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**Contents:**

SSJ: A Java library for stochastic simulation. Efficient simulation of gamma and variance-gamma processes Avramidis , L'ecuyer , Tremblay.

Scrambled net variance for integrals of smooth functions Art B. Random number generation and quasi-monte carlo methods H. Variance with alternative scramblings of digital nets Art B. Related Papers. Novak and H. Entacher, W. Schmid, A. Uhl 62 , Novak Journal of Complexity 19 , Hickernell and R. Yue Math.

Fang, F. Hickernell, H. Niederreiter Eds. Notes Comput. Novak, G.

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Niederreiter and J. Spanier, editors , pp. Complexity 14, No. Petersburg Workshop on Simulation, S. Ermakov, Y. Kashtanov and V. Melas, editors, pp. Frank Niederreiter, Harald ed. Proceedings of a conference at the University of Salzburg, Austria, July , Notes Stat. Frank J. Complexity 12, No. Frank and S. Pereverzev J.

Variance reduction for Monte Carlo methods by means of deterministic numerical computation. Monte Carlo Methods Appl. Frank Schock, Eberhard ed. Helmut Brakhage zu Ehren. Figure 3 a shows pseudo-random numbers sampled from a uniform distribution in the unit square. Figure 3 b shows the same number of points generated by using a Sobol sequence. It can be observed that the sampling space is filled in a more uniform manner in figure 3 b. Figure 3 c , d show, respectively, the spatial distribution of points with pseudo-randon numbers generation and Sobol sequences.

Various pseudo-random and Sobol sequences sampling over the unit square. Although there are currently many researchers using MLQMC, there are still very limited works most of them still in press in the literature [ 39 , 42 , 43 ].

6. Monte Carlo Simulation

In order to obtain unbiased estimators for the variances, we need to induce some randomness to the QMC points, this process is known as QMC randomization. There are several ways of QMC randomization, depending on the type of low-discrepancy sequence used.

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In this study, we use the digital scrambling technique described in [ 44 ]. The procedure followed for conducting the experiments is as follows: firstly, we check empirically from which level i. The three tolerances employed for all the comparisons are: 0. This enables one to get a minimal level of resolution of the problem [ 2 , 3 ].

The conditions of theorem 3. The dominant cost will rely on the PDE solution, and an algebraic multi-grid method is used as the iterative linear solver. As could be expected from similar works in the field and after reviewing the theory related to both methods, the MLMC method clearly outperforms the standard MC.

The MC results are given in table 4.

The last row of the first column shows the level at which the code stops. Performance plots for the expectation in the MLMC method. Performance plots for the variance in the MLMC method. In this section, we compare the performance of MC and QMC methods for the same tolerances used in the previous sections.

In this case, low-discrepancy sequences clearly outperform pseudo-random for all the tolerances. The reduction rate achieved at this level is 9. This could indicate that after the discretization error has been adequately reduced, and consequently, a fine resolution of the QoI is being obtained in each simulation, there is not much additional gain by reducing the sample variance or sampling error. The latter can be also deduced from figure 9 , where after level 4 or tolerance 0.

These results are within the logic of deterministic sequences generation, and they seem to be as one could expect a direct consequence of the ordered deterministic way in which the MLQMC estimator is built. The overall picture with the performance of all the methods is shown in figure 9.

The analysis was focused on employing the four methods to solve, under the same conditions, a stochastic model defined in a high-dimensional probability space, and in comparing the computational costs incurred by the four different approaches. The improvements were related to the use of low-discrepancy Sobol sequences for the space filling design QMC and variance reduction in the multi-grid schemes MLMC. On the other hand, in cases where the uncertain parameters are not smooth enough e.

In this case, the use of unbiased randomized QMC estimators as the one used in the MLQMC method might be an alternative, although this would lead to a loss of the deterministic control offered by the standard QMC.

A description of such randomized QMC methods is provided in [ 5 ]. We provided a detailed comparison of the accuracy and efficiency between the different methods. From the numerical results obtained in the model problem studied in this paper, the QMC and MLMC methods provided the same order of accuracy that the classical MC with considerably less computational runs. The combination of both methods led to the MLQMC method, which was proved to provide the optimal computational effort for the simulator while retaining the same accuracy in the calculations.

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