# On the impact of dimensionally-consistent and physics-based inner products
for POD-Galerkin and least-squares model reduction of compressible flows

Eric J. Parish  Francesco Rizzi  Sandia National Laboratories, Livermore, CA
Nex Gen Analytics, Sheridan, WY

###### Abstract

Model reduction of the compressible Euler equations based on proper orthogonal
decomposition (POD) and Galerkin orthogonality or least-squares residual
minimization requires the selection of inner product spaces in which to
perform projections and measure norms. The most popular choice is the vector-
valued $L^{2}(\Omega)$ inner product space. This choice, however, yields
dimensionally-inconsistent reduced-order model (ROM) formulations which often
lack robustness. In this work, we try to address this weakness by studying a
set of dimensionally-consistent inner products with application to the
compressible Euler equations. First, we demonstrate that non-dimensional inner
products have a positive impact on both POD and Galerkin/least-squares ROMs.
Second, we further demonstrate that physics-based inner products based on
entropy principles result in drastically more accurate and robust ROM
formulations than those based on non-dimensional $L^{2}(\Omega)$ inner
products.As test cases, we consider the following well-known problems: the
one-dimensional Sod shock tube, the two-dimensional Kelvin-Helmholtz
instability and two-dimensional homogeneous isotropic turbulence.

##  1 Introduction

Projection-based model reduction of fluid flows based on proper orthogonal
decomposition (POD) and, e.g., (Petrov-)Galerkin projection has been a
research area of great interest for the past several decades. A wealth of
literature exists examining applications to incompressible flows [4, 9, 47,
22, 29], and more recently POD-based reduced-order models (ROMs) have been
examined within the context of compressible and multiphysics compressible
flows (e.g., compressible flows closed with the Reynolds–averaged Navier-
Stokes (RANS) equations [13], reacting flows [23]). While these works have
demonstrated the potential of POD-based ROMs for more complex configurations,
a common trend that has emerged is a lack of robustness.

The challenges for constructing accurate POD ROMs for compressible flows are
many. Difficulties include efficiently handling non-polynomial nonlinearities
[3, 16, 18, 13], developing effective sampling techniques [17, 37], properly
accounting for boundary conditions [27, 20], and a slowly decaying Kolmogorov
$n$-width for convection-dominated problems [30, 36]. We emphasize that all of
these issues are significant and require attention. In this work we focus on a
fundamental challenge that underpins any ROM formulation: the selection of the
inner product space(s). We place a particular focus on dimensional consistency
of inner product spaces and their compatibility for use in model reduction of
both dimensional and non-dimensional problem configurations and solvers.

The choice of inner product used in POD and in the (Petrov)-Galerkin
projection plays a critical role on ROM accuracy and stability, and it is well
known111but perhaps not well appreciated, or commonly undertaken! that
significantly more care must be taken when choosing an inner product for
compressible flows (as opposed to incompressible flows). This aspect is
emphasized in Rowley et al. [42], where the authors highlight that, in an
incompressible flow with state variables $\mathbf{u}_{1}$, $\mathbf{u}_{2}$,
and $\mathbf{u}_{3}$, a standard vector-valued $L^{2}(\Omega)$ inner product
induces a norm that measures the total kinetic energy of the system. The same
cannot be said for a compressible flow with, e.g., conserved variables, where
the standard vector-valued $L^{2}(\Omega)$ inner product induces a norm that
not only does not measure any physically meaningful quantity, but is also
dimensionally inconsistent, i.e., it adds together quantities that have
different units. This challenge is amplified for more complex configurations,
such as reacting flows, where different state variables span numerous orders
of magnitude and computational fluid dynamics solvers are often written to
solve the dimensional form of the governing equations. As a result it is
unclear which inner product is most appropriate for compressible flows. We
further emphasize that the choice of inner product is important for both POD
and the projection step used to construct the ROM, and it is possible that
different inner products could be used in each step.

Several studies have examined the impact of the choice of inner product on the
ROM performance, and we aim to provide a brief review here. For starters,
Rowley et al. [42] proposed a new, dimensionally-consistent, energy-based
inner product for use in POD and Galerkin projections for the isothermal Euler
equations. The approach is shown to be effective, but is not directly
applicable to the non-isothermal Euler equations. In a similar spirit, Refs.
[2, 1, 27, 43] developed and investigated symmetrizing inner products and
stabilizing projections for the linearized Euler equations discretized in
primitive variables; recent work has adapted this inner product within the
context of control [38]. Although not emphasized, we point out here that the
inner products developed in these works are dimensionally consistent. Galerkin
ROMs based on these stabilizing inner products have been shown to be more
effective than traditional Galerkin methods. Follow on work by Tezaur (f.k.a
Kalashnikova) and Barone put forth a framework for developing Galerkin ROMs of
the compressible Navier-Stokes equations discretized in entropy variables via
a vector-valued $L^{2}(\Omega)$ inner product [26]. This choice of variables
is motivated by the fact that (1) it is a symmetrizing transformation and (2)
a Galerkin projection process in the vector-valued $L^{2}(\Omega)$ inner
product with entropy variables results in a statement of the second law of
thermodynamics, which is akin to energy for incompressible flows. Although
again not emphasized in Ref. [26], this formulation results in ROMs that are
dimensionally consistent. Results were only presented for scalar conservation
laws and the choice of inner product for POD was not discussed. On a similar
note, recent work has developed entropy-stable hyper-reduced ROMs [14] based
on conserved variables with the standard vector-valued $L^{2}(\Omega)$ inner
product and re-projection of the states after mapping to and back from entropy
variables. The choice of inner product employed for POD was not explicitly
discussed in [14], but it appears a standard $L^{2}(\Omega)$ inner product was
employed. Lastly, while we have focused the above review to the Euler
equations, similar ideas exist for other physics. Ref. [28], for instance,
develops POD-Galerkin ROMs for magnetohydrodynamics (MHD) based on an “MHD
inner product”. Similarly, Ref. [44] develops POD-Galerkin ROMs of the
rotating shallow water equations based on an energy inner product space.

Another body of work that can be closely tied to the choice of inner product
is the development of ROMs based on least-squares residual minimization[31,
32, 33, 7, 39, 10, 6, 8, 10, 13, 12, 11, 35]. Instead of performing Galerkin
projection, these techniques compute solutions that lie within a reduced
subspace that minimize the residual of the system of interest. Like the
aforementioned entropy-based ROMs, least-squares techniques symmetrize the
system and enhance stability. We additionally note that these techniques are
often interpreted as Petrov-Galerkin projections with a time-varying test
basis. Typically, least-squares ROMs are formulated to minimize the residual
in a standard $L^{2}(\Omega)$-norm222Strictly speaking, most least-squares
have been formulated at the fully discrete level and minimize an $\ell^{2}$
norm.. While this choice of norm has resulted in ROMs that are more robust
than their Galerkin counterparts, application of the standard
$L^{2}(\Omega)$-norm to compressible flows results in a ROM formulation that
is dimensionally inconsistent (e.g., the residual of the momentum equation has
different units than the residual of the energy equation, and the standard
$L^{2}(\Omega)$ norm simply adds these two quantities together). This has been
increasingly realized in recent years. Washabaugh ans Huang et al. [23], for
example, formulate least-squares ROMs that minimizes a scaled residual.
Provided that the scaling is done consistently, this process can result in a
ROM formulation that is dimensionally consistent. We additionally note that
Refs. [23] and [5] provide a similar scaling for POD. While this scaling
process results in dimensionally-consistent ROMs, their resulting performance
will be, of course, tied to the choice of scaling; such a drawback was pointed
out in Rowley et al. [42].

This work aims to provide a simple but comprehensive overview of these
aforementioned issues by studying a set of inner products for performing POD
and building Galerkin and least-squares ROMs within the context of the
compressible Euler equations. In our study we focus on the dimensional
consistency of existing formulations and highlight the advantages of physics-
based inner products. Using dimensional arguments, we additionally extend
existing ideas in entropy-based ROMs by proposing a new inner product in which
to perform POD for entropy variables as well as a new “entropy-least-squares”
ROM based on this entropy inner product. In summary, the contributions of this
work are as follows:

  1. 1.

A concise framework for building various dimensionally-consistent ROMs through
the language of inner products.

  2. 2.

A new inner product space in which to perform POD based on entropy principles
and dimensional arguments.

  3. 3.

A new least-squares formulation based on an entropy inner product space. This
formulation results in solutions that are more accurate, both qualitatively
and quantitatively, than solutions obtained from Galerkin and least-squares
ROMs obtained in the traditional $L^{2}(\Omega)$ inner product.

  4. 4.

Numerical experiments via the open-source software pressio-demoapps
demonstrating the impact of the choice of inner product/norm.

Lastly, we emphasize that we do not consider hyper-reduction in this work. Our
focus is not to create the most effective ROM, but rather to directly
highlight the impact of the inner product/norm on the ROM results.

This paper proceeds as follows. First, in Section 2 we outline the
compressible Euler equations. Next, in Section 3 we introduce POD, highlight
challenges for systems with multiple states, and introduce a candidate set of
inner products in which to perform POD. Next, Section 4 outlines several ROM
formulations obtained through Galerkin projection and least-squares residual
minimization in different inner products. Next Sections 6-9 present numerical
experiments demonstrating the importance of the choice of inner product. In
these experiments we build and deploy ROMs on dimensional and non-dimensional
versions of the same problem to evaluate the performance and consistency of
the various methods. Conclusions are then provided in Section 10.

##  2 The compressible Euler equations

In this work we consider model reduction of the compressible Euler equations
in one and two-dimensions. In what follows we provide the formulation for the
two-dimensional setting.

The compressible Euler equations on the two-dimensional spatial domain
$\Omega$ are given for $t\in[0,T]$ as

$\frac{\partial\boldsymbol{U}}{\partial
t}+\nabla\cdot\boldsymbol{F}(\boldsymbol{U})=\boldsymbol{0},$ (1)

where $\boldsymbol{U}:[0,T]\times\Omega\rightarrow\mathbb{R}^{4}$ comprise the
density, $\boldsymbol{x}_{1}$ and $\boldsymbol{x}_{2}$ momentum, and total
energy. The inviscid fluxes are given by
$\boldsymbol{F}=\left[\boldsymbol{F}_{1},\boldsymbol{F}_{2}\right]$ where
$\boldsymbol{F}_{1}(\cdot)\in\mathbb{R}^{4}$ is the inviscid flux in the
$\boldsymbol{x}_{1}$ direction and
$\boldsymbol{F}_{2}(\cdot)\in\mathbb{R}^{4}$ is the inviscid flux in the
$\boldsymbol{x}_{2}$ direction. The state and inviscid fluxes are given by

$\boldsymbol{U}=\begin{Bmatrix}\rho\\\ \rho u_{1}\\\ \rho u_{2}\\\ \rho
E\end{Bmatrix},\qquad\boldsymbol{F}_{1}(\boldsymbol{U})=\begin{Bmatrix}\rho
u_{1}\\\ \rho u_{1}^{2}+p\\\ \rho u_{1}u_{2}\\\
u_{1}(E+p)\end{Bmatrix},\qquad\boldsymbol{F}_{2}(\boldsymbol{U})=\begin{Bmatrix}\rho
u_{2}\\\ \rho u_{1}u_{2}\\\ \rho u_{2}^{2}+p\\\ u_{2}(E+p)\end{Bmatrix},$

where we have assumed a calorically perfect gas such that the thermodynamic
pressure is given by

$p=\left(\gamma-1\right)\left(\rho
E-\frac{1}{2}\rho\left(u_{1}^{2}+u_{2}^{2}\right)\right),$

with $\gamma=1.4$ being the heat-capacity ratio. We assume the existence of
initial and boundary conditions such that Eq. (1) is well-posed.

