Research
Research
For under-resolved regimes, we know that the standard ROM which is the Galerkin ROM (G-ROM) is not totally capable to model the main dynamics of the problem. Thus, to increase the computational accuracy of the numerical method, we either use more POD modes to construct the ROM which increases the computational cost or add a low-dimensional closure term that models the relationship between the resolved and unresolved POD modes and is. All the methods we present in this project are based on the latter approach. In other words, we aim to find an accurate and efficient closure term that models sub-scales.
In [1], the authors propose the DD-VMS-ROM which centers around the hierarchical structure of the VMS methodology and utilizes data to increase the ROM accuracy at a modest computational cost. Thanks to the VMS methodology, by using the ROM projection, they decompose the ROM basis into three categories: (i) resolved large scales, (ii) resolved sub-scales, and (iii) unresolved scales. Then, by using interactions among three types of scales, they propose the 2S-DD-VMS-ROM, which uses resolved and unresolved scales, and the 3S-DD-VMS-ROM, which uses all three scales to model the VMS-ROM closure term(s). The ROM closure problems in [1] are solved by using the following approximation:
The closure term is built by using the large and small length scale coefficient whereas the ansatz is only depends on the large scale coefficient. In [1], the numerical results show that the 3S-DD-VMS-ROM is more accurate than the 2S-DD-VMS-ROM and both of them are significantly more accurate than the G-ROM.
In [1], the closure term is modeled by using the ansatz which is defined above depending on the large-scale ROM coefficient which is often inconsistent. The question of whether we can construct a consistent ROM closure ansatz leads the authors in [2] to propose and create a new data-driven VMS-ROM (D2-VMS-ROM) by modeling sub-scales using the residual term, which is a consistent term. The authors construct two different residual-based D2-VMS-ROMs (R-D2-VMS-ROMs). The R1-D2-VMS-ROM is constructed by using only one ansatz whereas in R2-D2-VMS-ROM uses two different ansatz to model the effect of sub-scales. For both models, ansatzes depend on the residual terms. The numerical results are tested on the parameter-dependent problem by comparing the R1-D2-VMS-ROM and R2-D2-VMS-ROM with the coefficient-based D2-VMS-ROM (C-D2-VMS-ROM) which is proposed in [1]. The numerical results in [2] show that the R1-D2-VMS-ROM and R2-D2-VMS-ROM yield more consistent and accurate results than the C-D2-VMS-ROM.
References:
[1] Mou, C., Koc, B., San, O., Rebholz, L. G., & Iliescu, T. (2021). Data-driven variational multiscale reduced order models. Computer Methods in Applied Mechanics and Engineering, 373, 113470.
[2] Koc, B., Rebollo, T. C., & Iliescu, T. (2022). Residual Data-Driven Variational Multiscale Reduced Order Models for Parameter Dependent Problems. arXiv preprint arXiv:2208.00059.