verifiability

Numerical Analysis of Data-Driven Closure ROM

REFERENCE: Koc, B., Mou, C., Liu, H., Wang, Z., Rozza, G., & Iliescu, T. (2022). Verifiability of the data-driven variational multiscale reduced order model. Journal of Scientific Computing, 93(2), 1-26.

To attempt to glean insights into the answers to these questions, we investigate the ``verifiability'' concept which was originally defined in a large eddy simulation (LES) context. In particular, we defined a ROM closure model as verifiable if the ROM error is bounded (in some norm) by the ROM closure model error which is rigorously defined in Definition:

The right-hand side of above inequality does not depend on the ROM solution and can be evaluated before actually running the ROM. Thus, the ROM error bound is a priori error bound, just like the priori finite element error bound. Furthermore, the right-hand side of the inequality is the product of a generic constant that does NOT depend on the ROM discretization parameters, and a term that can be tuned by the user (i.e., the average ROM closure error term). Thus, as the average ROM closure error decreases, we expect the ROM error to decrease at the same rate.
We prove that the data-driven closure ROM with linear ansatz and physical constraint is verifiable and the numerical investigation with Burgers equation and 2D Navier-Stokes equations shows that this is indeed the case.