The Galerkin, ADJDS, and GLSDS FEM FOMs are the worst performing methods and
all result in oscillatory solutions.

  * 2.

SUPG, GLSST, and ADJST all provide solutions of a similar qualitative and
quantitative quality, and are all able to suppress the oscillations seen in
the Galerkin FEM solution.

  * 3.

The “space–time” GLSST and ADJST FEMs again perform much better than the
“discretize-then-stabilize” GLSDS and ADJDS FEM FOMs.

These results are similar to those obtained for example 1 (Section 7.1.5).

(a) H-Res

(b) Galerkin

(c) SUPG

(d) GLSDS

(e) ADJDS

(f) GLSST

(g) ADJST

Figure 12: Example 2, advecting front. FOM solutions to the CDR equation at $t=5$. Table 5: Example 2, advecting front. Integrated (relative) $L^{2}(\Omega)$ and $\mathcal{H}^{1}(\Omega)$ errors of various FOMs presented in Figure 12, along with the stabilization parameters at which the FOMs were executed. | Galerkin | SUPG | GLSDS | ADJDS | GLSST | ADJST
---|---|---|---|---|---|---
$\overline{e}_{L^{2}}$ | $2.759\,173\,028\,166\,726\,771\text{\times}{10}^{-02}$ | $6.620\,296\,672\,187\,269\,728\text{\times}{10}^{-03}$ | $2.239\,904\,833\,384\,482\,805\text{\times}{10}^{-02}$ | $2.192\,068\,632\,024\,630\,689\text{\times}{10}^{-02}$ | $6.657\,746\,643\,360\,291\,233\text{\times}{10}^{-03}$ | $6.587\,166\,969\,012\,498\,048\text{\times}{10}^{-03}$
$\overline{e}_{\mathcal{H}^{1}}$ | $9.300\,839\,286\,897\,473\,468\text{\times}{10}^{-02}$ | $5.371\,882\,630\,527\,806\,748\text{\times}{10}^{-03}$ | $7.530\,943\,471\,626\,841\,412\text{\times}{10}^{-02}$ | $7.319\,738\,029\,699\,919\,546\text{\times}{10}^{-02}$ | $5.806\,980\,897\,480\,060\,536\text{\times}{10}^{-03}$ | $5.020\,729\,023\,394\,916\,030\text{\times}{10}^{-03}$
$\tau$ | N/A | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-02}$ | $1.000\,000\,000\,000\,000\,048\text{\times}{10}^{-04}$ | $1.000\,000\,000\,000\,000\,048\text{\times}{10}^{-04}$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-02}$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-02}$
$\Delta t$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-03}$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-03}$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-03}$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-03}$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-03}$ | $1.000\,000\,000\,000\,000\,021\text{\times}{10}^{-03}$

####  7.2.3 Results as a function of RB dimension

Figure 13 shows the $L^{2}(\Omega)$ error and $\mathcal{H}^{1}(\Omega)$ error
for each of the ROMs considered as a function of RB size. Results are
presented for the value of $\tau$ and $\Delta t$ that led to the lowest
$L^{2}(\Omega)$ error; Figure 14 shows these optimal stabilization parameters
and time steps for each basis dimension.

Examining Figure 13, we make the following observations about the accuracy of
the various ROMs:

  * 1.

The APG-based, SUPG, ADJST, and GLSST ROMs are the best overall performing
methods.

  * 2.

The “discretize-then-stabilize” ROMs outperform the Galerkin ROM, but are
consistently worse than their space–time stabilized counterparts for all ROM
dimensions. In general, even when equipped with LSPG, the discretize-then-
stabilize methods perform quite poorly.

  * 3.

When applied to the standard Galerkin and GLSDS FEM models, LSPG leads to
slightly improved solutions. LSPG does not lead to improved solutions for
SUPG, GLSST, or ADJST.

  * 4.

It is again interesting to note the improved performance of G–APG over the
standard Galerkin method at high reduced basis dimensions, given that G–APG
will converge to the Galerkin method in the limit of a full reduced basis. The
improved performance is also seen for G–LSPG, but to a lesser extent.

  * 5.

Once again, a decrease in error in the $L^{2}(\Omega)$ norm does not always
correspond to a decrease in error in the $\mathcal{H}^{1}(\Omega)$ norm, and
vice versa.

  * 6.

Comparing Figure 13 to Table 5, we again observe that some ROMs are more
accurate than their corresponding FEM.

Examining Figure 14, where we show the optimal stabilization parameters and
time steps associated with Figure 13, we make the following observations about
the behavior of the stabilization parameters of the various ROMs:

  * 1.