The Euler equations are typically formulated in terms of conservative
variables, as in Eq. (1). An alternative path relevant to this work is to
express the Euler equations in terms of entropy variables. Specifically,
leveraging the formulation developed by Hughes [24], we introduce a set of
entropy variables as

$\boldsymbol{V}=\begin{Bmatrix}\frac{-s}{\gamma-1}+\frac{\gamma+1}{\gamma-1}-\frac{\rho
E}{p}\\\ \frac{\rho u_{1}}{p}\\\ \frac{\rho u_{2}}{p}\\\
-\frac{\rho}{p}\end{Bmatrix}$

with $s=\ln(p)-\gamma\log(\rho)$. The Euler equations can be written in terms
of the entropy variables as

$\boldsymbol{A}\frac{\partial\boldsymbol{V}}{\partial
t}+\frac{\partial\boldsymbol{F}_{1}}{\partial\boldsymbol{U}}\boldsymbol{A}\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{x}_{1}}+\frac{\partial\boldsymbol{F}_{2}}{\partial\boldsymbol{U}}\boldsymbol{A}\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{x}_{2}}=\boldsymbol{0},$
(2)

where
$\boldsymbol{A}\equiv\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\in\mathbb{S}^{4}$
is the Jacobian of the variable transformation and $\mathbb{S}^{N}$ is the
space of $N\times N$ symmetric matrices. The analytic form of the Jacobian
$\boldsymbol{A}$ is provided in A. Critically, $\boldsymbol{A}$,
$\frac{\partial\boldsymbol{F}_{1}}{\partial\boldsymbol{U}}\boldsymbol{A}$, and
$\frac{\partial\boldsymbol{F}_{2}}{\partial\boldsymbol{U}}\boldsymbol{A}$ are
all symmetric matrices, and thus Eq. (2) is “symmetrized”. We emphasize that,
while the Euler equations expressed in terms of entropy variables (2) is
analytically equivalent to the Euler equations expressed in terms of conserved
variables (1), the symmetric nature of the entropy formulation leads to
numerical formulations that enjoy, e.g., enhanced stability properties [21,
24].

######  Remark 2.1.

In this work we will consider several ROM formulations that employ entropy
variables. In our numerical experiments, these ROMs are constructed from a
flow solver that spatially discretizes the conservative from (1) rather than
the entropy form (2). We consider this case as most flow solvers do not
discretize in entropy variables. Consequences of this choice are a loss of
symmetry and entropy stability at the discrete level.333With special care
these properties can be retained, but we do not consider this case here.

##  3 Model reduction via POD

Projection-based reduced-order models generate approximate solutions to the
system of interest by first restricting the state to live within a low-
dimensional trial space. Various approaches exist for constructing this trial
space (e.g., the reduced-basis method, space–time methods) and here we focus
on the classic spatial-only POD-based approach [22, 4]. POD, which is closely
related to principle component analysis and the Karhunen Loeve expansion, is a
data-compression method that finds a $K$-dimensional orthonormal basis that
minimizes the projection error of a dataset in an inner product space
$\mathcal{I}$ equipped with the inner product
$\left(\cdot,\cdot\right)_{\mathcal{I}}$. For a scalar-valued system with a
state variable $u(\boldsymbol{x},t)$, POD results in the state being
approximated as444We note that some bodies of work consider an affine subspace
that, e.g., satisfies initial conditions or boundary conditions [20]; the
ideas presented here generalize to this setting

$u(\boldsymbol{x},t)=\sum_{i=1}^{K}\phi_{i}(\boldsymbol{x})\hat{u}_{i}(t),$

where $\phi_{i}:\Omega\rightarrow\mathbb{R}$, $i=1,\ldots,K$ are the POD basis
vectors and $\hat{u}_{i}:[0,T]\rightarrow\mathbb{R}$, $i=1,\ldots,K$ are
generalized coordinates.

The situation is more complicated for vector-valued systems with a state
variable
$\mathbf{u}(\boldsymbol{x},t):\Omega\times[0,T]\rightarrow\mathbb{R}^{M}$
where $M>1$ is the number of state variables (i.e., $M=4$ for the Euler
equations in two dimensions). Using the taxonomy of [42], two approaches exist
for applying POD to vector-valued systems like the Euler equations:

  1. 1.

Scalar-valued POD: A separate POD basis is constructed for each state
variable, i.e., $M$ separate POD problems are solved, one for each for
variable. The advantage of this approach is that each state variable is
treated separately and as a result the process is dimensionally consistent.
The disadvantages of this approach are that it cannot capture correlations
between the different state variables and results in larger ROMs.

  2. 2.

Vector-valued POD: A global basis is constructed for all the state variables.
In this approach one POD problem is solved and the result is used to construct
a basis for all the variables. Advantages of this approach are that it can
capture correlations between the different state variables and it leads to
smaller ROMs. The primary disadvantage of the approach is that it results in
dimensionally-inconsistent formulations if special care is not taken when
defining the inner product.

In this work we exclusively consider vector-valued POD. The reasons for this
are several, including (1) that vector-valued POD appears to be the more
common approach, (2) past works have not demonstrated that scalar-valued POD
outperforms vector-valued POD [42, 40, 41], and (3) scalar-valued POD results
in ROMs that are of a significantly higher dimension than vector-valued POD.

We now outline vector-valued POD within the context of the compressible Euler
equations in two-dimensions. Specifically, given a dataset of solution
“snapshots” $\boldsymbol{S}:\Omega\rightarrow\mathbb{R}^{4\times n_{S}}$,
where $n_{S}$ is the number of samples, vector-valued POD solves the
minimization problem

$\boldsymbol{\Phi}=\underset{\\{\boldsymbol{\Phi}^{*}:\Omega\rightarrow\mathbb{R}^{4\times
K}|\left(\boldsymbol{\Phi}^{*},\boldsymbol{\Phi}^{*}\right)_{\mathcal{I}}=\mathbf{I}\\}}{\text{arg
min}}\sum_{i=1}^{n_{S}}\|\boldsymbol{S}_{i}-\boldsymbol{\Phi}^{*}\left(\boldsymbol{\Phi}^{*},\boldsymbol{S}_{i}\right)_{\mathcal{I}}\|_{\mathcal{I}},$
(3)

where $\|\cdot\|\equiv\sqrt{\left(\cdot,\cdot\right)_{\mathcal{I}}}$ and, to
simplify the notation, we use $\boldsymbol{S}_{i}$ to denote the $i$th column
of the snapshot matrix, i.e., the $i$th sample. Assuming that the POD process
is employed on snapshots of the conservative variables, the conservative
variables are then approximated as

$\boldsymbol{U}(\boldsymbol{x},t)\approx\boldsymbol{\Phi}(\boldsymbol{x})\hat{\boldsymbol{U}}(t),$

where $\boldsymbol{\Phi}:\Omega\rightarrow\mathbb{R}^{4\times K}$ is the ROM
basis and $\hat{\boldsymbol{U}}:[0,T]\rightarrow\mathbb{R}^{K}$ are the
generalized coordinates. We note that a similar expression exists if POD was
performed on a different variable set, e.g., entropy variables.

The accuracy of the POD basis and the resulting ROM clearly depends on the
choice of the inner product space $(\cdot,\cdot)_{\mathcal{I}}$. This is
particularly true for vector-valued POD, where the minimization problem (3)
combines variables with different units. The remainder of this section
outlines various inner products within the context of the compressible Euler
equations to study this issue.

###  3.1 Vector-valued $L^{2}(\Omega)$ inner product in conserved variables

We first study the classic vector-valued $L^{2}(\Omega)$ inner product with
conserved variables. For
$\boldsymbol{u},\boldsymbol{v}:\Omega\rightarrow\mathbb{R}^{4}$ we define the
vector-valued $L^{2}(\Omega)$ inner product as

$\left(\boldsymbol{u},\boldsymbol{v}\right)_{L^{2}(\Omega)}=\int_{\Omega}\boldsymbol{u}(\boldsymbol{x})^{T}\boldsymbol{v}(\boldsymbol{x})d\boldsymbol{x}\equiv\sum_{i=1}^{4}\int_{\Omega}\boldsymbol{u}_{i}(\boldsymbol{x})\boldsymbol{v}_{i}(\boldsymbol{x})d\boldsymbol{x}.$

This inner product is not dimensionally consistent for the compressible Euler
equations with solution vectors defined in terms of the conserved variables as

$\left(\boldsymbol{U},\boldsymbol{U}\right)_{L^{2}(\Omega)}\equiv\int_{\Omega}\rho(\boldsymbol{x})\rho(\boldsymbol{x})d\boldsymbol{x}+\int_{\Omega}\rho
u_{1}(\boldsymbol{x})\rho
u_{1}(\boldsymbol{x})d\boldsymbol{x}+\int_{\Omega}\rho
u_{2}(\boldsymbol{x})\rho
u_{2}(\boldsymbol{x})d\boldsymbol{x}+\int_{\Omega}\rho E(\boldsymbol{x})\rho
E(\boldsymbol{x})d\boldsymbol{x},$

which does not make sense on dimensional grounds. Despite this inconsistency,
this inner product has been widely employed for model reduction of
compressible flows (particularly in the Galerkin/least-squares projection
process; see, for example, the residual-minimization formulation in the
seminal work [13]). The vector-valued $L^{2}(\Omega)$ inner product results in
a trial basis defined as the solution to the minimization problem

$\boldsymbol{\Phi}^{\boldsymbol{U}}=\underset{\\{\boldsymbol{\Phi}^{*}:\Omega\rightarrow\mathbb{R}^{4\times
K}|\left(\boldsymbol{\Phi}^{*},\boldsymbol{\Phi}^{*}\right)_{L^{2}(\Omega)}=\mathbf{I}\\}}{\text{arg
min}}\sum_{i=1}^{n_{S}}\big{|}\big{|}\boldsymbol{S}^{\boldsymbol{U}}_{i}-\boldsymbol{\Phi}^{*}\left(\boldsymbol{\Phi}^{*},\boldsymbol{S}^{\boldsymbol{U}}_{i}\right)_{L^{2}(\Omega)}\big{|}\big{|}_{L^{2}(\Omega)}$
(4)

where $\boldsymbol{S}^{\boldsymbol{U}}:\Omega\rightarrow\mathbb{R}^{4\times
n_{S}}$ is the sample matrix of the conserved state variables.

