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# Nonlinear Estimation

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

### Smolyak's rule

A numerical approximation for the following integral can be expressed as [Wasilkowski 1995] [Jia 2012a]. $$\begin{split} &I_{n,L}(f)=\int_{\Re^n}f(x)\aleph(x;0,I_n)\\&\approx \sum_{q=L-n}^{L-1}(-1)^{L-1-q}C^{n-1}_{L-1-q}\sum_{\Xi\in\textbf{N}_q^n}(I_{l_1} \otimes I_{l_2} \otimes...\otimes I_{l_n})(f) \end{split}$$ where $I_{n,L}$ represents the numerical evaluation of $n$-dimensional system with the accuracy level $L$. This means that the approximation is exact for all the polynomials having degree upto $(2L-1)$. $C$ stands for the binomial coefficient i.e. $C_k^n=n!/k!(n-k)!$. $I_{l_j}$ is the single dimensional quadrature rule with the accuracy level $l_j\in\Xi$ and $\Xi \triangleq (l_1,l_2,...,l_n)$, $\otimes$ stands for the tensor product and $N_q^n$ is set of possible values of $l_j$ given as

Above Equation can be written as \begin{align} \begin{split} I_{n,L}(f)&\approx \sum_{q=L-n}^{L-1}(-1)^{L-1-q}C^{n-1}_{L-1-q}\sum_{\Xi\in\textbf{N}_q^n}\sum_{q_{s_1}\in X_{l_1}}\sum_{q_{s_2}\in X_{l_2}} \\ &...\sum_{q_{s_n}\in X_{l_n}}f(q_{s_1},q_{s_2},...,q_{s_n})w_{s_1}w_{s_2}...w_{s_n} \end{split} \end{align}

where $X_{l_j}$ is the set of quadrature points for the single dimensional quadrature rule $I_{l_j}$, $[q_{s_1},q_{s_2},...,q_{s_n}]^T$ is a Sparse-grid quadrature (SGQ) point. $q_{s_j}\in X_{l_j}$ and $w_{s_j}$ is the weight associated with $q_{s_j}$. Some SGQ points occure multiple times, that could be counted once by adding their weight.