The optimal stabilization parameters for the SUPG, ADJST, GLSST, and APG ROMs
increase for $R<4$. This result is consistent with analyses performed for SUPG
in [52] and APG in [113], which suggested that the optimal stabilization
parameter decreases with increasing ROM dimension.

  * 2.

The optimal stabilization parameter for GLSDS is quite high for all ROM
dimensions.

  * 3.

The optimal time step for all LSPG ROMs occurs at an intermediate time step
larger than the FEM FOM.

  * 4.

Interestingly, the optimal time step for G–LSPG is the same as the optimal
time step for GLSDS.

  * 5.

The reader may observe that the optimal time step for SUPG, GLSST, and ADJDS
decreases for moderately high ROM dimensions; we note that this improvement is
very minor as can will be seen in Figure 16.

(a) $L^{2}(\Omega)$ error.

(b) $L^{2}(\Omega)$ error.

(c) $\mathcal{H}^{1}(\Omega)$ error.

(d) $\mathcal{H}^{1}(\Omega)$ error.

Figure 13: Example 2, advecting front. $L^{2}(\Omega)$ (top) and
$\mathcal{H}^{1}(\Omega)$ (bottom) error as a function of RB dimension for the
various ROMs evaluated. We note that the left and right figures show the same
quantities, but with different visualization techniques. Results are shown for
optimal values of $t,\tau$ as discussed in Section 7.1.4.

(a) Optimal parameter, $\tau$.

(b) Optimal parameter, $\tau$.

(c) Optimal time step, $\Delta t$.

(d) Optimal time step, $\Delta t$.

Figure 14: Example 2, advecting front. Optimal stabilization parameter (top)
and time step (bottom) as a function of RB dimension. We note that the left
and right figures show the same quantities, but with different visualization
techniques. Results are shown for optimal values of $t,\tau$ as discussed in
Section 7.1.4.

Next, Figure 15 presents physical space solution profiles for the various ROMs
for a reduced basis dimension of $R=5$ at the final time instance, $t=2.0$. We
observe that all methods yield qualitatively accurate solutions with the
exception of Galerkin, G–LSPG, and GLSDS; these three methods under-predict
the magnitude of the solution in the lower-left quadrant of the domain
($x_{1},x_{2}\leq 0.4$). It is interesting to note that, although Figure 13
showed that ADJDS and the non-Galerkin LSPG ROMs clearly perform less well
than the other formulations, Figure 15 shows that their physical space
solutions still show a significant improvement over the standard Galerkin ROM.

(a) H-Res ($L^{2}(\Omega)$ best fit)

(b) H-Res ($H_{1}(\Omega)$ best fit)

(c) Galerkin

(d) SUPG

(e) GLSDS

(f) ADJDS

(g) GLSST

(h) ADJST

(i) G–LSPG

(j) SUPG–LSPG

(k) GLSDS–LSPG

(l) ADJDS–LSPG

(m) GLSST–LSPG

(n) ADJST–LSPG

(o) G–APG

(p) SUPG–APG

(q) GLSST–APG

(r) ADJST–APG

Figure 15: Example 2, advecting front. ROM solutions to the CDR equation at
$t=2.0$. Results are shown for $R=5$.

####  7.2.4 Sensitivity to time step and stabilization parameters

We now quantify the sensitivity of the various methods to their stabilization
parameters and the time step. Figure 16 presents results for the continuous
and LSPG ROM solutions obtained at $R=5$ on the parameter grid $(\tau,\Delta
t)\in\boldsymbol{\tau}\times\boldsymbol{\Delta}\boldsymbol{t}$, while Figure
17 presents results for the G–APG ROM solutions obtained at $R=5$ on the
parameter grid $(\tau_{\mathrm{APG}},\Delta
t)\in\boldsymbol{\tau}\times\boldsymbol{\Delta}\boldsymbol{t}$ and the
remaining APG ROM solutions (which depend on three parameters, $\Delta
t,\tau,$ and $\tau_{\mathrm{APG}}$) obtained at $R=5$ on the parameter grid
$(\tau,\tau_{\mathrm{APG}})\in\boldsymbol{\tau}\times\boldsymbol{\tau}$ with a
fixed time step $\Delta t=10^{-3}$. As a reference, Figure 18 shows the same
sensitivities but for full-order FEM simulations executed on the FOM trial
space. We make the following observations:

  * 1.