###  3.2 Non-dimensional vector-valued $L^{2}$ inner product in conserved
variables

The next inner product we consider is similar to the $L^{2}(\Omega)$ inner
product, but this time includes a scaling function that non-dimensionalizes
the variables. Specifically, we define a non-dimensional $L^{2}(\Omega)$ inner
product for $\boldsymbol{u},\boldsymbol{v}:\Omega\rightarrow\mathbb{R}^{4}$ as

$\left(\boldsymbol{u},\boldsymbol{v}\right)_{L^{2}_{*}(\Omega)}=\int_{\Omega}\boldsymbol{u}(\boldsymbol{x})^{T}\mathbf{D}(\boldsymbol{x})\boldsymbol{v}(\boldsymbol{x})d\boldsymbol{x}$
(5)

where $\mathbf{D}:\Omega\rightarrow\mathbb{R}^{4\times 4}$ is a diagonal
matrix given by

$\mathbf{D}=\begin{bmatrix}\frac{1}{\rho_{\infty}^{2}}&0&0&0\\\
0&\frac{1}{(\rho u_{1,\infty})^{2}}&0&0\\\ 0&0&\frac{1}{(\rho
u_{2,\infty})^{2}}&0\\\ 0&0&0&\frac{1}{(\rho E_{\infty})^{2}}\end{bmatrix}.$

In the above $\rho_{\infty},\rho u_{1,\infty},\rho u_{2,\infty}$, and $\rho
E_{\infty}:\Omega\rightarrow\mathbb{R}$ are the reference states for the
density, momentum, and energy, and may or may not depend on the spatial
location. This inner product is dimensionally consistent for the compressible
Euler equations with solution vectors defined in terms of the conserved
variables as

$\left(\boldsymbol{U},\boldsymbol{U}\right)_{L^{2}_{*}(\Omega)}\equiv\int_{\Omega}\frac{1}{\rho_{\infty}(\boldsymbol{x})^{2}}\rho(\boldsymbol{x})\rho(\boldsymbol{x})d\boldsymbol{x}+\int_{\Omega}\frac{1}{(\rho
u_{1,\infty})^{2}}\rho u_{1}(\boldsymbol{x})\rho
u_{1}(\boldsymbol{x})d\boldsymbol{x}+\\\ \int_{\Omega}\frac{1}{(\rho
u_{2,\infty}(\boldsymbol{x}))^{2}}\rho u_{2}(\boldsymbol{x})\rho
u_{2}(\boldsymbol{x})d\boldsymbol{x}+\int_{\Omega}\frac{1}{(\rho
E_{\infty})^{2}}\rho E(\boldsymbol{x})\rho E(\boldsymbol{x})d\boldsymbol{x}.$
(6)

This non-dimensional $L^{2}(\Omega)$ inner product results in a trial basis
defined as the solution to the minimization problem

$\boldsymbol{\Phi}^{\boldsymbol{U}}=\underset{\\{\boldsymbol{\Phi}^{*}:\Omega\rightarrow\mathbb{R}^{4\times
K}|\left(\boldsymbol{\Phi}^{*},\boldsymbol{\Phi}^{*}\right)_{L^{2}_{*}(\Omega)}=\mathbf{I}\\}}{\text{arg
min}}\sum_{i=1}^{n_{S}}\|\boldsymbol{S}^{\boldsymbol{U}}_{i}-\boldsymbol{\Phi}^{*}\left(\boldsymbol{\Phi}^{*},\boldsymbol{S}^{\boldsymbol{U}}_{i}\right)_{L^{2}_{*}(\Omega)}\|_{L^{2}_{*}(\Omega)}$
(7)

where $\boldsymbol{S}^{\boldsymbol{U}}:\Omega\rightarrow\mathbb{R}^{4\times
n_{S}}$ is the sample matrix of the conserved state variables.

We emphasize that scaled vector-valued POD processes like the one outlined
above have been pursued in the literature. Ref. [23], for instance, performs
POD on the snapshots where each state variable is scaled by its
$L^{2}(\Omega)$ norm on the training set. Ref. [5] pursues a similar approach,
but instead scales by the maximum value of each state variable.

######  Remark 3.1.

Selection of the reference state. The non-dimensional inner product requires
specification of the reference state. Numerous choices exist, like those
described above. To the best of the authors’ knowledge, no approach has been
demonstrated to be “superior”.

###  3.3 Energy inner product in transformed variables

An interesting possibility for creating dimensionally-consistent ROMs is based
off of the “energy” inner products proposed in [42]. Unfortunately, the energy
inner products proposed in Ref. [42] are only applicable to the isentropic
Euler equations, while the energy inner products proposed in Ref. [19] are not
formal inner products, mathematically speaking (positivity is not guaranteed).
As a result, we don’t consider these energy inner products.

###  3.4 Entropy inner product in entropy variables

The next inner product we consider is an “entropy-based” inner product that
acts on solution vectors defined in terms of entropy variables. We start by
noting that the product of the entropy variables and the time derivative of
the conserved variables results in [24]555We use the $\sim$ notation as
numerous definitions of the entropy exist, and different definitions may have
constants of proportionality.

$\boldsymbol{V}^{T}\frac{\partial}{\partial
t}\boldsymbol{U}\sim\frac{\partial}{\partial t}\rho s.$

We observe that this product is dimensionally consistent and results in a
physically meaningful quantity. Via the chain rule we can write the above as

$\boldsymbol{V}^{T}\frac{\partial}{\partial
t}\boldsymbol{U}=\boldsymbol{V}^{T}\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\frac{\partial}{\partial
t}\boldsymbol{V}.$

On dimensional grounds we propose an inner product space for use with entropy
variables that is based of an inner product of the type
$\boldsymbol{V}^{T}\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\boldsymbol{V}$.
We note the following important points:

  * 1.

The matrix $\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}$ is
symmetric positive definite so can be used to define an inner product.

  * 2.

To create an inner product space that is of use for POD, the weighting matrix
$\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}$ needs to be constant
(e.g., defined for some background state).

Thus, to create our entropy inner product space, we start by defining the
matrix

$\boldsymbol{A}_{\infty}(\boldsymbol{x})\equiv\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\left(\boldsymbol{V}_{\infty}(\boldsymbol{x})\right)$

where $\boldsymbol{V}_{\infty}$ is a reference state. Leveraging the fact that
$\boldsymbol{A}_{\infty}:\Omega\mapsto\mathbb{S}^{4}$ is a symmetric positive
definite matrix, we can define the inner product

$\left(\boldsymbol{u},\boldsymbol{v}\right)_{\boldsymbol{A}_{\infty}}=\int_{\Omega}\boldsymbol{u}(\boldsymbol{x})^{T}\boldsymbol{A}_{\infty}(\boldsymbol{x})\boldsymbol{v}(\boldsymbol{x})d\boldsymbol{x}.$
(8)

This inner product is naturally dimensionally consistent for the compressible
Euler equations with solution vectors defined in terms of the entropy
variables, and as a result does not require any additional scaling when
deployed on dimensional problems. This entropy-based inner product results in
a trial basis for the entropy variables defined as the solution to the
minimization problem

$\boldsymbol{\Phi}^{\boldsymbol{V}}=\underset{\\{\boldsymbol{\Phi}^{*}:\Omega\rightarrow\mathbb{R}^{4\times
K}|\left(\boldsymbol{\Phi}^{*},\boldsymbol{\Phi}^{*}\right)_{\boldsymbol{A}_{\infty}}=\mathbf{I}\\}}{\text{arg
min}}\sum_{i=1}^{n_{S}}\|\boldsymbol{S}^{\boldsymbol{V}}_{i}-\boldsymbol{\Phi}^{*}\left(\boldsymbol{\Phi}^{*},\boldsymbol{S}^{\boldsymbol{V}}_{i}\right)_{\boldsymbol{A}_{\infty}}\|_{L^{2}_{\boldsymbol{A}_{\infty}}(\Omega)}$
(9)

where $\boldsymbol{S}^{\boldsymbol{V}}:\Omega\rightarrow\mathbb{R}^{4\times
n_{S}}$ is the sample matrix of the entropy variables.

###  3.5 Entropy inner product in conserved variables

The last inner product we consider is an entropy-based inner product that acts
on solution vectors defined in terms of the conserved variables. Similar to
the entropy inner product space presented in the previous section, we derive
this inner product based on dimensional arguments. We start by recalling the
dimensional consistency of
$\boldsymbol{V}^{T}\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\boldsymbol{V}$,
and use this dimensional consistency to create an analogous inner product for
use with the conserved variables. On dimensional grounds this results in the
inner product
$\boldsymbol{U}^{T}\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}\boldsymbol{U}$,
which has the same units as
$\boldsymbol{V}^{T}\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\boldsymbol{V}$.
Similarly to our entropy inner product, we make the following comments:

  * 1.

The matrix $\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}$ is positive
definite and thus can be used to define an inner product and norm.

  * 2.

To create an inner product space that is of use for POD, the weighting matrix
$\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}$ needs to be constant.

Thus, to create this entropy inner product space we start by defining the
matrix

$\boldsymbol{B}_{\infty}(\boldsymbol{x})\equiv\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}\left(\boldsymbol{U}_{\infty}(\boldsymbol{x})\right)=\boldsymbol{A}_{\infty}^{-1}(\boldsymbol{x})$

where $\boldsymbol{U}_{\infty}$ is a reference state. Leveraging the fact that
$\boldsymbol{B}_{\infty}:\Omega\mapsto\mathbb{S}^{4}$ is a symmetric positive
definite matrix, we can define the inner product

$\left(\boldsymbol{u},\boldsymbol{v}\right)_{\boldsymbol{B}_{\infty}}=\int_{\Omega}\boldsymbol{u}(\boldsymbol{x})^{T}\boldsymbol{B}_{\infty}(\boldsymbol{x})\boldsymbol{v}(\boldsymbol{x})d\boldsymbol{x}.$
(10)

This inner product is dimensionally consistent for the compressible Euler
equations with solution vectors defined in terms of the conserved variables,
and as a result does not require any additional scaling when deployed on
dimensional problems. This entropy-based inner product results in a trial
basis for the conserved variables defined as the solution to the minimization
problem

$\boldsymbol{\Phi}^{\boldsymbol{U}}=\underset{\\{\boldsymbol{\Phi}^{*}:\Omega\rightarrow\mathbb{R}^{4\times
K}|\left(\boldsymbol{\Phi}^{*},\boldsymbol{\Phi}^{*}\right)_{\boldsymbol{B}_{\infty}}=\mathbf{I}\\}}{\text{arg
min}}\sum_{i=1}^{n_{S}}\|\boldsymbol{S}^{\boldsymbol{U}}_{i}-\boldsymbol{\Phi}^{*}\left(\boldsymbol{\Phi}^{*},\boldsymbol{S}^{\boldsymbol{U}}_{i}\right)_{\boldsymbol{B}_{\infty}}\|_{L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)}$
(11)

where $\boldsymbol{S}^{\boldsymbol{V}}:\Omega\rightarrow\mathbb{R}^{4\times
n_{S}}$ is the sample matrix of the entropy variables.

