Moment set

We now complete the relation between our new formalism and the conventional expression given in Eq. (65). First note that the term with k = 0 in gives a contribution which is the same for all states. We shall usually set o = 0, which means in fact that the centre of gravity of the whole set is zero, because we are only interested in relative energies. Also, for the moment, set = 0 for k > 2. Then writing... 

The concept of realizability or consistency of a moment set will be introduced in Section 3.1.4. 

Theorem 3.4 A moment set is said to be realizable if the Hankel-Hadamard determinants (Gautschi, 2004 Shohat G Tamarkin, 1943) are all non-negative ... 

It is now sufficient to modify the second-order moment to 25 (instead of 26, corresponding to a difference of only 4%) to make the moment set unrealizable. The Hankel-Hadamard determinants are now equal to -179352 and -12 362 344, respectively, whereas the difference table (see Table 3.2) presents negative elements in the column containing the second-order differences (namely A2). If this moment set is fed to the PD algorithm, the resulting quadrature is unable to reproduce the moment set. 

In general, simulations carried out starting from realizable moment sets should result in realizable moment sets. Nevertheless, the moment-transport equations are integrated numerically with some finite discretization errors. As Wright (2007) clearly reports, most of the problems are caused by the approximation of the convective term, in particular with higher-order discretization schemes, which can turn a realizable set of moments into... 

Exercise 3.4 Use the moment-correction algorithm to modify the following moment set ... 

In this case the moments that have been changed, which are responsible for the invalidity of the set, are mi and mo. When this moment set is fed to the correction algorithm, the following answer is obtained ... 

It can be seen that all the moments have been changed and, although the final moment set is realizable, it is not the original one. 

This second correction algorithm consists of calculating the new moment set from those corresponding to log-normal distributions. A log-normal distribution has the following functional form ... 

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. 

An optimal moment set includes all linearly independent moments of a particular order y before adding moments of higher order in order to result in a perfectly symmetric treatment of the internal coordinates. 

The last property ensures that all the lower-order mixed moments are included in the set. This turns out to be very important since generally neglecting mixed moments leads to abscissas that lie on lower-dimensional subspaces of the M-dimensional phase space. Whenever the dynamics of the investigated problem is not confining the abscissas on lower-dimensional supports, it is appropriate to choose a moment set to define A that is not restricted to generating such behavior. Therefore, it is suggested for many applications that one should use moment sets that treat all the M internal coordinates equally. If we relax this third condition a valid (but not optimal) moment set is instead obtained. 

The methodology proposed for finding optimal moment sets for a given M and N is as follows. 

The procedure outlined above is applied for a given value of A > 2. If it fails, then no optimal moment set can be found for that value of N, so the procedure must be repeated with the next larger N. 

Table 3.5. The optimal moment set used to build a bivariate quadrature approximation (M = 2) for At = 4...

At = 4) for building a bivariate quadrature approximation (M = 2), the optimal moment set constituted by 12 moments and reported in Table 3.5 is obtained. It can be seen that the moment set is symmetric with respect to the two internal coordinates and all the lower-order mixed moments are included in the set, resulting in a non-singular Jacobian matrix. Another optimal moment set of 27 moments is identified for M = 2 when using At = 9 as reported in Table 3.6. 

The system formed by Eqs. (3.48)-(3.51) can be solved to find the weights. It can be seen that the four-point bivariate quadrature approximation accommodates eight moments, namely the moment of order zero with respect to both the internal coordinates, six pure moments, and one mixed moment. Table 3.8 reports in matrix form these eight moments, and comparison with Table 3.5 clearly shows that this is a subset of the optimal moment set. 

This eight-point tensor-product QMOM fixes the moment of order zero with respect to all the internal coordinates, nine pure moments plus four mixed moments, for a total of 14 moments. The highest-order moment accommodated by this particular choice is mi,i,i of global order 7 = 3. Comparison of this moment set with the optimal moment set reported in Table 3.7 for M = 3 and N = 8 clearly shows that is a subset of the 32 optimal moments. 

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