(f) Single domain ROM

Figure 17: Relative error (40) of the partitioned solution for select basis
sizes as a function of time in the predictive regime for the TP example. The
error plots for the FR-rLM and FR-fLM formulations in subfigures (c,d), and
the error plots for all couplings in subfigure (e), are indistinguishable.

(a) All coupled problems

(b) Provably well-posed Schur complement

Figure 18: Condition number of Schur complement matrix for each coupled
formulation as a function of the composite reduced basis size
$d_{i,C}=d_{i,0}+d_{i,\gamma}$ size in the predictive regime for the TP
example. Subfigure (a) reports results for all methods evaluated, whereas
subfigure (b) focuses only on methods with provably well-posed Schur
complements.

(a) RR-rLM, 15/10 modes

(b) FR-fLM, 15/10 modes

(c) FR-rLM, 15/10 modes

(d) RR-rLM, 60/40 modes

(e) FR-fLM, 60/40 modes

(f) FR-rLM, 60/40 modes

(g) RR-rLM, 90/60 modes

(h) FR-fLM, 90/60 modes

(i) FR-rLM, 90/60 modes

Figure 19: Comparison of the interface states at $T_{f}$ for the partitioned
schemes with provably well-posed Schur complements vs. the single domain
(monolithic) solution of the model problem. Predictive test in the TP setting.
The oscillations in the FR-rLM formulations with “small” RB sizes are due to
accumulation of interface errors during the time integration caused by the
approximate enforcement (45) of the coupling condition. The legend “$m/n$
modes” corresponds to $m$ interior and $n$ interface modes.

##  9 Conclusions

The main contributions of this paper are (i) extension of the IVR scheme to
the partitioned solution of coupled ROM-ROM and ROM-FOM problems, and (ii)
identification of the trace compatibility condition as a key factor for the
well-posedness of the IVR extensions. The IVR extension requires each coupled
problem to have a non-singular Schur complement in order to compute accurate
estimates of the interface flux needed to independently solve the subdomain
equations at each time step. Moreover, the Schur complement must be well-
conditioned for the partitioned scheme to be robust and accurate.

Our previous numerical studies [4] revealed that the Schur complement of a
coupled ROM-ROM problem, based on full subdomain bases and a full Lagrange
multiplier space, can become severely ill-conditioned and compromise the
accuracy of the coupling. The key issue with this formulation is the lack of
trace compatibility for the Lagrange multiplier space. However, the full
subdomain basis does not provide a satisfactory setting to secure this
property. In this paper, we demonstrated that a robust and effective solution
to ensure well-posed Schur complements is to consider an alternative composite
reduced basis comprising independently defined sets of basis vectors for the
interface and interior variables, respectively. The interface reduced basis
then provides a trace-compatible Lagrange multiplier space for coupled ROM-ROM
problems. For coupled ROM-FOM problems, one can use either the reduced
interface basis from the ROM side or the interface finite element space from
the FOM side.

Using variational techniques, we proved rigorously that these choices of the
Lagrange multiplier space lead to coupled problems with non-singular Schur
complements whose condition numbers are independent of the underlying FOM mesh
size and the dimension of the composite reduced basis. We performed numerical
experiments in two distinct simulations settings. In both cases, numerical
results confirmed the theoretical analysis and demonstrated the accuracy of
the partitioned schemes. The consistent behavior of the well-posed
formulations in both simulation settings further underscores the importance of
the trace compatibility condition and the robustness of the schemes.

Our future work will focus on extension of the partitioned schemes in this
paper to nonlinear problems where one has to consider techniques such as DEIM
[66], gappy POD [67] or the ECSW method [68] to handle the nonlinear terms.

## Acknowledgments

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research,
Mathematical Multifaceted Integrated Capability Centers (MMICCs) program,
under Field Work Proposal 22-025291 (Multifaceted Mathematics for Predictive
Digital Twins (M2dt)), Field Work Proposal 20-020467, and the Laboratory
Directed Research and Development program at Sandia National Laboratories. The
writing of this manuscript was funded in part by the fourth author’s (Irina
Tezaur’s) Presidential Early Career Award for Scientists and Engineers
(PECASE).

This article has been authored by an employee of National Technology &
Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the
U.S. Department of Energy (DOE). The employee owns all right, title and
interest in and to the article and is solely responsible for its contents. The
United States Government retains and the publisher, by accepting the article
for publication, acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this article or allow others to do so, for United States
Government purposes. The DOE will provide public access to these results of
federally sponsored research in accordance with the DOE Public Access Plan
https://www.energy.gov/downloads/doe-public-access-plan.

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greater number of luminous HMXBs, including ULXs, at lower metallicities (e.g.
Dray (2006); Linden et al. (2010); Fragos et al. (2013)). The bright end of
the SMC XLF above $10^{37}$ erg s-1 appears to flatten in a way that is
consistent with the XLFs of other low-metallicity galaxies (Lehmer et al.,
2021).

Studies of HMXBs in the Milky Way and Magellanic Clouds have thus demonstrated
that both metallicity and age impact the properties of HMXB populations. These
effects are being investigated in more detail by both theoretical simulations
(e.g.Linden et al. (2010); Fragos et al. (2013)) and observational studies of
HMXBs in other galaxies (e.g. Lehmer et al. (2021)). Studying the impact of
both metallicity and age on HMXB properties is relevant to understanding the
gravitational wave sources that may evolve from HMXBs as well as the
contribution of the first generations of HMXBs in the Universe to the heating
of intergalactic gas during the Epoch of Reionization.

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