###  3.6 Summary of inner products

In summary, we considered four potential inner products: the classic
$L^{2}(\Omega)$ inner product, a non-dimensional $L^{2}_{*}(\Omega)$ inner
product, and two entropy-based inner products: one for use with solution
vectors defined in terms of entropy variables and the other for use with
solution vectors defined in terms of conservative variables. The classic
$L^{2}(\Omega)$ inner product is dimensionally inconsistent when used with
solution vectors defined in terms of conservative variables, while the non-
dimensional $L^{2}_{*}(\Omega)$ inner product,
$L^{2}_{\boldsymbol{A}_{\infty}}(\Omega)$ inner product, and
$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ inner product are dimensionally
consistent when used with solution vectors defined in terms of the
conservative variables, entropy variables, and conservative variables,
respectively. Table 1 summarizes these properties.

| $L^{2}(\Omega)$ | $L^{2}_{*}(\Omega)$ | $L^{2}_{\boldsymbol{A}_{\infty}}(\Omega)$ | $L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$
---|---|---|---|---
Dimensionally consistent | ✗ | ✓ | ✓ | ✓
Solution variables | $\boldsymbol{U}$ | $\boldsymbol{U}$ | $\boldsymbol{V}$ | $\boldsymbol{U}$
Table 1: Summary of potential inner products for POD

######  Remark 3.2.

A comment on orthogonality for Galerkin ROMs. POD results in bases that are
orthonormal in the inner product in which they were constructed. This
orthonormality is typically leveraged in a Galerkin projection process to
simplify the construction of the ROM. If the POD bases are constructed in an
inner product which differs from the inner product used in Galerkin
projection, then the typical simplifications stemming from orthonormality are
lost. We emphasize here that this is not a significant issue as (1) ROM
performance relies on the subspace described by the basis, not orthonormality
(e.g., consider two ROMs: one with an orthonormal basis and one with a non-
orthonormal basis. So long as the bases have the same range, Galerkin
projection will result ROMs that yield the same solution), and (2) the basis
can always be re-orthogonalized in the desired inner product via, e.g., a QR
decomposition.

##  4 Reduced-order models

Thus far we have outlined the construction of a reduced basis for the state
variables through POD in different inner products. In addition to construction
of this reduced-basis, a ROM requires a projection process applied to the
governing equations. This section outlines several such approaches. First,
Section 4.1 outlines the construction of the Galerkin ROM through various
inner products. Next, Section 4.2 outlines several least-squares ROMs that are
obtained from residual minimization in norms induced by different inner
product spaces.

###  4.1 Galerkin-based ROMs

Galerkin ROMs rely on a residual orthogonalization process where the residual
of the full-order model is restricted to be orthogonal to the trial space in
some inner product. Similar to POD, the performance of the Galerkin ROM
depends on the definition of the inner product. We now outline the
construction of several Galerkin-type ROMs via the inner products defined
above.

####  4.1.1 Galerkin ROM

The Galerkin ROM is arguably the most popular ROM formulation and, in general,
results in accurate and stable solutions for symmetric systems; in this case
Galerkin projection can be interpreted to be optimal in a system-specific
norm. In the present context the Galerkin ROM with solution vectors defined in
terms of the conservative variables is given by

$\left(\boldsymbol{\Phi}^{\boldsymbol{U}}_{i},\boldsymbol{\Phi}^{\boldsymbol{U}}\frac{\partial}{\partial
t}\hat{\boldsymbol{U}}\right)_{L^{2}(\Omega)}+\left(\boldsymbol{\Phi}^{\boldsymbol{U}}_{i},\nabla\cdot\boldsymbol{F}(\boldsymbol{\Phi}^{\boldsymbol{U}}\hat{\boldsymbol{U}})\right)_{L^{2}(\Omega)}=\boldsymbol{0}$
(12)

for $i=1,\ldots,K$. We will refer to this ROM formulation as the
Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$ ROM. As the compressible Euler
equations are not symmetric with respect to the conservative variables, we
expect the Galerkin ROM to, in general, perform poorly. Further, this Galerkin
ROM is not dimensionally consistent for the same reasons outlined above. We
further emphasize that, even if the POD bases are defined via a dimensionally-
consistent inner product, this ROM formulation is still dimensionally
inconsistent. We expect the Galerkin ROM to be the least accurate method we
explore.

####  4.1.2 Non-dimensional Galerkin reduced-order model

To remedy the dimensional inconsistency of the Galerkin ROM outlined above, we
can instead define a non-dimensional Galerkin ROM with solution vectors
defined in terms of the conservative variables as

$\left(\boldsymbol{\Phi}^{\boldsymbol{U}}_{i},\boldsymbol{\Phi}^{\boldsymbol{U}}\frac{\partial}{\partial
t}\hat{\boldsymbol{U}}\right)_{L^{2}_{*}(\Omega)}+\left(\boldsymbol{\Phi}^{\boldsymbol{U}}_{i},\nabla\cdot\boldsymbol{F}(\boldsymbol{\Phi}^{\boldsymbol{U}}\hat{\boldsymbol{U}})\right)_{L^{2}_{*}(\Omega)}=\boldsymbol{0}$
(13)

for $i=1,\ldots,K$. We will refer to this ROM formulation as the
Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROM. Unlike Eq. (12), Eq.
(13) is dimensionally consistent. Despite being dimensionally consistent,
however, we still expect this ROM to perform poorly as the Euler equations are
not symmetric with respect to the conserved variables.

####  4.1.3 Galerkin reduced-order model in entropy variables

We can develop a ROM addressing both of the deficiencies of the above Galerkin
ROMs, namely dimensional consistency and a lack of symmetry, by creating a
Galerkin ROM with solution vectors defined in terms of the entropy variables
as

$\left(\boldsymbol{\Phi}^{\boldsymbol{V}}_{i},\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\boldsymbol{\Phi}^{\boldsymbol{V}}\frac{\partial}{\partial
t}\hat{\boldsymbol{V}}\right)_{L^{2}(\Omega)}+\left(\boldsymbol{\Phi}^{\boldsymbol{V}}_{i},\nabla\cdot\boldsymbol{F}\left(\boldsymbol{U}\left(\boldsymbol{\Phi}^{\boldsymbol{V}}\hat{\boldsymbol{V}}\right)\right)\right)_{L^{2}(\Omega)}=\boldsymbol{0}$
(14)

for $i=1,\ldots,K$. We will refer to this ROM as the
Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$ ROM. This ROM is both
dimensionally consistent and symmetric with respect to the entropy variables.
As a result we expect this ROM to be more accurate and more stable than both
the ROMs outlined above.

###  4.2 Least-squares reduced-order models

Least-squares ROMs comprise a popular alternative to Galerkin-based ROMs.
Unlike Galerkin ROMs, which rely on residual orthogonalization, least-squares
ROMs rely on residual minimization, i.e., they compute solutions that lie
within a low-dimensional trial subspace that minimize the residual of the
system of interest in some norm. A variety of least-squares ROM formulations
exist, and here we formulate our least-squares ROMs within the context of the
windowed least-squares (WLS) approach [35], which addresses addresses several
deficiencies of the more standard least-squares Petrov-Galerkin (LSPG)
approach.

To describe WLS ROMs, we first define the residual of the compressible Euler
equations expressed in terms of the conserved variables as

$\mathcal{R}^{\boldsymbol{U}}:\boldsymbol{U}\mapsto\frac{\partial}{\partial
t}\boldsymbol{U}+\nabla\cdot\boldsymbol{F}(\boldsymbol{U}).$ (15)

Analogously, we define a residual for the compressible Euler equations in
terms of entropy variables as

$\mathcal{R}^{\boldsymbol{V}}:\boldsymbol{V}\mapsto\frac{\partial}{\partial
t}\boldsymbol{U}(\boldsymbol{V})+\nabla\cdot\boldsymbol{F}\left(\boldsymbol{U}\left(\boldsymbol{V}\right)\right).$
(16)

WLS ROMs operate by partitioning the time domain into a set of $N_{w}$ non-
overlapping time windows $[t_{s}^{n},t_{f}^{n}]\subset[0,T]$ of length $\Delta
T^{n}=t_{f}^{n}-t_{s}^{n}$, $n=1,\ldots,N_{w}$ such that $t_{s}^{1}=0$ and
$t_{f}^{N_{w}}=T$ and $t_{s}^{n+1}=t_{f}^{n}$, $n=1,\ldots,N_{w}-1$; see
Figure 1.

Figure 1: Depiction of the decomposition of the time domain into windows for
WLS.

WLS then sequentially seeks approximate solutions that lie within the trial
subspace that minimize the residual over each window in some norm defined by
the inner product space $\mathcal{I}$, i.e., WLS solves the sequence of
optimization problems for $n=1,\ldots,N_{w}$

$\tilde{\boldsymbol{U}}(t)=\underset{{\boldsymbol{U}}^{*}(t)\in\text{Range}(\boldsymbol{\Phi}^{\boldsymbol{U}})}{\text{arg
min}}\int_{t_{s}^{n}}^{t_{f}^{n}}\left(\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right),\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right)\right)_{\mathcal{I}}dt$
(17)

subject to the initial condition constraints
$\tilde{\boldsymbol{U}}(t_{s}^{n})=\tilde{\boldsymbol{U}}(t_{f}^{n-1})$ for
$n=2,\ldots,N_{w}$ and $\tilde{\boldsymbol{U}}(t_{s}^{1})=\boldsymbol{U}_{0}$,
where $\boldsymbol{U}_{0}$ is, e.g., projection of the initial conditions onto
the trial subspace. The analogous ROM formulation for a solution vector
defined in terms of entropy variables is given by

$\tilde{\boldsymbol{V}}(t)=\underset{{\boldsymbol{V}}^{*}(t)\in\text{Range}(\boldsymbol{\Phi}^{\boldsymbol{V}})}{\text{arg
min}}\int_{t_{s}^{n}}^{t_{f}^{n}}\left(\mathcal{R}^{\boldsymbol{V}}\left({\boldsymbol{V}}^{*}\right),\mathcal{R}^{\boldsymbol{V}}\left({\boldsymbol{V}}^{*}\right)\right)_{\mathcal{I}}dt$
(18)

subject to analogous initial condition constraints.