The SUPG (Figure 16(b)), GLSST (Figure 16(e)), ADJST (Figure 16(f)), and G–APG
(Figure 17(a)) ROMs again all display a dependence on the time step and
stabilization parameter that is similar to the first example. Optimal results
are obtained for an intermediate value of $\tau$, and the solutions all
converge in the limit of $\Delta t\rightarrow 0$.

  * 2.

G–LSPG (Figure 16(g)), which contains no dependence on $\tau$, yields optimal
results at an intermediate time step. LSPG’s optimality at an intermediate
time step is well documented in the community [28, 113].

  * 3.

For sufficiently large $\tau$, the dependence of GLSDS (Figure 16(c)) on the
time step is similar to that of LSPG (Figure 16(g)).

  * 4.

Neither GLSDS nor ADJDS ROMs have optimal results at the minimum time step,
suggesting that these methods are not well behaved in the low time-step limit.
This result reinforces those presented in Figure 14(d).

  * 5.

All continuous ROMs (Figures 16(a)-16(f)) display a similar dependence to the
stabilization parameter and time step as their corresponding FOMs (Figures
18(a)-18(f)).

  * 6.

Lastly, for APG ROMs built on top of a stabilized FEM (Figures 17(c)-17(e)),
optimal results are obtained for either an intermediate value of
$\tau_{\mathrm{APG}}$ and low value of $\tau$, or vice versa. It is
interesting to note that the solutions are almost symmetric with respect to
these two parameters.

(a) Galerkin

(b) SUPG

(c) GLSDS

(d) ADJDS

(e) GLSST

(f) ADJST

(g) G–LSPG

(h) SUPG–LSPG

(i) GLSDS–LSPG

(j) ADJDS–LSPG

(k) GLSST–LSPG

(l) ADJST–LSPG

Figure 16: Example 2, advecting front. Time integrated $L^{2}(\Omega)$ error
as a function of time step and stabilization parameter for the various ROMs
evaluated. Note that Galerkin and LSPG display no dependence on the
stabilization parameter. White regions indicate solutions that diverged to
NaN.

(a) G–APG

(b) G–APG

(c) SUPG–APG

(d) GLSST

(e) ADJST

Figure 17: Example 2, advecting front. Time integrated $L^{2}(\Omega)$ error
as a function of time step and stabilization parameter for the various ROMs
evaluated. Note that Galerkin and LSPG display no dependence on the
stabilization parameter. White regions indicate errors higher than the color
limit.

(a) Galerkin-FEM

(b) SUPG-FEM

(c) GLSDS-FEM

(d) ADJDS-FEM

(e) GLSST-FEM

(f) ADJST-FEM

Figure 18: Example 2, advecting front. Time integrated $L^{2}(\Omega)$ error
as a function of time step and stabilization parameter. Results are shown for
full-order FEM solutions executed on the FOM trial space. White regions
indicate errors higher than the color limit.

###  7.3 Summary of numerical experiments and empirical findings

Sections 7.1.4 and 7.2 presented results for stabilized ROMs applied to the
CDR system for two different configurations. Across both cases, we observed
that the “space–time” stabilized continuous ROM formulations were superior to
their “discretize-then-stabilize” counterparts: the space–time stabilization
formulations had lower errors, were well-behaved in the low time-step limit
(the discretize-then-stabilized methods were not), and had a smoother, more
intuitive dependence on the time step and stabilization parameter.

ROMs constructed from LSPG projection had mixed results in terms of accuracy.
In the first numerical experiment, ROM solutions computed via LSPG projection
tended to be slightly better than their non-stabilized (e.g., discrete
Galerkin) ROM counterparts. In the second example, however, the performance
was mixed. While G–LSPG and GLSDS–LSPG led to better solutions than Galerkin
and GLSDS, respectively, SUPG–LSPG, ADJDS–LSPG, GLSST–LSPG, and ADJST–LSPG all
performed worse than SUPG, ADJDS, GLSST, and ADJST, respectively. Further, all
LSPG ROMs were optimal for intermediate time steps and, when built on top of a
stabilized FEM solution, displayed a complex sensitivity to both the
stabilization parameter and time step. The a priori selection of an optimal
time step appears difficult, and is likely problem dependent.

Constructing ROMs from APG stabilization tended to have a positive result in
terms of accuracy. In the first example, APG ROMs outperform their non-
stabilized (i.e., discrete Galerkin) counterpart for all cases with the
exception of GLSST. In the second example, APG-based ROMs were better than
their non-stabilized counterparts for all FEM formulations. Further, unlike
LSPG, APG was well behaved in the low time step limit. However, building an
APG ROM on top of a stabilized FEM solution led to complex behavior for the
stabilization parameter: high values of $\tau$ for the stabilized FEM models
required low values of $\tau_{\mathrm{APG}}$ for the ROM, and vice versa. The
a priori selection of these parameters again appears difficult.

