Tensor-product QMOM bivariate

Using tensor-product QMOM, the final four-point quadrature approximation is centered on the following bivariate nodes ... 

Table 3.8. The moment set used to build a bivariate quadrature approximation (M = 2) for N = A with the tensor-product QMOM...

Figure 3.1. Positions in the internal-coordinate plane of the four nodes of the bivariate tensor-product QMOM (M = 2) obtained with two-point univariate quadratures N =...

Figure 3.5. Quadrature approximations for bivariate Gaussian distributions with = 10, 2 = 20, cTi = ct2 = 2, and p = 0 (top) and p = 0.5 (bottom) for // = 4 (left) and for N = 9 (right), namely brute-force QMOM (diamonds), tensor-product QMOM (circles) and CQMOM (squares).

Below a Matlab script implementing the tensor-product QMOM for a simple bivariate case described in this section is reported. The required inputs are the number of nodes for the first (Nl) and for the second (N2) internal coordinates. Since in the formulation described above the moments used for the calculation of the quadrature approximation are defined by the method itself, no exponent matrix is needed. The moments used are passed though a matrix variable m, whose elements are defined by two indices. The first one indicates the order of the moments with respect to the first internal coordinates (index 1 for moment 0, index 2 for moment order 1, etc.), whereas the second one is for the order of the moments with respect to the second internal coordinate. The final matrix is very similar to that reported in Table 3.8. The script returns the quadrature approximation in the usual form the weights are stored in the weight vector w of size N = Mi M2, whereas the nodes are stored in a matrix with two rows (corresponding to the first and second internal coordinate) and M = M1M2 columns (corresponding to the different nodes). 

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