Various approaches exist to solve the sequence of WLS minimization problems,
including optimize-then-discretize approaches that involve solving the
Euler–Lagrange equations associated with the residual minimization principle,
and discretize-then-optimize approaches that involve discretizing the
objective function in space and time and solving the resulting nonlinear
least-squares problem with a discrete least-squares solver (e.g.,
Gauss–Newton).666In this case WLS is essentially LSPG, but with the residual
defined over multiple time steps. We consider the latter here due to its
simplicity and ease of implementation. In particular, we will consider a
discretize-then-optimize scheme where the temporal derivative in Eq. (17) (or
Eq. (18)) is discretized with the Crank-Nicolson method and the spatial
derivatives are discretized with the finite volume method.

Similar to POD, the choice of inner product dictates the accuracy of WLS ROMs.
The remainder of this section will outline several possible inner products for
WLS ROMs within the context of the Euler equations to investigate this issue.

####  4.2.1 $L^{2}(\Omega)$ inner product

The most straight-forward inner product to employ in the optimization problems
(17) and (18) is the standard vector-valued $L^{2}(\Omega)$ inner product. In
this case our ROM formulation with a basis for the conserved variables becomes

$\tilde{\boldsymbol{U}}(t)=\underset{{\boldsymbol{U}}^{*}(t)\in\text{Range}(\boldsymbol{\Phi}^{\boldsymbol{U}})}{\text{arg
min}}\int_{t_{s}^{n}}^{t_{f}^{n}}\left(\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right),\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right)\right)_{L^{2}(\Omega)}dt.$
(19)

A similar definition exists if we discretize in entropy variables. We will
refer to WLS ROMs that employ the above inner product and discretize the
conserved variables as WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$ ROMs.

We emphasize that, like the classic $L^{2}(\Omega)$ inner product employed in
POD, this choice of inner product is dimensionally inconsistent for the
residual minimization problems (17) and (18) as it results in a norm that,
e.g., adds the residual of the energy equation to the residual of the momentum
equations. Regardless, this inner product as been widely employed in least-
squares model reduction (see again Ref. [13]). Due to their symmetric nature
and tie to an underlying residual minimization problem, least-squares ROMs
based on the $L^{2}(\Omega)$ inner product have, in general, been observed to
be more robust than their Galerkin counterparts.

####  4.2.2 Non-dimensional $L^{2}_{*}(\Omega)$ inner product

A more consistent approach is to employ a non-dimensional $L^{2}_{*}(\Omega)$
inner product like that defined in Eq. (5). With this inner product, and
discretizing in conserved variables, our ROM formulation becomes

$\tilde{\boldsymbol{U}}(t)=\underset{{\boldsymbol{U}}^{*}(t)\in\text{Range}(\boldsymbol{\Phi}^{\boldsymbol{U}})}{\text{arg
min}}\int_{t_{s}^{n}}^{t_{f}^{n}}\left(\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right),\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right)\right)_{L^{2}_{*}(\Omega)}dt.$
(20)

A similar definition again exists if we discretize in entropy variables. We
will refer to WLS ROMs that employ the above inner product and discretize the
conserved variables as WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROMs.

This choice of inner product is dimensionally consistent for the residual
minimization problems (17) and (18). We emphasize that non-dimensional inner
products have been considered in the literature, see, e.g., Refs. [23, 5], and
typically have resulted in more robust ROMs than those obtained with direct
$L^{2}(\Omega)$ residual minimization. Despite being dimensionally consistent,
this inner product is arguably not a “physics-based” norm as it simply adds
scaled variables of different units.

####  4.2.3 Entropy inner product

Lastly, we consider WLS ROMs based on entropy inner products. Noting that the
residual of the Euler equations has units of $\frac{d}{dt}\boldsymbol{U}$, we
employ the inner product $L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ described
in Section 3.5. With this inner product, and discretizing in conserved
variables, our ROM formulation becomes

$\tilde{\boldsymbol{U}}(t)=\underset{{\boldsymbol{U}}^{*}(t)\in\text{Range}(\boldsymbol{\Phi}^{\boldsymbol{U}})}{\text{arg
min}}\int_{t_{s}^{n}}^{t_{f}^{n}}\left(\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right),\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right)\right)_{\boldsymbol{B}_{\infty}}dt.$
(21)

We will refer to WLS ROMs that employ the
$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ inner product and discretize the
conserved variables as
WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ ROMs.
A similar definition exists if we discretize in entropy variables, in which
case we obtain the ROM formulation

$\tilde{\boldsymbol{V}}(t)=\underset{{\boldsymbol{V}}^{*}(t)\in\text{Range}(\boldsymbol{\Phi}^{\boldsymbol{V}})}{\text{arg
min}}\int_{t_{s}^{n}}^{t_{f}^{n}}\left(\mathcal{R}^{\boldsymbol{V}}\left({\boldsymbol{V}}^{*}\right),\mathcal{R}^{\boldsymbol{V}}\left({\boldsymbol{V}}^{*}\right)\right)_{\boldsymbol{B}_{\infty}}dt.$
(22)

We will refer to ROMs based on the above formulation as
WLS${}_{\text{entropy}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ ROMs. This
choice of inner product is dimensionally consistent for the residual
minimization problems (21) and (22), and as a result does not require any
additional scaling if deployed on a dimensional problem.

######  Remark 4.1.

We chose to construct our “entropy norm” based on a fixed background state
$\boldsymbol{U}_{\infty}$. We note that an alternative option would be to left
multiply our residual by the Cholesky decomposition of
$\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}$ evaluated about the
current state and minimize the residual in the standard $L^{2}(\Omega)$-norm,
i.e., we could solve

$\tilde{\boldsymbol{U}}(t)=\underset{{\boldsymbol{U}}^{*}(t)\in\text{Range}(\boldsymbol{\Phi}^{\boldsymbol{U}})}{\text{arg
min}}\int_{t_{s}^{n}}^{t_{f}^{n}}\left(\sqrt{\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}({\boldsymbol{U}}^{*})}\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right),\sqrt{\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}({\boldsymbol{U}}^{*})}\mathcal{R}^{\boldsymbol{U}}\left({\boldsymbol{U}}^{*}\right)\right)_{L^{2}(\Omega)}dt$

where we use $\sqrt{\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}}$ to
denote the Cholesky decomposition of
$\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}$ such that
$\sqrt{\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}}^{T}\sqrt{\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}}=\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{U}}$.
In this case our ROM formulation could be interpreted as a “preconditioned”
ROM [34]. We choose a fixed background state as (1) it directly relates to an
inner product space (e.g., that could be used in POD) and (2) it simplifies
the numerical implementation, but note that employing the “preconditioned”
version may have advantages.

###  4.3 Summary of considered ROMs

In summary we outlined the construction of 6 types of ROMs. The first three
ROMs comprised Galerkin-type methods that were obtained by restricting the
residual of the governing equations to be orthogonal to the trial space in a
standard $L^{2}(\Omega)$ inner product, a non-dimensional $L^{2}(\Omega)$
inner product, and in a standard $L^{2}(\Omega)$ inner product but with a POD
basis for entropy variables. The second three ROMs considered were least-
squares ROMs that were obtained via residual minimization in a norm induced by
the standard $L^{2}(\Omega)$ inner product, a non-dimensional
$L^{2}_{*}(\Omega)$ inner product, and an “entropy”
$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ inner product. All of the ROMs
considered are dimensionally consistent except for the Galerkin and WLS ROMs
employing the $L^{2}(\Omega)$ inner product. Only the “entropy” ROMs employ
“physics-based” inner products. Tables 2 and 3 summarize these properties.
Lastly, we note that additional ROMs could be constructed from the described
inner products that we do not consider here. For instance, we could create a
Galerkin ROM discretized in conserved variables based on the
$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ inner product; this ROM would be
dimensionally consistent and may have better stability properties than the
standard Galerkin ROM.

| Gal${}_{\text{cons.}}$-$L^{2}(\Omega)$ | Gal${}_{\text{cons.}}$-$L^{2}_{*}(\Omega)$ | Gal${}_{\text{ent.}}$-$L^{2}(\Omega)$
---|---|---|---
Dimensionally consistent | ✗ | ✓ | ✓
Physics-based | ✗ | ✗ | ✓
Table 2: Summary of Galerkin ROMs considered. | WLS${}_{\text{cons.}}$-$L^{2}(\Omega)$ | WLS${}_{\text{cons.}}$-$L^{2}_{*}(\Omega)$ | WLS${}_{\text{cons.}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ | WLS${}_{\text{ent.}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$
---|---|---|---|---
Dimensionally consistent | ✗ | ✓ | ✓ | ✓
Physics-based | ✗ | ✗ | ✓ | ✓
Table 3: Summary of WLS ROMs considered.

##  5 Current results for numerical analyses of ROMs for the compressible
Euler equations

Before proceeding, we briefly summarize available results (to the best of the
authors’ knowledge) for numerical analyses of the above ROMs applied to the
compressible Euler equations. We note that analyses for the Galerkin ROMs
deployed on simplified versions of the Euler equations (e.g., the isothermal
Euler equations, the linearized Euler equations) exist in the literature [27,
42], but here we restrict our discussion to the fully nonlinear Euler
equations.

  * 1.

Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$. There are presently no
theoretical results guaranteeing stability of the Galerkin ROM with a basis
for the conserved variables applied to the compressible Euler equations
(numerical results have demonstrated that the Galerkin ROM is often unstable).
There are additionally no practical, sharp a priori or a posteriori error
bounds that exist. We note that analyses for the Galerkin ROM have been
undertaken for a generic dynamical system in Ref. [11]; these analyses rely on
the availability of difficult-to-compute Lipshitz constants and sharpness was
not addressed.

  * 2.

Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$. The non-dimensional
Galerkin ROM, to the best of the authors knowledge, has not been considered in
the literature. As a result there are no available analyses; we fully expect
the non-dimensional Galerkin ROM to suffer from the same shortcomings of the
standard Galerkin ROM.

  * 3.

Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$. The Galerkin ROM with a basis
for the entropy variables can leverage (in part) past analyses by e.g.,
Hughes, Tadmor, and Harten [24, 45, 46], that was performed in the context of
finite elements. The Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$ ROM has
been analyzed within the ROM context in Ref. [26], where Tezaur and Barone
demonstrate that the Galerkin ROM discretized in entropy variables is “entropy
stable”, meaning that it can be shown to satisfy the second law of
thermodynamics (the Clausius-Duhem inequality). In order to be entropy-stable
in practice the Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$ ROM must employ
an entropy stable discretization of the governing equations (e.g., via entropy
conserving fluxes [25]). We remark that, even if the
Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$ ROM is entropy stable, it is not
totally clear if this guarantees numerical stability. Further, no error bounds
have been derived for the Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$ ROM.