Figure 19 summarizes the performance of the various ROMs by tabulating the
number of times a given ROM formulation led to the lowest errors in the
$L^{2}(\Omega)$ and $\mathcal{H}^{1}(\Omega)$-norm. Results are compiled for
ROMs of basis dimensions $R=1,\ldots,20$ across both numerical experiments. We
observe that SUPG–APG was the best performing ROM for both error metrics. The
next best performing ROMs were ADJST–APG and GLSST–APG, followed closely by
GLSST, SUPG, and ADJST. It is interesting to note that the two different error
measures lead to slightly different measure of optimality (SUPG is never a
top-performing method in $L^{2}(\Omega)$ but is consistently a top-performing
method in $\mathcal{H}^{1}(\Omega)$).

(a) $L^{2}(\Omega)$-norm

(b) $\mathcal{H}^{1}(\Omega)$-norm

Figure 19: Summary of numerical experiments. Number of times a given ROM
formulation led to lowest errors in the $L^{2}(\Omega)$-norm (left) and
$\mathcal{H}^{1}(\Omega)$-norm (right). Results are compiled across both
numerical examples for all basis dimensions.

Lastly, Figure 20 attempts to rank the various ROM formulations by scoring
their performance. For a given basis dimension, we scored a ROM on a scale of
$1-N_{\text{ROMS}}$, where $N_{\text{ROMS}}=16$ is the number of ROM
formulations considered. The best ROM gets a score of $N_{\text{ROMS}}$, the
second best ROM gets a score of $N_{\text{ROMS}}-1$, and so on until the
worst-performing ROM gets a score of 1. The total score for each ROM
formulation is computed by summing the individual scores across all basis
dimensions and both numerical experiments. By this scoring system, we find
that SUPG–APG is the best performing ROM in both error measures, followed
closely by ADJST–APG, GLSST, and GLSST–APG. The GLSST ROM is the best
continuous ROM considered in this work, while the Galerkin ROM performs the
worst.

We end the discussion with a brief note regarding the computational cost of
the various ROMs and methods evaluated. While we did not report wall-clock
times for running the various models considered in this work, we remark that,
for linear problems, all ROM methods considered herein have similar online
computational costs. The costs will differ for nonlinear problems, and will
depend on the choice of hyper-reduction method used to handle the online
evaluation of the nonlinearities in the governing problem. Extension of the
analysis presented herein to nonlinear problems will be the subject of a
subsequent publication.

(a) Score based on the $L^{2}(\Omega)$-norm

(b) Score based on the $\mathcal{H}^{1}(\Omega)$-norm

Figure 20: Summary of numerical experiments. Score as measured by the
$L^{2}(\Omega)$-norm (left) and $\mathcal{H}^{1}(\Omega)$-norm (right). For a
given basis dimension, we scored a ROM on a scale of $1$ to $N_{\text{ROMS}}$,
where $N_{\text{ROMS}}=16$ is the number of ROM formulations considered. The
best ROM gets a score of $N_{\text{ROMS}}$, the second best ROM gets a score
of $N_{\text{ROMS}}-1$, and so on until the worst ROM gets a score of 1. The
total score for each ROM formulation is computed by summing the individual
scores across all basis dimensions and both numerical experiments.

##  8 Conclusions

The development of robust ROMs for time-critical and many-query scenarios
remains an active research area. This work outlined the construction of
stabilized ROMs for the transient convection-diffusion-reaction equation via
two differing approaches that have emerged within the community: discrete and
continuous projection. We outlined the standard Galerkin, SUPG, GLSDS, ADJDS,
GLSST, and ADJST continuous ROMs developed via traditional stabilized finite
elements. We additionally outlined the construction of the discrete Galerkin,
LSPG, and APG ROMs. These discrete ROMs can be constructed from a standard
Galerkin FEM model, or they can also be constructed from a stabilized FEM
model. We highlighted the well-established equivalences between constructing
ROM basis vectors at the discrete and continuous levels. We additionally
highlighted the established equivalence conditions between discrete ROM
formulations and continuous ROM formulations. Lastly, a brief summary of
existing numerical analyses was provided, where we discussed consistency,
stability, and error bounds of the various methods.

