This is my first time posting my question here, and I hope I won’t violate any rules or post duplicate questions.
I am currently implementing a parameterization method of models to reduce the computing time. $$\int_0^1f(x)\,\Bbb dx$$ I need to get the integral of $f(x)$ for $x$ between 0 and 1. The value of $f(x)$ is obtained from running a model, which takes significant time. Therefore, I wanted to use the Gaussian Quadrature Method for the integration. Specifically, we can use $x_k$ (abscissa) and $w_k$ (weights).
Therefore, the integral of $f(x)$ can be approximated as $$\sum_{k=1}^n w_k f(x_k)$$
However, as shown in the figure, the function $f(x)$ we are dealing with rapidly increases as $x$ approaches 1, close to the exponential function. Due to the shape of $f(x)$, $n=5$ (shown in this example) does not capture very large values of $f(x)$ near $x = 1$ since it assumes a smooth polynomial (blue line in the figure). As a result, the integral using the Gaussian Quadrature method gives an underestimated integral value. Increasing $n$ in the Gaussian Quadrature method would certainly give a better answer of the integral, but more data points mean much longer model computing time. Therefore, without increasing $n$ numbers significantly, I am wondering what is the best approach to obtain the integral of $f(x)$ in this case. I found variation formula of the Gaussian Quadrature method (e.g., Gauss-Jacobi quadrature), or adaptive Gaussian quadrature method, but not fully understood which method is expected to give a better accuracy with the fixed $n$ number for the rapidly increasing function at the upper limit. The Adaptive Gaussian method divides the interval of $x$ (0 and 1) into two parts (0-0.5 and 0.5-1), but I am wondering if we can consider different lengths of $x$ intervals (e.g., 0-0.9 and 0.9-1.0) to describe the rapid increase of $f(x)$ near $x=1$.
In addition, if we know the truth of the integral of $f(x)$ between 0 and 1, and we can get the error term, $\epsilon$ (the solution from Gaussian Quadrature minus truth), can the error term ($\epsilon$) be applied to the Gaussian Quadrature formula as an additive term as follows? $$\sum_{k=1}^n w_k f(x_k)+\epsilon $$
If you can give any insights, that would be great.
Example of $f(x)$
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