Quadrature multidimensional integrals

Clearly, in the stochastic collocation technique, unlike in the Galerkin s method, one does not require transforming the original equations into any other form. Instead, the focus is on evaluating the multidimensional integrals. An inspection of Eq. 30 reveals that these integrals can be evaluated using suitable quadrature rules. 

As it was seen in the previous section, numerical evaluation of multidimensional integrals using quadrature rules is not computationally feasible for high dimensions. In such situations, Monte Carlo simulations(MCS) provide a method for... 

In this section, the accuracy of the sparse grid collocation technique for computing numerically multidimensional integrals is examined. A test function is examined whose exact numerical integral is available. For the one-dimensional quadrature rules used in the Smolyak s algorithm, only the nested quadrature rules, namely, the Clenshaw-Curtis(CC) rule and the Gauss-Patterson(GP) rule, have been used. 

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