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11.4 Sparse Grid Approximation for High Dimension/High Polynomial Order

As noted above, the number of grid points needed for the Gauss–Hermite filter is given by

(11.95)

where m is the order of the Hermite polynomial used in the approximation and nx is the dimension of the state vector. As the dimension of the state vector increases, the number of Sigma points required grows exponentially resulting in the “curse of dimensionality.” The fully populated square grid of Sigma points becomes so large at dimensions above 4 that the GHKF is rarely used. In the previous section of this chapter, we considered Hermite expansions of second order (m = 3) that required 3nx sigma points. The only way to reduce the number of points for this case is to use the UKF instead of the GHKF at a small cost in accuracy.

As noted just before (11.30), the Hermite interpolation method can be expanded to higher order. Of course, higher order interpolation requires more sigma points making the curse of dimensionality even worse. Because higher order Hermite interpolation requires the solution to polynomial equations of the same order as the Hermite polynomial, numerical methods are generally used to generate the eigenvalues, resulting in weights and sigma points that are irrational numbers vice the rational numbers shown in (11.48). It is these irrational numbers that are tabulated in such books as Ref. [7]. ...

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