  * 4.

WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$. Analyses for WLS ROMs have been
carried out for a generic dynamical system in Ref. [35], but similar to the
Galerkin ROM these analyses rely on difficult-to-compute Lipshitz constants
and may not be sharp. These analyses indicated that minimizing the residual
over larger windows resulted in stable and more accurate ROMs, but in general
it is not clear if stability can be guaranteed. A similar ROM formulation that
preceded WLS is the least-squares Petrov Galerkin (LSPG) ROM, and analyses of
LSPG applied to a generic dynamical system were again presented in Ref. [11].
Stability for LSPG is not guaranteed, and the error bounds suffer from the
same issues of sharpness and difficult-to-compute Lipshitz constants.

  * 5.

WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$,
WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$, and
WLS${}_{\text{entropy}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$. The WLS
ROM in these norms has not been considered in the literature, and as a result
there are no available analyses.

In summary, none of the ROMs described in this work are equipped with
guarantees of stability when they are applied to the compressible Euler
equations, nor are they equipped with a priori or a posteriori error bounds.
We believe this should be an area for future work.

##  6 Description of numerical experiments

We now present numerical experiments studying the impact of the choice of
inner product on the POD bases and the resulting ROM formulation. We consider
three numerical experiments: the Sod shock tube in one dimension, the Kelvin
Helmholtz instability in two dimensions, and homogeneous isotropic turbulence
in two dimensions. For each problem we create a “non-dimensional” and
“dimensional” configuration. These configurations are designed to operate at
the same Mach number and non-dimensional time step, and are defined over the
same non-dimensional time horizon. Methods that are dimensionally consistent
should produce identical results on the two configurations, while methods that
are not dimensionally consistent should produce different results on the two
configurations. All ROMs considered are reproductive, and our aim is only to
study the impact of the inner product.

All three problems are simulated in Python using pressio-demoapps (PDA). PDA
is an open-source project developed at Sandia National Laboratory that is
aimed at providing benchmark ROM cases, and is part of the Pressio ecosystem.
Features of PDA include a cell-centered finite volume-based solver equipped
with several flux schemes, exact Jacobians, sample mesh capabilities, and
benchmark cases in one, two, and three dimensions. We emphasize that PDA’s
Euler solver discretizes the governing equations in their conserved form with
conserved variables. For ROMs employing entropy variables, we simply employ
the required change of variables before evaluating the spatial fluxes in PDA.
We now describe the specific cases studied in this work.

###  6.1 Sod shock tube

The first problem considered is the Sod shock tube in one dimension, which is
a typical benchmark problem for the compressible Euler equations. The problem
is governed by the compressible Euler equations on the 1D domain
$\Omega\in[-0.5,0.5]$. For the spatial discretization, we use a finite volume
scheme with 500 uniformly sized cells and Weno5 flux reconstruction. For the
temporal discretization, we employ a low-storage fourth-order Runge-Kutta
(RK4) scheme. The initial conditions are

$\left[\rho,\;u_{1},\;p\right]=\begin{cases}\left[\rho_{\infty},\;0,\;\gamma
p_{\infty}\right]&x<0\\\
\left[\frac{1}{8}\rho_{\infty},\;0,\frac{\gamma}{10}p_{\infty}\right]&x\geq
0\end{cases}.$

The PDA webpage for this case can be found here.

We perform two FOM simulations: a “non-dimensional” FOM and a dimensional FOM.
The non-dimensional FOM employs $\rho_{\infty}=1$ and
$p_{\infty}=\frac{1}{\gamma}$, while the dimensional FOM employs
$\rho_{\infty}=1.225$ and $p_{\infty}=101325$. The FOMs are simulated for
$t\in[0,0.25/a_{\infty}]$, where $a_{\infty}=\sqrt{\gamma
p_{\infty}/\rho_{\infty}}$ with a time step of ${\Delta t}=\frac{0.25\Delta
x}{a_{\infty}}$ where $\Delta x=\frac{1}{500}$. We collect solution snapshots
every time step for construction of the POD bases. Figure 2 shows
representative plots of the density at three different times simulated using
the “non-dimensional” FOM.

Figure 2: Representative plots of the density field at $t=0$ (left), $t=0.125$
(middle) and $t=0.25$ (right) obtained for the “non-dimensional” Sod 1D FOM.

###  6.2 Kelvin Helmholtz instability

The second example we consider is the Kelvin Helmholtz instability. The
problem is governed by the compressible Euler equations on the periodic domain
$\Omega\in[-5,5]\times[-5,5]$ and is run for 2.5 flow through times. For
spatial discretization we again employ a finite volume scheme with Weno5777We
note that in the Weno reconstruction, a small number is typically added to the
denominator of the nonlinear weights. We found that the standard value of
$10^{-6}$ led to inconsistencies between our dimensional and non-dimensional
configurations for the second two numerical examples. For these two cases we
employed a value of $10^{-20}$. flux reconstruction on a discrete domain
comprising 256x256 uniformly spaced volumes. Time integration is performed
with a low-storage RK4 scheme. The initial conditions are:

$\left[\rho,\;u_{1},\;u_{2},\;p\right]=\begin{cases}\left[2\rho_{\infty},\;\frac{1}{2}a_{\infty},\;0,\;\frac{7}{2}p_{\infty}\right]&\boldsymbol{x}\in\Omega_{1}\\\
\left[\rho_{\infty},\;-\frac{1}{2}a_{\infty},\;0,\;\frac{7}{2}p_{\infty}\right]&\boldsymbol{x}\in\Omega_{2}\end{cases}$

where $\Omega_{2}=[-5,5]\times[-2+\cos(0.8\pi x),2+\cos(0.8\pi x)]$ and
$\Omega_{1}=\Omega-\Omega_{2}$. The PDA webpage for this case can be found
here.

We again perform two FOM simulations: a “non-dimensional” FOM with
$\rho_{\infty}=1,\;p_{\infty}=\frac{1}{\gamma},\;a_{\infty}=1$, and a
“dimensional” FOM with $\rho_{\infty}=1.225,\;p_{\infty}=101325$ and
$a_{\infty}=\sqrt{\frac{\gamma p_{\infty}}{\rho_{\infty}}}$. The FOMs are
simulated for $t\in[0,50/a_{\infty}]$ at a time step of $\Delta
t=\frac{0.25\Delta x}{a_{\infty}}$ with $\Delta x=\frac{10}{256}$. We export
solution snapshots every five time steps for construction of the POD bases.
Figure 3 shows representative plots of the density at three different times
simulated using the “non-dimensional” FOM.

(a) $t=0.0$

(b) $t=24.4$

(c) $t=48.8$

Figure 3: Representative plots of the density at $t=0.0$ (left), $t=24.4$
(middle) and $t=48.8$ (right) obtained for the “non-dimensional” 2D Kelvin
Helmholtz FOM.

###  6.3 Two-dimensional homogeneous turbulence at an infinite Reynolds number

The final problem setup we consider comprises homogeneous turbulence at an
infinite Reynolds number in two dimensions. The problem is again governed by
the compressible Euler equations on the periodic domain
$\Omega\in[-5,5]\times[-5,5]$. For spatial discretization we employ a finite
volume scheme with Weno5 reconstruction on a discrete domain comprising
512x512 uniformly spaced volumes. Time integration is performed with a low-
storage RK4 scheme. 888This problem is not currently available in PDA, but it
will be included in the next release. The problem is initialized in the
frequency domain based on the kinetic energy spectrum

$e(\mathbf{k})=\frac{50}{k_{p}}au_{0}^{2}\left(\frac{\|\mathbf{k}\|}{k_{p}}\right)^{2s+1}\exp\left(-\left(s+\frac{1}{2}\right)\left(\frac{\|\mathbf{k}\|}{k_{p}}\right)^{2}\right)$

where $\mathbf{k}=\left[k_{1},k_{2}\right]$ is the wave number,
$\|\mathbf{k}\|=\sqrt{k_{1}^{2}+k_{2}^{2}}$ is the wave number magnitude, and
$u_{0}=25a_{\infty}$, $k_{p}=25$, and $s=3$. The initial density and pressure
are set to be uniform with values of $\rho_{\infty}$ and $p_{\infty}$,
respectively. The problem is initialized in physical space by first
prescribing a vorticity field with the desired energy spectrum and then
constructing the $x_{1}$ and $x_{2}$ velocity fields based on the vorticity
field; see Ref. [15] for details.

We again perform two FOM simulations: a “non-dimensional” FOM with
$\rho_{\infty}=1,\;p_{\infty}=\frac{1}{\gamma},\;a_{\infty}=1$, and a
“dimensional” FOM with $\rho_{\infty}=1.225,\;p_{\infty}=101325$ and
$a_{\infty}=\sqrt{\frac{\gamma p_{\infty}}{\rho_{\infty}}}$. The FOMs are
simulated for $t\in[0,20/a_{\infty}]$ at a time step of $\Delta
t=\frac{0.25\Delta x}{a_{\infty}}$ with $\Delta x=\frac{10}{512}$. We export
solution snapshots every five time steps for construction of the POD bases.
Figure 4 shows representative plots of the $x_{1}$ momentum at three different
times simulated using the “non-dimensional” FOM.

(a) $t=0.0$

(b) $t=9.8$

(c) $t=19.5$

Figure 4: Representative plots of the $x_{1}$ momentum at $t=0.0$ (left),
$t=9.8$ (middle) and $t=19.5$ (right) obtained for the “non-dimensional” 2D
homogeneous turbulence FOM.

###  6.4 Reduced-order models studied

In our experiments we study the performance of 7 ROM configurations using the
various inner products described above. Tables 4 and 5 summarize these ROMs
and identify the variables that the basis represents, the POD problem that is
solved to obtain the bases, and the formulation used to construct the ROM. All
ROMs employ the same finite volume discretization used by the FOMs. All
Galerkin ROMs employ the same RK4 time stepping scheme used by the FOMs, while
all WLS ROMs employ the Crank Nicolson method for time discretization.999The
error incurred by temporal discretization is small for the CFL numbers we
consider in the experiments, and the difference between the Crank Nicolson and
RK4 time schemes is minimal. We employ Crank Nicolson for convenience of
evaluating the residual in the least-squares minimization problem arising in
WLS. The time step for all ROMs is set to be the same as the FOM time step.
For all WLS ROMs we employ a uniform window size. For the 1D Sod problem the
window size is set to be $\Delta T=10\Delta t$. For the 2D Kelvin Helmholtz
and homogeneous turbulence problems the window size is set to be $\Delta
T=2\Delta t$. The nonlinear least-squares problem resulting from the WLS
formulation is solved using scipy’s scipy.optimize.least_squares solver with
default settings. We emphasize again that ROM run time is not an aspect we are
considering in this work; we are focused on the impact of the inner product
space on the ROM performance. Lastly, for each problem we study ROMs of
varying basis dimensions. For the 1D Sod problem we study ROMs of basis
dimensions $K=5,10,15,20,25,30,35,40,45,50$, while for the 2D Kelvin Helmholtz
and homogeneous turbulence problems we study ROMs of dimensions $K=25,50,100,$
and $150$.