Numerical experiments were conducted for two configurations of the CDR system.
These experiments demonstrated that all stabilized ROMs result in superior
performance over a standard Galerkin ROM built via continuous projection. APG
and GLSST-based ROMs proved to be the best performing methods, while GLSDS,
ADJDS, and G–LSPG proved to be the worst performing stabilized methods. In
particular, we found that equipping a stabilized FEM model with APG or LSPG
projection can result in more accurate solutions. This improvement in accuracy
comes at the cost of a more complex dependence on the stabilization parameters
and time step. In the case of LSPG, results are optimal at an intermediate
time step which is hard to select a priori. Further, the accuracy of LSPG-
based methods degrades in the small time step limit. In the case of APG, we
observed a non-trivial relationship between $\tau$ in the stabilized FEM model
and $\tau_{\mathrm{APG}}$ in the APG projection. Low values of $\tau$ in the
stabilized FEM model required high values of $\tau_{\mathrm{APG}}$, and vice
versa. There exist minimal methods for the a priori selection of these optimal
stabilization parameters. However, APG-based ROMs were robust for small time
steps. Lastly, we observed that ROMs built via continuous projection from the
“space–time” approach were clearly superior to ROMs built via the “discretize-
then-stabilize” approach: the space–time stabilization formulations had lower
errors, were well-behaved in the low time-step limit (the discretize-then-
stabilized methods were not), and had a smoother, more intuitive dependence on
the time step and stabilization parameter.

In addition, our study highlighted several points (most of which are well-
established in the literature) which we reiterate here:

  * 1.

In the case of discrete projection, the POD basis must be obtained in an inner
product that is $\mathbf{M}$-orthogonal to recover the POD basis obtained via
continuous projection. This is well established in the literature.

  * 2.

Both the Galerkin and stabilized ROMs obtained via continuous projection can
be obtained via the Galerkin ROM obtained via discrete projection. This is
additionally well established.

  * 3.

G–LSPG approximates a continuous minimization principle if the discrete norm
is minimized in the $\mathbf{M}^{-1}$ inner product.

  * 4.

All stabilized methods depend on a stabilization parameter, $\tau$, the time
step, $\Delta t$, or both. The dependency on these parameters is complex, and
more work needs to be done for proper a priori selection of the stabilization
parameters.

Future work should focus on three aspects. First, theory for discretely
stabilized ROMs such as LSPG and APG is lacking for linear problems: minimal
analyses exist studying the stability of these methods as well as their
accuracy, and future work should address this. Second, future work should
focus on the a priori selection of the stabilization parameters and optimal
time steps (or, alternatively, the development of methods that eliminate this
sensitivity as in Ref [112]). Minimal analyses exists for the appropriate a
priori selection of these parameters within the context of ROMs: SUPG is
studied in Refs. [52, 77], LSPG is studied in [28], and APG is studied in
[113]. Lastly, future work should focus on extension to nonlinear and vector-
valued systems. Here, hyper-reduction is important and the extension of
stabilization techniques to this setting offers numerous interesting and
important questions.

## Acknowledgments

E. Parish acknowledges funding from the John von Neumann Postdoctoral
Fellowship, ASC V&V 103723/05.30.02, and ASC V&V 131792/04.09.02. I. Tezaur
acknowledges funding from her Presidential Early Career Award for Scientists
and Engineers (PECASE), awarded by the U.S. Department of Energy (DOE), as
well as support from the U.S. Office of Science, Office of Advanced Scientific
Computing Research, Mathematical Multifaceted Integrated Capability Centers
(MMICCS) program, under Field Work Proposal 22025291 and the Multifaceted
Mathematics for Predictive Digital Twins (M2dt) project. T. Iliescu
acknowledges funding from the NSF grants DMS-2012253 and CDS&E-MSS-1953113.
This paper describes objective technical results and analysis. Any subjective
views or opinions that might be expressed in the paper do not necessarily
represent the views of the U.S. Department of Energy or the United States
Government. Sandia National Laboratories is a multimission laboratory managed
and operated by National Technology & Engineering Solutions of Sandia, LLC, a
wholly owned subsidiary of Honeywell International Inc., for the U.S.
Department of Energy’s National Nuclear Security Administration under contract
DE-NA0003525.

