Moment-inversion algorithms

Four necessary properties of the moment-inversion algorithm are the following. 

The moment-inversion algorithm described above for a Gaussian distribution can be applied for other choices of 6a- (e.g. for NDF defined on a semi-infinite or a finite interval). The first step is to compute the integer moments of Saix, y) with respect to x analytically. For example, with the definition in Eq. (3.84) for x [0, oo) (Yuan et al, 2012), these integer moments with y = are... 

In this expression, the weights and abscissas are found from the moments at every time step and at every spatial location using a moment-inversion algorithm. Following the method described in Marchisio Fox (2005), the DQMOM equations corresponding to Eq. (3.143) are... 

The moment-inversion algorithms for bivariate and multivariate problems are discussed in Section 3.2. 

When the first procedure is adopted, the CQMOM should be used as the moment-inversion algorithm (Yuan Fox, 2011). When the second procedure is adopted, the evolution of the M-dimensional, iV-point quadrature approximation is directly tracked. In the case of spatially homogeneous systems the two methods are equivalent. 

The quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM) were introduced in Chapter 3 as equivalent methods for solving a homogeneous GPBE. In fact, the DQMOM was derived by Marchisio Fox (2005) primarily for the purpose of solving spatially inhomogeneous multivariate moment-transport equations. Unlike for the univariate case, where the moment-inversion algorithm is uniquely defined for a given set of moments, the QMOM in the multivariate case is much... 

In this appendix Matlab scripts for the moment-inversion algorithms discussed in Chapter 3 are reported. 

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