime}\) and \(\mathrm{s}\). Adding Equations (13) and (14), we obtain  
\[ \begin{array}{rcl}f(\xi)+f(\xi^{\prime})=2f(\mathrm{s})&+&\frac{1}{2}\left\{ (\xi-\mathrm{s})^{T}[V^{2}f(\eta)](\xi-\mathrm{s})\ +\right.\\ &+&(\xi^{\prime}-\mathrm{s})^{T}[V^{2}f(\eta^{\prime})](\xi^{\prime}-\mathrm{s })\right\}.\end{array} \]  
Due to the symmetry, there is no term involving the gradient \(Vf(\mathrm{s})\) in the previous formula. If we examine the variance \(\mathbf{V}[f(\xi)+f(\xi^{\prime})]\), keeping in mind that the variance of the constant \(2f(\mathrm{s})\) is zero, we arrive at  
\[