We now discuss construction of the inner product spaces. The above experiments
were all constructed such that the non-dimensional configurations have a
reference speed of sound $a_{\infty}=1$ based on a thermodynamic state of
$\rho_{\infty}=1,p_{\infty}=\frac{1}{\gamma}$. Similarly, the “dimensional”
configurations were constructed to have a reference speed of sound of
$a_{\infty}=340.3$ based on a thermodynamic state of $\rho_{\infty}=1.225$ and
$p_{\infty}=101325$. We use these reference variables for the non-dimensional
inner product (5). For the non-dimensional configuration we set
$\rho_{\infty}=1$, $\rho_{\infty}u_{1,\infty}=\rho_{\infty}a_{\infty}$,
$\rho_{\infty}u_{2,\infty}=\rho_{\infty}a_{\infty}$, and $\rho
E_{\infty}=\rho_{\infty}a_{\infty}^{2}$ with $a_{\infty}=1$. For the
dimensional configuration we set $\rho_{\infty}=1.225$, $\rho
u_{1,\infty}=\rho_{\infty}a_{\infty}$, $\rho
u_{2,\infty}=\rho_{\infty}a_{\infty}$, and $\rho
E_{\infty}=\rho_{\infty}a_{\infty}^{2}$ with $a_{\infty}=340.3$. This
selection of reference states is not unique, and it is possible the other
choices may give better results. We additionally note that this choice of
reference states makes the $L^{2}_{*}(\Omega)$ inner product equivalent to the
$L^{2}(\Omega)$ inner product when deployed on the non-dimensional
configuration, which is intentional. Lastly, for the entropy inner products
(8) and (10) we define the background states $\boldsymbol{V}_{\infty}$ and
$\boldsymbol{U}_{\infty}$ as their mean over the training snapshots101010We
note that this choice of reference state cannot be used to select the values
of, e.g., $\rho_{\infty}u_{1,\infty}$ for use in the $L^{2}_{*}(\Omega)$ inner
product as it can result in the inner product not being positive definite if,
e.g., the mean value of $\rho u_{1}$ is negative. While there are other
logical choices, such as employing the norm of the snapshot training data or
the mean absolute value, this selection is ultimately subjective and it is not
clear if one approach is favored. The entropy inner product is not subject to
these issues as positivity is guaranteed, even if the mean value is negative.
used to compute the basis (e.g., no spatial dependence). Again, it is possible
that a different choice of background state could yield better results.

| Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$ | Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ | Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$ |
---|---|---|---|---
Basis variables | $\boldsymbol{U}$ | $\boldsymbol{U}$ | $\boldsymbol{V}$ |
POD method | Eq. (4) | Eq. (7) | Eq. (9) |
ROM method | Eq. (12) | Eq. (13) | Eq. (14) |
Table 4: Summary of Galerkin ROMs investigated. | WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$ | WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ | WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ | WLS${}_{\text{entropy}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$
---|---|---|---|---
Basis variables | $\boldsymbol{U}$ | $\boldsymbol{U}$ | $\boldsymbol{U}$ | $\boldsymbol{V}$
POD method | Eq. (4) | Eq. (7) | Eq. (7) | Eq. (9)
ROM method | Eq. (19) | Eq. (20) | Eq. (21) | Eq. (22)
Table 5: Summary of WLS ROMs investigated.

###  6.5 Error metrics

We employ the relative, time-integrated mean-squared error as our error metric
for the following experiments.111111We note that we investigated other error
metrics, e.g., the $\ell^{\infty}$ error, and found that they resulted in
similar conclusions. We compute separate errors for each conserved solution
variable, and thus have four (three in one-dimension) errors that we monitor.
For, e.g., the density field, this error metric is defined by

$e_{\rho}=\frac{\int_{0}^{T}\int_{\Omega}\left(\tilde{\rho}(\boldsymbol{x},t)-\rho(\boldsymbol{x},t)\right)^{2}d\boldsymbol{x}dt}{\int_{0}^{T}\int_{\Omega}\rho(\boldsymbol{x},t)^{2}d\boldsymbol{x}dt}$

where $\tilde{\rho}(\boldsymbol{x},t)$ is the ROM approximation to the density
field. We define analogous errors for the $x_{1}$-momentum, $x_{2}$-momentum,
and energy fields as $e_{\rho u_{1}}$, $e_{\rho u_{2}}$, and $e_{\rho E}$,
respectively.

###  6.6 Stability

In our numerical experiments we will use the terms “stable” and “unstable”.
Numerous definitions of stability exist in the literature. In our work, we
will refer to a ROM as “unstable” if it results in a solution containing a
NaN, and thus the solver cannot continue. Similarly, we will refer to a ROM as
“stable” if it results in a real-valued solution.

##  7 Non-dimensional vs. dimensional $L^{2}(\Omega)$ inner products

We first explore the impact of constructing POD-Galerkin and POD-WLS ROMs
based on standard non-dimensional and dimensional $L^{2}(\Omega)$ inner
products; we will consider the physics-based “entropy” inner products in
subsequent sections. We emphasize that we consider this setting first as
Galerkin and least-squares ROMs based on the standard $L^{2}(\Omega)$ inner
products are by far the most studied ROMs.

We first study POD in the various inner products. Figure 5 presents the error
associated with projection of the FOM training data onto the trial subspace as
a function of basis dimension for the various inner products examined in
Section 3. We note that projection is done in the same inner product as which
the corresponding basis is constructed. Results are presented for the inner
products applied to the dimensional and non-dimensional FOM configurations. In
all cases we observe that the $L^{2}(\Omega)$ inner product results in
different bases when applied to the dimensional data vs. non-dimensional data.
This difference is very clear for the Kelvin Helmholtz problem (Figure 5,
center column) as well as for the $x_{1}$ and $x_{2}$ momentum fields for the
homogeneous turbulence problem (Figure 5, right column). The discrepancy is
very minor for the Sod problem (Figure 5, left column) and is not visibly
obvious, but is still present. For the Kelvin Helmholtz problem we observe
that, when compared to the non-dimensional $L^{2}_{*}(\Omega)$ inner product,
the $L^{2}(\Omega)$ inner product applied to the dimensional data results in
bases that have a poor approximation power for the density, $x_{1}$, and
$x_{2}$ momentum fields, but a good approximation power for the total energy
field. This result is intuitive as the total energy snapshots are orders of
magnitude larger than the density and velocity snapshots and hence dominate
the $L^{2}(\Omega)$ inner product. We next observe that, as expected, the non-
dimensional $L^{2}_{*}(\Omega)$ inner product results in bases with equivalent
approximation power when applied to the dimensional and non-dimensional data.
Overall, we observe that the non-dimensional $L_{*}^{2}(\Omega)$ inner product
is slightly “better” for bases construction.

Figure 5: Relative errors associated with projection of the snapshot data on
to the trial subspace for the Sod 1D problem (left column), Kelvin Helmholtz
problem (center column), and homogeneous turbulence problem (right column).
Results are shown for the mass (top row), $x_{1}$ momentum (second row),
$x_{2}$ momentum (where applicable, third row), and energy (last row) state
variables as a function of basis dimension. Results are shown for both the
dimensional and non dimensional data.

Next we consider the results of the
Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$,
Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$,
WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$, and
WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROMs. Figure 6 provides a
summary of the performance of these ROMs on the three exemplars. First,
Figures 6(a)-6(c) depict the error for the density field predicted by the
various ROMs at dimensions of $K=50,50,$ and $100$ as a function of time for
the Sod 1D, Kelvin Helmholtz, and homogeneous turbulence problems,
respectively. Results are shown for both the dimensional and non-dimensional
configurations. Next, Figure 6(d) summarizes the overall ROM performance by
tabulating (1) the number of times a ROM formulation gave solutions with the
lowest errors for a state variable (among the
Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$,
Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$,
WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$, and
WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROMs) and (2) the percentage of
the time a ROM formulation was stable. The results in Figure 6(d) are computed
by aggregating across all ROM runs on the dimensional configurations. We make
the following observations:

  * 1.

(Figures 6(a)-6(c).) The Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$
and WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROMs, which are based on
non-dimensional inner products, yield the same results when deployed on the
dimensional and non-dimensional configurations. This result is expected and
demonstrates that these methods are dimensionally consistent.

  * 2.

(Figures 6(a)-6(c).) The Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$ and
WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$ ROMs, which are not based on
dimensionally-consistent inner products, yield different results for the
dimensional and non-dimensional configurations. This result is very minor for
the Sod 1D problem, but is very evident for the Kelvin Helmholtz and
homogeneous turbulence examples. This result is again expected, and
demonstrates that these ROM formulations are dimensionally inconsistent.

  * 3.

(Figures 6(a)-6(d).) When applied to dimensional problems, the ROMs based on
the non-dimensional $L^{2}_{*}(\Omega)$ inner product consistently outperform
their counterparts that are based on the standard dimensional $L^{2}(\Omega)$
inner product.

  * 4.

(Figure 6(d).) The Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$ and
Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROMs are rarely stable;
the Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$ ROM was stable less than
40% of the time, while the Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$
ROM was stable approximately 40% of the time.

  * 5.

(Figure 6(d).) The WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROM was
always stable and always resulted in solutions with the lowest errors for each
conserved variable.

This section demonstrated that constructing ROMs on dimensionally-consistent
inner products results in more stable ROMs that produce more accurate
solutions. As a result we will no longer focus on the
Galerkin${}_{\text{conserved}}$-$L^{2}(\Omega)$ and
WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$ ROMs in the remainder of this work
as they are dimensionally inconsistent (which makes them theoretically
unappealing) and, as just seen, result in inaccurate/unstable ROMs.

(a) Sod 1D problem

(b) Kelvin Helmholtz problem

(c) Homogeneous turbulence problem

(d) ROM summary

Figure 6: Comparison of ROM results obtained with dimensional $L^{2}(\Omega)$
and non-dimensional $L^{2}_{*}(\Omega)$ inner products. Figures 6(a)-6(c)
depict errors in the density field as a function of time for the Sod 1D,
Kelvin Helmholtz, and homogeneous turbulence problems, respectively. Figure
6(d) summarizes the percentage of time a ROM formulation was optimal and the
percentage of time a ROM formulation was stable.

