## 1 Introduction: Problem Setting
### Randomized Quasi-Monte Carlo (RQMC)  
Instead of employing randomized (Monte Carlo) algorithms for computing the mentioned sensitivity parameters, one can consider deterministic quasi-Monte Carlo algorithms or randomized quasi-Monte Carlo methods [18; 19]. Randomized (Monte Carlo) algorithms have proven highly efficient in solving multidimensional integrals over composite domains [9; 11]. Simultaneously, quasi-Monte Carlo (QMC) methods based on well-distributed Sobol' sequences present a compelling alternative to Monte Carlo algorithms, particularly for smooth integrands and relatively low effective dimensions (up to \(d=15\)) [20; 21; 22]. Sobol' \(\Lambda\Pi_{\tau}\)-sequences are strong contenders for efficient QMC algorithms. Although these algorithms are deterministic, they emulate the pseudo-random sequences used in Monte Carlo integration. A significant limitation of \(\Lambda\Pi_{\tau}\)-sequences is their potential for suboptimal two-dimensional projections, implying that the distribution of points can deviate substantially from uniformity. If the computational problem involves such projections, the lack of uniformity could result in a considerable loss of accuracy. To mitigate this issue, randomized QMC methods can be employed. Several randomization techniques exist, with scrambling being a notable example. The primary motivation behind scrambling [6; 23]was to enhance the uniformity of quasi-random sequences in high dimensions, as assessed through two-dimensional projections. Moreover, scrambling offers a simple and unified way to generate quasi-random numbers for parallel, distributed, and grid-based computational environments. Essentially, scrambled algorithms can be regarded as Monte Carlo algorithms with a specific choice in the density function. Thus, it is logical to compare two classes of algorithms: _deterministic_ and _randomized_. Various versions of scrambling methods exist, based on digital permutations, and their differences lie in the definitions of the permutation functions. Examples include [PERSON]'s nested scrambling [6; 24], [PERSON]'s generalized Faure sequences [25], and [PERSON]'s linear scrambling [26]. Following the introduction of Niederreiter sequences [13], [PERSON] [6] and [PERSON] [25] independently developed two influential scrambling methods for \((t,s)\)-sequences in 1994. [PERSON] specifically highlighted that scrambling can be employed to generate error estimates for quasi-Monte Carlo (QMC) methods. Numerous other techniques for scrambling \((t,s)\)-sequences have since been proposed, many of which represent modifications or simplifications of the original [PERSON] and [PERSON] schemes. [PERSON]'s method is particularly effective for \((t,s)\)-sequences, whereas the [PERSON] algorithm was shown to be efficient for \((0,s)\)-sequences. Most existing scrambling methods involve randomizing a single digit at a time. In contrast, the approach presented in [27] randomizes multiple digits within a single point simultaneously, offering enhanced efficiency when utilizing standard pseudo-random number generators as scramblers. [PERSON]'s scrambling [6], also known as nested scrambling, was developed to provide a practical error estimate for QMC by treating each scrambled sequence as a separate and independent random sample from a family of randomly scrambled quasi-random numbers. Let \(x^{(i)}=(x_{i,1},x_{i,2},\ldots,x_{i,s})\), \(i=1,\ldots,n\) be quasi-random numbers in \([0,1)^{s}\), and let \(z^{(i)}=(z_{i,1},z_{i,2},\ldots,z_{i,s})\) be the scrambled version of the point \(x^{(i)}\). Suppose that each \(x_{i,j}\) can be represented in base \(b\) as \(x_{i,j}=(0.x_{i1,j}\ x_{i2,j}\ldots x_{iK,j}\ldots)_{b}\), with \(K\) being the number of digits to be scrambled.