Numerical Investigation of Bounds for Model Reduction

In Model Order Reduction, the goal is to estimate solutions to high dimensional models of physical systems using a lower dimension reduced order model (ROM) that is faster to compute. A common approach is to build ROMs on a linear subspace of the solution space of the high dimensional model. For ROMs on n-dimensional linear subspaces, the lowest achievable approximation error is given by the Kolmogorov n-width. ROMs on linear subspaces can be extended to ROMs on polynomially mapped manifolds. There is an analogue to the Kolmogorov n-width for ROMs on polynomially mapped manifolds of degree p, called the polynomial Kolmogorov (n,p)-width. In most cases, it is not possible to compute these widths exactly. We propose two methods for estimating both the Kolmogorov n-width and polynomial Kolmogorov (n,2)-width and compare their performance on an example setting. A theoretical approximation bound (lower and upper) can be given for the polynomial Kolmogorov (n,p)-width, formulated in terms of the Kolmogorov n-width. Using our estimation methods for the two widths, we investigate this approximation bound numerically on an example setting.