Brute-force QMOM

Brute-force QMOM, or the direct-inversion method, has been proposed by Wright et al. (2001b) in the context of particle coalescence and sintering, and is referred to as a multivariate QMOM approximation. This method is based on the very simple idea of 

The system of N(M + 1) equations is then solved by using standard nonlinear solvers such as the Newton-Raphson method or the conjugate-gradient minimization algorithm, both of which are described in Press et al. (1992). 

Since the matrix has to be inverted (when using the Newton-Raphson method), it must be non-singular throughout the entire calculation.  

We define an optimal moment set for a given number of internal coordinates M to have the following properties (Fox, 2009b). 

An optimal moment set will result in a full-rank square matrix A for all possible sets of N distinct, non-degenerate abscissas. 

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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).

In principle, the EQMOM introduced in Section 3.3.2 can be generalized to include multiple internal coordinates. However, depending on the assumed form of the kernel density functions, it may be necessary to use a multivariate nonlinear-equation solver to find the parameters (i.e. similar to the brute-force QMOM discussed in Section 3.2.1). An interesting alternative is to extend the CQMOM algorithm described in Section 3.2.3. Here we consider examples using both methods. 

In the literature (Chalons et al, 2010), only a bivariate EQMOM with four abscissas represented by weighted Gaussian distributions with a diagonal covariance matrix has been considered. However, it is likely that brute-force QMOM algorithms can be developed for other distribution functions. Using the multi-Gaussian representation as an example, the approximate NDF can be written as... 

As has already been mentioned, this coefficient matrix, also indicated as A, corresponds to the Jacobian of the nonlinear system reported in Eq. (3.43) and employed in brute-force QMOM for calculating the quadrature approximation. 

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