##  8 Non-dimensional inner products vs physics-based inner product

The previous section demonstrated the importance of having a dimensionally-
consistent inner product when working with dimensional problems. However, even
though it is dimensionally consistent, the $L^{2}_{*}(\Omega)$ inner product
is arguably not a “physics-based” inner product as it simply results in a
(weighted) sum of the different state variables and carries no physical
meaning. In this section, we investigate the performance of ROMs built on the
physics-based “entropy” inner products introduced in Section 3 and compare
them to ROMs built using the non-dimensional $L^{2}_{*}(\Omega)$ inner
products. We emphasize that these entropy inner products are dimensionally
consistent.

We again first study POD in these different inner products. Figure 7 presents
the projection error121212Each projection is performed in the relevant inner
product. of the FOM data onto the trial subspace as a function of the basis
dimension for the non-dimensional $L^{2}_{*}(\Omega)$ and entropy
$L^{2}_{\boldsymbol{A}_{\infty}}(\Omega)$ inner products examined in Section
3; we note that we don’t consider bases constructed from the
$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ inner product. We observe that all
bases have equivalent approximation power for both the dimensional and non-
dimensional configurations; this result is expected as all methods are
dimensionally consistent. In addition, we observe that the trial subspaces
obtained from the entropy inner product have a slightly different
approximation power than the bases obtained from the non-dimensional
$L^{2}_{*}(\Omega)$ inner product, but the same trends are present. We observe
the same convergence in the relative error for both methods and, in general,
no method appears significantly “better” than the other.

Figure 7: Relative errors associated with projection of the snapshot data onto
the trial subspace in the $L^{2}_{\boldsymbol{A}_{\infty}}(\Omega)$ and
$L^{2}_{*}(\Omega)$ inner products for the Sod 1D problem (left column),
Kelvin Helmholtz problem (center column), and homogeneous turbulence problem
(right column). Results are shown for the mass (top row), $x_{1}$ momentum
(second row), $x_{2}$ momentum (where applicable, third row), and energy (last
row) state variables as a function of basis dimension. Results are shown for
both the dimensional and non dimensional data.

We next consider the various ROM formulations described above applied to the
dimensional and non-dimensional FOMs. Figures 9-13 present results for the
various ROMs applied to the 1D Sod problem, Kelvin Helmholtz problem, and 2D
homogeneous turbulence problem, respectively. The left columns of Figures 9-13
present the convergence of the relative errors for mass, momentum, and energy
as a function of the basis dimension, while the right columns present
physical-space solutions. Results are presented for a basis dimension of
$K=50$, $150$, and $150$ for the Sod 1D, Kelvin Helmholtz, and homogeneous
turbulence problems, respectively. Across all solutions, we make the following
observations:

  * 1.

(Figures 9-13.) WLS ROMs based on the entropy inner product consistently out-
perform the standard WLS ROMs based on the $L^{2}_{*}(\Omega)$ inner product.
We observe the entropy-based WLS ROMs to (1) yield solutions that are
qualitatively more accurate than the
WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROM and (2) display better
convergence properties. This is particularly evident in the field solutions
for the Sod and Kelvin Helmholtz problems: the
WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROM results in poor solutions
in both cases, and simply changing the ROM formulation to measure the residual
in the $L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ inner product results in a
substantial increase in accuracy. We emphasize that the
WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$ ROM
employs the same basis as the WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$
ROM, and thus the improvements are solely due to the ROM formulation.

  * 2.

(Figures 9-13.) The Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$ ROM based on
the entropy inner product is stable for all ROM dimensions, while the
Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROM based on the non-
dimensional $L^{2}_{*}(\Omega)$ inner product is rarely stable. In particular,
we observe that the Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROM
often becomes less stable as the basis dimension grows, while convergence is,
in general, observed for the Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$
ROM.

  * 3.

(Figures 9-13.) All methods yield the same results on the dimensional and non-
dimensional configurations; this is again expected as all methods are
dimensionally consistent.

(a) FOM

(b) Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$

(c) WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$

(d) WLS${}_{\text{entropy}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$

(e) WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$

Figure 9: Sod 1D problem. Convergence of ROM errors as a function of basis
dimension (left) and physical space solutions for the density field at a non-
dimensional time of $t=0.25$ (right) for a ROM dimension of $K=50$.

(a) FOM

(b) Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$

(c) WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$

(d) WLS${}_{\text{entropy}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$

(e) WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$

Figure 11: Kelvin Helmholtz problem. Convergence of ROM errors as a function
of basis dimension (left) and physical space solutions for the density field
at a non-dimensional time of $t=19.5$ (right) with a ROM dimension of $K=150$.

(a) FOM

(b) Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$

(c) WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$

(d) WLS${}_{\text{entropy}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$

(e) WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$

Figure 13: Homogeneous turbulence problem. Convergence of ROM errors as a
function of basis dimension (left) and physical space solutions for the
density field at a non-dimensional time of $t=14.7$ (right) with a ROM
dimension of $K=100$.

##  9 Summary of numerical results

To summarize our numerical results, Figure 14 presents the percentage of time
a given ROM formulation resulted in the most accurate solution for a state
variable and the number of times that a given ROM formulation was stable.
Results are presented for both the dimensional configuration (left) and non-
dimensional configuration (right). The results are aggregated across all ROM
dimensions and all state variables. We make the following remarks:

  * 1.

ROMs based on the entropy inner products are always stable for both the
dimensional and non-dimensional configurations.

  * 2.

The WLS-Entropy ROMs using either a conserved basis or entropy basis always
result in the most accurate predictions.

  * 3.

The Galerkin ROMs are rarely stable. Employing a non-dimensional
$L^{2}_{*}(\Omega)$ inner product helps slightly, but performance is still
poor.

  * 4.

The non-dimensional WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$ ROM is
always stable while the dimensional WLS${}_{\text{conserved}}$-$L^{2}(\Omega)$
is not.

  * 5.

The WLS-Entropy ROM with an entropy basis results in the most accurate
prediction for a state variable $80\%$ of the time.

(a) Dimensional

(b) Non-dimensional

Figure 14: Percentage of the time a ROM formulation gave a solution with the
lowest mean-squared-error and percentage of time a given ROM formulation was
stable. Results are compiled over all bases dimensions and state variables.
The dimensional configuration is shown on the left while the non-dimensional
configuration is shown on the right.

Lastly, compiling results across our examples, we can rank the performance of
our dimensionally-consistent ROMs in terms of solution accuracy from best-to-
worst as:

  1. 1.

WLS${}_{\text{entropy}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$,

  2. 2.

WLS${}_{\text{conservative}}$-$L^{2}_{\boldsymbol{B}_{\infty}}(\Omega)$,

  3. 3.

Galerkin${}_{\text{entropy}}$-$L^{2}(\Omega)$,

  4. 4.

WLS${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$,

  5. 5.

Galerkin${}_{\text{conserved}}$-$L^{2}_{*}(\Omega)$,

where we note it is a toss-up between (3) and (4).

##  10 Conclusions

This work investigated the impact of the choice of inner product on the
performance of POD Galerkin and POD least-squares reduced-order models within
the context of the compressible Euler equations. We demonstrated that
employing the classic $L^{2}(\Omega)$ inner product is inappropriate for POD,
Galerkin, and least-squares ROMs discretized with conserved variables. This
inner product resulted in ROMs that were dimensionally inconsistent and
resulted in different solutions when applied to non-dimensional and
dimensional versions of the same problem. Further, these formulations were
often unstable. We showed that this issued can be alleviated by employing a
non-dimensional $L^{2}_{*}(\Omega)$ inner product, in which case a
dimensionally-consistent POD process and Galerkin/least-squares ROM can be
formulated. Our numerical experiments demonstrated that employing this non-
dimensional inner product led to ROMs that were consistently more accurate and
more stable. However, even when employing this non-dimensional inner product,
the Galerkin ROM was still often unstable and least-squares ROMs, while
stable, were not particularly accurate. We demonstrated that the use of a
physics-based “entropy” inner product addressed this issue and vastly improved
the ROM performance. We demonstrated that employing an entropy inner product
in the POD process resulted in a dimensionally-consistent POD formulation that
provided bases with a similar approximation power to a non-dimensional
$L^{2}_{*}(\Omega)$ inner product. Next, we demonstrated that the Galerkin ROM
built off of entropy variables led to significant improvements over the
standard Galerkin ROM. Lastly, we demonstrated that least-squares ROMs which
minimize the residual in an entropy-based norm were significantly more
accurate than least-squares ROMs that minimize the residual in a non-
dimensional $L^{2}_{*}(\Omega)$ norm. Overall, the most accurate ROM
formulations were the least-squares ROMs formulated in the entropy inner
product. These methods provided solutions that were qualitatively accurate and
associated with the lowest errors across every exemplar.

The results of this work demonstrated the promise of ROMs based on physics-
based inner products, and in particular demonstrated the efficacy of entropy
inner products for the compressible Euler equations. While we believe the
presented results are promising, we emphasize that important aspects of a
practical ROM formulation, such as online efficiency through hyper-reduction
and certification via error analysis, were outside the scope this study. As a
result, future work will investigate theoretical stability properties of
entropy ROMs, integrating them with hyper-reduction and/or other techniques to
enhance their online efficiency, and application of physics-based inner
products to other domain areas.

##  11 Acknowledgments

The authors thank Irina Tezaur, Chad Sockwell, Patrick Blonigan, and John
Tencer for conversations from which this work benefited. This work was
sponsored by Sandia’s Advanced Simulation and Computing (ASC) Verification and
Validation (V&V) Project/Task #103723/05.30.02 and 103723/06.20.05. 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 and 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-
NA-0003525.

##  Appendix A Entropy Jacobian

The Jacobian of the conserved variables with respect to the entropy variables
is

$\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{V}}\equiv\boldsymbol{A}=\begin{bmatrix}\boldsymbol{U}_{1}&\boldsymbol{U}_{2}&\boldsymbol{U}_{3}&\boldsymbol{U}_{4}\\\
\text{sym.}&\frac{\boldsymbol{U}_{2}^{2}}{\boldsymbol{U}_{1}}+p&-\frac{\boldsymbol{U}_{2}\boldsymbol{V}_{3}}{\boldsymbol{V}_{4}}&-\frac{\boldsymbol{U}_{2}}{\boldsymbol{V}_{4}}-\frac{\boldsymbol{V}_{2}\boldsymbol{U}_{4}}{\boldsymbol{V}_{4}}\\\
\text{sym.}&\text{sym.}&\frac{\boldsymbol{U}_{3}^{2}}{\boldsymbol{U}_{0}}+p&-\frac{\boldsymbol{U}_{3}}{\boldsymbol{V}_{4}}-\frac{\boldsymbol{V}_{3}\boldsymbol{U}_{4}}{\boldsymbol{V}_{4}}\\\
\text{sym.}&\text{sym.}&\text{sym.}&\boldsymbol{U}_{1}H^{2}-\frac{a^{2}p}{\gamma-1}\end{bmatrix}$

where $H=\frac{\boldsymbol{U}_{4}+p}{\rho}$ and $a^{2}=\frac{\gamma p}{\rho}$.

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