Tensor-product QMOM

The construction of the A -point quadrature approximation requires the definition of the following abscissa basis vectors  

There are M different vectors composed of N, N2, , Nm components, with Dirac delta functions centered on the univariate abscissas (in each of the M directions) corresponding to the basis function set used to approximate the functional form of the multivariate NDF. The final set of N multivariate abscissas is obtained by using the following tensor prod- 

Exercise 3.6 Consider a bivariate distribution (M = 2) with two internal coordinates and 2, and let us construct a four-point quadrature approximation, resulting from univariate quadratures of order N = N2 = 2. Knowledge of the first, 2N = 2N2 = 4, pure moments with respect to the first, f, and second, 2, internal coordinates, suffices for 

Nm equations can be written by forcing agreement with the univariate + Nm - 1) are linearly independent. 

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

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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 =...

Since for velocity distributions (in three spatial dimensions) three internal coordinates (M = 3) are needed, we discuss in the following example the construction of a quadrature approximation for a trivariate tensor-product QMOM. 

Exercise 3.7 Consider a trivariate distribution (M = 3) of three internal coordinates i, 2, and fs. Let us construct an eight-point tensor-product QMOM resulting from univariate... 

This eight-point tensor-product QMOM fixes the moment of order zero with respect to all the internal coordinates, nine pure moments plus four mixed moments, for a total of 14 moments. The highest-order moment accommodated by this particular choice is mi,i,i of global order 7 = 3. Comparison of this moment set with the optimal moment set reported in Table 3.7 for M = 3 and N = 8 clearly shows that is a subset of the 32 optimal moments. 

To conclude this section on tensor-product QMOM, it is important to highlight that, although it is not necessary, the formulation of the problem in terms of translated (i.e. centered on the mean) and rotated (i.e. with diagonal covariance matrix) internal coordinates can be advantageous. In fact, if a change of variables is implemented so that the distribution is rewritten with respect to its principal coordinates, the calculations for the derivation of the quadrature approximation are simplified. These concepts will be illustrated in Exercise 3.8. A Matlab script implementing a tensor-product QMOM can be found in Section A.3.2 of Appendix A. 

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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