Multivariate EQMOM

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 

In order to be consistent with the univariate EQMOM discussed in Section 3.3.2, the diagonal components of H are determined from the pure moments of order 2n (e.g. in 2D phase space m(2n,0) and m 0,2n) are used to find (Th and 0-22, respectively). The ofF-diagonal components of H can then be determined by using the cross moments of total order 2n, but of order n in the i and j components. For example, in 2D phase space, the moment m(n, n) is used to find 0-12, and in 3D phase space the moment m n, 0, n) is used to And 0-13. The reader can verify that none of these moments is contained in the optimal moment set and, hence, they are available for use in the extended optimal moment set. The multivariate moments m(k) computed from Eq. (3.124) can be written as 

In these expressions, we have used the index notation i = (fi./ ) such that = Note that, in the limit where PI 0, only the zeroth-order moment mi (0) = 1 is nonzero, and Eq. (3.126) reverts to the QMOM moments. The central Gaussian moments m-z(i) are known functions of the covariance matrix E. For a given moment order 7 = ki + + kM, Eq. (3.126) has a lower triangular form that can be inverted using forward substitution  

The algorithm for computing H is analogous to the one used in the univariate case. Using a root-finding (or minimization) scheme, we must vary the components of E until the objective function is minimized. The components of the objective function found from the extended optimal moments are 

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The application of multivariate EQMOM to evaluate integrals is very similar to that of univariate EQMOM. For the multi-Gaussian EQMOM, we have... 

As a final case, we will again consider the joint velocity-scalar NDF governed by Eq. (8.122), but without specifying a functional form for the scalar-conditioned velocity. The moment-transport equation is again given by Eq. (8.122). However, we will now use a scalar-conditioned multivariate EQMOM to reconstruct the joint NDE ... 

Even when multivariate EQMOM is used with kernel density functions, a Dirac delta function (dualquadrature) representation is employed to close the terms in the GPBE. See Section 3.3.4 for more details. 

In many systems (e.g. those involving to collisions), the velocity NDE will be continuous and thus will not be well represented by an A -point distribution function. In such cases, in order to have a more accurate representation of the spatial fluxes, the multivariate EQMOM described in Section 3.3.4 can be used to reconstruct the NDF nf.fiiy) from the transported moment set M . Note that, unlike with regular quadrature, this NDF will be a continuous (known) function of v. Nevertheless, for evaluating the spatial fluxes in Eq. (B.35), it will still be advantageous to construct a regular quadrature using the moments of n ifiy). As... 

To conclude this section, we discuss three technical points that arise when applying the realizable high-order scheme for free transport. The first point is how to choose the value of N given that iV = is used for multivariate EQMOM. To answer this question, we first note that the highest-order transported moment in any one direction is of order 2n (which corresponds to 2n -t-1 pure moments in any one direction). The extra (even-order) moment is used in the EQMOM to determine the spread of the kernel density functions (see Section 3.3.2 for details). When free transport is applied to the moment of order 2n, the flux function involves the moment of order 2n -i- 1. Therefore, in order to exactly predict the flux function for free transport using the half-moment sets, we must have fV > (n-i-1). The obvious choice for N is thus fV = (n -i- 1). ... 

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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