# Nonlinear Estimation

## 2 Literature review on quadrature based nonlinear estimation (cont'd...)

### Smolyak's rule (cont'd...)

The final set of the SGQ poits are given as
\begin{equation}
X_{n,L}=\bigcup_{q=L-n}^{L-1}\bigcup_{\Xi \in \textbf{N}_q^n}(X_{l_1}\otimes X_{l_2} \otimes...\otimes X_{l_n})
\end{equation}
where $\bigcup$ represents union of the individual SGQ points.

In [Gerstner 1998],
multivariate quadrature formulas based on sparse grids using Newton-Cotes,
Clenshaw-curtis, Gauss and Gauss-Patterson formulas were discussed. Comparison of these methods based on
computational cost and error bounds were done. It was found out that Gauss-Patterson method performed best
in comparison to other methods. Sparse-grid Quadrature Filter for nonlinear filtering problems was introduced
in [Jia 2012a].
Univariate quadrature integrals were solved using moment matching methods and then using
sparse-grid theory, these were extended for multi-dimensional systems, *i.e* filter uses weighted
sparse-grid quadrature points to evaluate the multi-dimensional integrals in the nonlinear Bayesian estimation algorithm.

This algorithm gives flexibility in defining the accuracy level of estimation and significantly reduces the computational cost of the algorithm.
Number of sparse-grid quadrature points is a polynomial of the dimension of the system. Hence, *curse of dimensionality problem*
of GHF is alleviated in this algorithm. A comparison between sparse-grid quadrature filter and cubature kalman filter was studied in
[Jia 2012b]. This study defined the relation between the two filters. It was found out that the projection of
sparse-grid quadrature rule results in arbitrary degree cubature rules.
Application of SGHF to various real time problems was studied in [Jia 2010]
[Jia 2012c].

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