##  Appendix A Correspondence of LSPG to a continuous minimization principle

This section outlines the equivalence between LSPG and a continuous
minimization principle. We define the time-discrete, spatially continuous
residual of the PDE (1) as

$\mathsf{R}_{\mathrm{cdr}}:(\mathsf{w};\mathsf{z})\mapsto\frac{\mathsf{w}-\mathsf{z}}{\Delta
t}-\nu\nabla^{2}\mathsf{w}+\mathbf{b}\cdot\nabla\mathsf{w}+\sigma\mathsf{w}-f.$

Under the assumption that the state is sufficiently regular121212We remark
that this assumption does not hold for standard $\mathcal{C}^{0}(\Omega)$ FEM
discretizations, in which case the state is not twice differentiable., the
residual of the Galerkin FOM O$\Delta$E can be written as

$[\mathbf{r}_{\mathrm{G}}]_{i}:(\mathbf{w};\mathbf{z})\mapsto
m\left(\mathsf{v}_{i},\mathsf{R}_{\mathrm{cdr}}(\mathbf{\mathsf{v}}\mathbf{w};\mathbf{\mathsf{v}}\mathbf{z}\right)),\qquad\forall\mathsf{v}\in\mathcal{V}_{\mathrm{h}}.$

The FEM coefficients of the $L^{2}(\Omega)$ orthogonal projection of this
residual onto the trial space $\mathcal{V}_{\mathrm{h}}$ are given by

$\mathbf{r}^{\parallel}_{\mathrm{G}}:(\mathbf{w};\mathbf{z})\mapsto\mathbf{M}^{-1}\mathbf{r}_{\mathrm{G}}(\mathbf{w};\mathbf{z}).$

Analogously, at the spatially continuous level

$\mathsf{R}^{\parallel}_{\mathrm{G}}:(\mathsf{w};\mathsf{z})\mapsto\mathbf{\mathsf{v}}\mathbf{M}^{-1}m\left(\mathbf{\mathsf{v}},\mathsf{R}_{\mathrm{cdr}}(\mathsf{w};\mathsf{z})\right).$

The square residual integrated over the domain is then given by

$\int_{\Omega}\left(\mathsf{R}^{\parallel}_{\mathrm{G}}(\mathsf{u}^{n};\mathsf{u}^{n-1})\right)^{2}dx=\left[\mathbf{M}^{-1}\mathbf{r}_{\mathrm{G}}(\mathbf{a}_{\mathrm{h}}^{n};\mathbf{a}_{\mathrm{h}}^{n-1})\right]^{T}\mathbf{M}\mathbf{M}^{-1}\mathbf{r}_{\mathrm{G}}(\mathbf{a}_{\mathrm{h}}^{n};\mathbf{a}_{\mathrm{h}}^{n-1}).$
(37)

Setting $\mathbf{r}\leftarrow\mathbf{r}_{\mathrm{G}}$ and
$\mathbf{P}\leftarrow\mathbf{M}^{-1}$ in optimization problem (24), LSPG
corresponds to the continuous minimization principle for
$\mathsf{u}_{\mathrm{r}}^{n}$, $n=1,\ldots,N_{t}$

$\mathsf{u}_{\mathrm{r}}^{n}=\underset{\mathsf{u}\in\mathcal{V}_{\mathrm{r}}}{\text{arg\,min}}\;\int_{\Omega}\left(\mathsf{R}^{\parallel}_{\mathrm{G}}(\mathsf{u};\mathsf{u}_{\mathrm{r}}^{n-1})\right)^{2}dx.$
(38)

LSPG computes the solution $\mathsf{u}_{\mathrm{r}}^{n}$ within the ROM trial
space $\mathcal{V}_{\mathrm{r}}$ that minimizes the $L^{2}(\Omega)$-norm of
the time-discrete, spatially continuous residual projected onto the finite
element trial space $\mathcal{V}_{\mathrm{h}}$. Analogously, LSPG can be
defined on the stabilized FOM O$\Delta$E (15). Setting
$\mathbf{r}\leftarrow\mathbf{r}_{\mathrm{S}}$, the optimality conditions
become

$m_{d}\left(\frac{\mathbf{M}\mathbf{\Psi}}{\Delta
t}+\mathbf{B}\mathbf{\Psi}+\mathbf{Q}\mathbf{\Psi},\mathbf{r}_{\mathrm{S}}(\mathbf{\Psi}\hat{\mathbf{y}};\mathbf{\Psi}\hat{\mathbf{x}}^{n-1})\right)=\mathbf{0}.$

##  Appendix B Derivation of the Adjoint Petrov–Galerkin method

The APG method was derived in Ref. [113] in the case where the coarse- and
fine-scale bases are orthogonal in a standard $\ell^{2}$ discrete inner
product. In FEM discretizations it is more appropriate to construct the spaces
to be orthogonal in an $\mathbf{M}$ inner product. As such, in this section we
derive the APG method for the case where the fine and coarse scales are
$\mathbf{M}$-orthogonal. We consider application to the dynamical system given
by

