Univariate quadrature

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... 

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 =...

The weights w can be calculated by solving a linear system, where the first N +(N2-l) + (A 3 - 1) = 4 equations are obtained by forcing agreement with the univariate quadratures ... 

The construction of the multivariate quadrature begins with the calculation of the univariate quadrature of order N for the first internal coordinate by using the Wheeler algorithm with the first 2N - 1 moments ... 

By using the definition reported in Eq. (3.61) and exploiting the univariate quadrature over the first internal coordinate, the generic mixed moment of order k = [ki,lc2,..., Icm-i, can be calculated as... 

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). 

Nested Univariate Quadrature Ruies Some of the quadrature schemes have sets of nodal points which are nested. This means the set of nodal points at i + 1) level, X,+i, includes all points in the setX,. Symbolically this is written as X, C X,+i. Therefore, while moving to a higher level, only the points in the difference set, X,+i — X need to be considered for function... 

Even though the computational cost increases with the number of periods, this increase is not significant compared to the computational cost added by the number of scenarios considered in the stochastic formulation. For that reason, a decomposition scheme is applied to tackle medium size problems. It is envisaged that the use of parallel computing platforms can help to reduce the time required for the solution of this kind of problems. Also, note that the use of multivariate/univariate quadrature rules to approximate the performance expectation are an alternative to reduce the number of scenarios while maintaining the solution accuracy (Lafnez-Aguirre and Reklaitis 2013). 

The theory of Gaussian quadrature applies only to univariate distributions. However, in practical cases, the study of distributions with multiple internal coordinates is often necessary. In these cases the closure problem generally assumes the following form ... 

As has already been mentioned, Eq. (3.36) is not a Gaussian quadrature approximation its degree of accuracy is not known a priori (and strongly depends on the choice of moments on which the formula is constructed) and the algorithms for its derivation, from the moments of the NDF, are not well known (unlike for the univariate case). 

The Gaussian quadrature algorithm introduced in Section 3.1.1 is equivalent to approximating the univariate NDF by a sum of Dirac delta functions ... 

By extending the concepts of the previous section, it is possible to derive an approximation (which is no longer a real Gaussian quadrature) for the bivariate system. The resulting Appoint quadrature approximation transforms the definition of the general bivariate moment into (Wright et at, 2001b) m = Za=i where, as for the univariate case, Wa... 

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... 

The multivariate GHQ rule extends the univariate /n-point set to the n-dimen-sional point set by the tensor product mle [29, 30]. It is exact for all polynomials of the form xj X2 - x with ltotal number of points Np = increases exponentially with the dimension n. Hence, it is hard to use for high dimensional problems. To alleviate this problem, the sparse Gauss-Hermite quadrature can be used [32]. In this paper, the conventional Gauss-Hermite quadrature is used since the dimension of this problem is three. 

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