Simplex Stochastic Collocation for Piecewise Smooth Functions with Kinks
This work provides a practical method for uncertainty quantification in gas networks and similar applications where kinks occur, though the approach is domain-specific and incremental.
The paper addresses the challenge of approximating piecewise smooth functions with kinks in high dimensions, such as those arising in gas network uncertainty quantification. By exploiting an indicator that assigns each sample to its smooth region, the proposed simplex stochastic collocation method achieves a global convergence order of (p+1)/d.
Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for non-smooth functions with kinks. For example, such kinks can arise in the uncertainty quantification of quantities of interest for gas networks. This is due to the regulation of the gas flow, pressure, or temperature. But, one can exploit that for each sample in the parameter space it is known if a regulator was active or not, which can be obtained from the result of the corresponding numerical solution. This information can be exploited in a stochastic collocation method. We approximate the function separately on each smooth region by polynomial interpolation and obtain an approximation to the kink. Note that we do not need information about the exact location of kinks, but only an indicator assigning each sample point to its smooth region. We obtain a global order of convergence of $(p+1)/d$, where $p$ is the degree of the employed polynomials and $d$ the dimension of the parameter space.