$\mathbf{M}\frac{d\mathbf{x}}{dt}+\mathbf{A}\mathbf{x}-\mathbf{f}=\boldsymbol{0},$

where $\mathbf{M}\in\mathbb{S}^{N}$ is the mass matrix,
$\mathbf{x}:[0,T]\rightarrow\mathbb{R}^{N}$ is the state,
$\mathbf{A}\in\mathbb{R}^{N\times N}$ is the system matrix, and $\mathbf{f}$
is a forcing term. The derivation begins by decomposing $\mathbb{R}^{N}$ into
a coarse-scale space $\textit{V}_{r}$ and a fine-scale space
$\textit{V}_{r}^{\prime}$ such that
$\textit{V}_{r}\oplus\textit{V}_{r}^{\prime}\equiv\mathbb{R}^{N}$. The coarse-
scale space $\textit{V}_{r}\subset\mathbb{R}^{N}$ corresponds to the standard
ROM space and is of dimension $\text{dim}(\textit{V}_{r})=R$, while the fine-
scale space $\textit{V}_{r}^{\prime}$ comprises the $\mathbf{M}$-orthogonal
complement of the coarse-scale space and has dimension
$\text{dim}(\textit{V}_{r}^{\prime})=N-R$. We equip the coarse- and fine-scale
spaces with $\mathbf{M}$-orthogonal bases $\mathbf{\Psi}\in\mathbb{R}^{N\times
R}$ and $\mathbf{\Psi}^{\prime}\in\mathbb{R}^{N\times(N-R)}$. Note that
$\mathbf{\Psi}^{T}\mathbf{M}\mathbf{\Psi}^{\prime}=\boldsymbol{0}$ by
definition.

The APG derivation proceeds by expressing the dynamical system in terms of the
generalized coordinates associated with the coarse and fine-scale bases. This
process results in a coupled system for the coarse and fine scales

$\begin{split}&\mathbf{\Psi}^{T}\mathbf{M}\frac{d}{dt}\mathbf{\Psi}\hat{\mathbf{x}}+\mathbf{\Psi}^{T}\mathbf{A}\left[\mathbf{\Psi}\hat{\mathbf{x}}+\mathbf{\Psi}^{{}^{\prime}}\hat{\mathbf{x}}^{{}^{\prime}}\right]-\mathbf{\Psi}^{T}\mathbf{f}=\boldsymbol{0}\\\
&\left[\mathbf{\Psi}^{{}^{\prime}}\right]^{T}\mathbf{M}\frac{d}{dt}\mathbf{\Psi}^{{}^{\prime}}\hat{\mathbf{x}}^{{}^{\prime}}+\left[\mathbf{\Psi}^{{}^{\prime}}\right]^{T}\mathbf{A}\left[\mathbf{\Psi}\hat{\mathbf{x}}+\mathbf{\Psi}^{{}^{\prime}}\hat{\mathbf{x}}^{{}^{\prime}}\right]-\left[\mathbf{\Psi}^{{}^{\prime}}\right]^{T}\mathbf{f}=\boldsymbol{0},\end{split}$

where $\hat{\mathbf{x}}:[0,T]\rightarrow\mathbb{R}^{R}$ are the coarse-scale
generalized coordinates and
$\hat{\mathbf{x}}^{{}^{\prime}}:[0,T]\rightarrow\mathbb{R}^{N-R}$ are the
fine-scale generalized coordinates. The APG method proceeds to approximate the
fine-scales via the Mori–Zwanzig formalism and a perturbation analysis, which
here results in the quasi-static aproximation

$\hat{\mathbf{x}}^{{}^{\prime}}(t)\approx-\tau\left[\mathbf{\Psi}^{{}^{\prime}}\right]^{T}\mathbf{A}\mathbf{\Psi}\hat{\mathbf{x}}(t),$

where $\tau\in\mathbb{R}_{>0}$ is a stabilization constant. Injecting in the
approximation to the fine-scale state into the coarse-scale equation results
in

$\mathbf{\Psi}^{T}\mathbf{M}\frac{d}{dt}\mathbf{\Psi}\hat{\mathbf{x}}+\mathbf{\Psi}^{T}\mathbf{A}\left[\mathbf{\Psi}\hat{\mathbf{x}}-\tau\mathbf{\Psi}^{{}^{\prime}}\left[\mathbf{\Psi}^{{}^{\prime}}\right]^{T}\mathbf{A}\mathbf{\Psi}\hat{\mathbf{x}}\right]-\mathbf{\Psi}^{T}\mathbf{f}=\boldsymbol{0}.$

