Multivariate quadrature DQMOM

The extension of the DQMOM to bivariate and multivariate systems uses a quadrature approximation of order N. The simple case of a bivariate distribution uses... 

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... 

There are undoubtedly other examples for which the DQMOM fails but the QMOM works. However, as a general rule, the more dominant the diffusion component of the spatial fluxes, the more likely it is that the abscissas will exhibit smooth variations in space and the DQMOM will be applicable. In cases for which the DQMOM is applicable, the problem of realizable moment sets is fairly simple to address because the only requirement is that the weights be nonnegative. Nevertheless, it is still necessary in multivariate cases to verify that the abscissas are physically realizable, i.e. they must take on values in the phase space of the internal coordinates (see Section 8.1.1). With the DQMOM (as with the QMOM), there is no guarantee that the solutions will represent realizable quadratures for a multivariate NDF. 

Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

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