Engg. tutorials

For multidimensional problems, the above mentioned single
dimensional quadrature rule can be extended using the product rule.
This results in an $n$ dimensional integral of the form
\begin{equation}
I_{N}=\int_{R_{n}}f(s)\dfrac{1}{2\pi^{n/2}} e^{-\dfrac{1}{2} \vert s \vert^{2}}ds
\end{equation}
It can be approximately evaluated as \begin{equation}
I_{N}=\sum_{j_{1}}^{N}...\sum_{j_{n}}^{N}f(q_{j_{1}},q_{j_{2}},...,q_{j_{n}})w_{j_{1}}w_{j_{2}}...w_{j_{n}}
\end{equation}

If we consider N-point GHF for an $n$ dimensional system, a total of $N^{n}$ number
of quadrature points and their corresponding weights have to be generated.
As an example, a three point GHF for a three dimensional system generates twenty seven
quadrature points and its weights. This can be expressed as ${q_{i},q_{j},q_{k}}$ and ${w_{i}w_{j}w_{k}}$
respectively for $i=1,2,3;j=1,2,3;k=1,2,3$ [Singh 2013] . Since number of quadrature points increases
exponentially with increase in dimension of systems, GHF suffers from the *curse of dimensionality problem*.
Upto some limited extent, this problem can be minimized by neglecting the quadrature points on the diagonal because
weights associated with them are very small. So their contribution to the computation of integrals can be ignored.

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