Next we use the property
$\mathbf{\Psi}\mathbf{\Psi}^{T}\mathbf{M}+\mathbf{\Psi}^{{}^{\prime}}\left[\mathbf{\Psi}^{{}^{\prime}}\right]^{T}\mathbf{M}=\mathbf{I}$
to remove the dependence on the fine-scale basis functions and get

$\mathbf{\Psi}^{T}\mathbf{M}\frac{d}{dt}\mathbf{\Psi}\hat{\mathbf{x}}+\mathbf{\Psi}^{T}\mathbf{A}\left[\mathbf{\Psi}\hat{\mathbf{x}}-\tau\mathbb{A}^{\prime}\mathbf{A}\mathbf{\Psi}\hat{\mathbf{x}}\right]-\mathbf{\Psi}^{T}\mathbf{f}=\boldsymbol{0},$

where $\mathbb{A}^{\prime}=\mathbf{M}^{-1}-\mathbf{\Psi}\mathbf{\Psi}^{T}$.
Next, assuming the forcing to be zero on the fine-scale space such that
$\mathbb{A}^{\prime}\mathbf{f}=\boldsymbol{0}$, we write the above in a
Petrov–Galerkin form

$\left[\left(\mathbf{I}-\tau\mathbb{A}^{\prime
T}\mathbf{A}^{T}\right)\mathbf{\Psi}\right]^{T}\left[\mathbf{M}\frac{d}{dt}\mathbf{\Psi}\hat{\mathbf{x}}+\mathbf{A}\mathbf{\Psi}\hat{\mathbf{x}}-\mathbf{f}\right]=\boldsymbol{0}$

where we have leveraged
$\mathbb{A}^{\prime}\mathbf{M}\mathbf{\Psi}=\boldsymbol{0}$. This
Petrov–Galerkin projection is what we refer to as the Adjoint Petrov–Galerkin
method.

##  Appendix C Proper orthogonal decomposition algorithm

Algorithm 2 presents the algorithm for computing the trial basis via proper
orthogonal decomposition.

Input: Snapshot matrix,
$\mathbf{S}_{\mathbf{a}_{\mathrm{h}}}\in\mathbb{R}^{N\times N_{s}}$ ; cutoff
energy tolerance, $\epsilon_{\mathrm{c}}$ ; symmetric positive definite inner
product matrix, $\mathbf{P}$

Output: POD Basis $\mathbf{\Psi}\in\mathbb{R}^{N\times R}$ ;

Steps:

  1. 1.

Compute the “covariance matrix”

$\mathbf{K}=\mathbf{S}_{\mathbf{a}_{\mathrm{h}}}^{T}\mathbf{P}\mathbf{S}_{\mathbf{a}_{\mathrm{h}}}.$

  2. 2.

Compute the eigenvalue decomposition

$\mathbf{K}=\mathbf{E}_{\mathbf{a}_{\mathrm{h}}}{\boldsymbol{\Lambda}}_{\mathbf{a}_{\mathrm{h}}}[\mathbf{E}_{\mathbf{a}_{\mathrm{h}}}]^{-1}.$

  3. 3.

Compute the statistical energy

$\epsilon_{K}=\frac{\sum_{i=1}^{K}\Lambda_{i}}{\sum_{i=1}^{N_{s}}\Lambda_{i}},$

where $K\leq N_{s}$.

  4. 4.

Determine basis dimension from cutoff criterion

$R=\text{Card}(\\{\epsilon_{i}\\}_{i=1}^{N_{s}}|\epsilon_{i}\leq\epsilon_{\mathrm{c}}).$

  5. 5.

Compute the ROM bases as

$\mathbf{\Psi}=\mathbf{S}_{\mathbf{a}_{\mathrm{h}}}\mathbf{E}_{\mathbf{a}_{\mathrm{h}}}^{R}\sqrt{[{\boldsymbol{\Lambda}}_{\mathbf{a}_{\mathrm{h}}}^{R}]^{-1}},$

where $\mathbf{E}_{\mathbf{a}_{\mathrm{h}}}^{R}$ and
${\boldsymbol{\Lambda}}_{\mathbf{a}_{\mathrm{h}}}^{R}$ comprise the first $R$
columns of $\mathbf{E}_{\mathbf{a}_{\mathrm{h}}}$ and the first $R$ columns
and rows of ${\boldsymbol{\Lambda}}_{\mathbf{a}_{\mathrm{h}}}$, respectively.

Algorithm 2 Algorithm for generating POD Basis.

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