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The Kolmogorov $n$-width is an established benchmark to judge the performanceof reduced basis and similar methods that produce linear reduced spaces.Although immensely successful in the elliptic regime, this width, showsunsatisfactory slow convergence rates for transport dominated problems. Whilethis has triggered a large amount of work on nonlinear model reductiontechniques, we are lacking a benchmark to evaluate their optimal performance.  Nonlinear benchmarks like manifold/stable/Lipschitz width applied to thesolution manifold are often trivial if the degrees of freedom exceed theparameter dimension and ignore desirable structure as offline/onlinedecompositions. In this paper, we show that the same benchmarks applied to thefull reduced order model pipeline from PDE to parametric quantity of interestprovide non-trivial benchmarks and we prove lower bounds for transportequations.

Performance bounds for Reduced Order Models with Application to Parametric Transport