## 2 Monte Carlo Algorithms Based on Modified Sobol' Sequences  
An algorithm following a _shaking_ technique was proposed recently in [40]. This concept involves taking a Sobol' \(\Lambda\Pi_{\tau}\) point x of dimension \(d\). The point is treated as the center of a sphere with a radius \(\kappa<1\). A random point \(\xi\in U^{d}\) is selected uniformly on the sphere. A random variable \(\theta\) is defined as a value of the corresponding integrand at that random point, i.e., \(\theta=f(\xi)\). Next, the random points \(\xi^{(i)}(\kappa)=\text{x}^{(i)}+\kappa\omega^{(i)}\in U^{d},i=1,\ldots,n\) are chosen, where \(\omega^{(i)}\) is a unique uniformly distributed vector in \(U^{d}\). The radius \(\kappa\) is relatively small \(\kappa\ll\frac{1}{2^{d}}\), so \(\xi^{(i)}(\kappa)\) remains within the same elementary \(i^{th}\) interval \(E_{i}^{d}=\prod_{j=1}^{d}\left[\frac{a_{j}^{(i)}}{2^{d}},\frac{a_{j}^{(i)}+1} {2^{d}j}\right]\), where the pattern \(\Lambda\Pi_{\tau}\) point \(\text{x}^{(i)}\) resides. Using a subscript \(i\) in \(E_{i}^{d}\) indicates that the \(i\)-th \(\Lambda\Pi_{\tau}\) point \(\text{x}^{(i)}\) is within it. Thus, if \(\text{x}^{(i)}\in E_{i}^{d}\), then \(\xi^{(i)}(\kappa)\in E_{i}^{d}\) too. It was proven in [40] that the mathematical expectation of the random variable \(\theta=f(\xi)\) coincides with the value of the integral (3); that is, \(\mathbf{E}\theta=S(f)=\int_{\text{J}\text{J}^{d}}f(\kappa)d\kappa\). It allows for defining a randomized algorithm. One can take the Sobol' \(\Lambda\Pi_{\tau}\) point \(\text{x}^{(i)}\) and _slake_ it somewhat. _Shaking_ means to define random points \(\xi^{(i)}(\kappa)=\text{x}^{(i)}+\kappa\omega^{(i)}\) according to the procedure described above. For simplicity, this algorithm is abbreviated as MCA-MSS. The probability error of the MCA-MSS was examined in [2]. It was demonstrated that, for integrands with continuous and bounded first derivatives, specifically \(f\in\mathbf{W}^{1}(L;U^{d})\), where \(L=\|f\|\), the following relationship applies:  
\[err(f,d)\leq c_{d}^{\prime}\|f\|n^{-\frac{1}{2}-\frac{1}{d}}\quad\text{and} \quad r(f,d)\leq c_{d}^{\prime\prime}\|f\|n^{-\frac{1}{2}-\frac{1}{d}},\]  
where the constants \(c_{d}^{\prime}\) and \(c_{d}^{\prime\prime}\) do not depend on \(n\). Here, a modification of the MCA-MSS is proposed and analyzed. The new algorithm will be called MCA-MSS-S. It is assumed that \(n=m^{d}\), \(m\geq 1\). The unit cube \(U^{d}\) is divided into \(m^{d}\) disjoint sub-domains, such that they coincide with the elementary \(d\)-dimensional subintervals defined in Section 1.2\(U^{d}=\bigcup_{j=1}^{m^{d}}K_{j}\), where \(K_{j}=\prod_{i=1}^{d}[a_{i}^{(j)},b_{i}^{(j)})\), with \(b_{i}^{(j)}-a_{i}^{(j)}=\frac{1}{m}\) for all \(i=1,\ldots,d\). Thus, in each \(d\)-dimensional sub-domain \(K_{j}\), there is precisely one \(\Lambda\Pi_{\tau}\) point \(\mathrm{x}^{(j)}\). If the random point remains within \(K_{j}\), i.e., \(\xi^{(j)}(\kappa)=\mathrm{x}^{(j)}+\kappa\omega^{(j)}\in K_{j}\) after _shaking_, one may attempt to use the smoothness of the integrand \(f\) if \(f\in\mathbf{W}^{2}(L;U^{d})\). Then, if \(p(x)\) is a probability density function, such that \(\int_{U^{d}}p(\mathrm{x})\mathrm{d}\mathrm{x}=1\), then  
\[\int_{K_{j}}p(\mathrm{x})\mathrm{d}\mathrm{x}=p_{j}\leq\frac{c_{1}^{(j)}}{n},\]  
where \(c_{1}^{(j)}\) are constants. If \(d_{j}\) is the diameter of \(K_{j}\), then  
\[d_{j}=\sup_{x_{1},x_{2}\in K_{j}}|x_{1}-x_{2}|\leq\frac{c_{2}^{(j)}}{n^{1/d}},\]  
where \(c_{2}^{(j)}\) are other constants. In the particular case when the subintervals are with edge \(1/m\) for all constants, we have \(c_{1}^{(j)}=1\) and \(c_{2}^{(j)}=\sqrt{d}\). In each sub-domain \(K_{j}\), the central point is denoted by \(\mathrm{s}^{(j)}\), where \(\mathrm{s}^{(j)}=(s_{1}^{(j)},s_{2}^{(j)},\ldots,s_{d}^{(j)})\). Let us say we choose two points, \(\xi^{(j)}\) and \(\xi^{(j)^{\prime}}\).