# Nonunique Stationary States and Broken Universality in Birth Death Diffusion
Processes

Kartik Chhajed1 and P. K. Mohanty2 1Department of Physical Sciences IISER
Mohali Knowledge city Sector 81 SAS Nagar Manauli 140306 India
2Department of Physical Sciences IISER Kolkata Mohanpur West Bengal 741246
India

###### Abstract

Systems with absorbing configurations usually lead to a unique stationary
probability measure called quasi stationary state (QSS) defined with respect
to the survived samples. We show that the birth death diffusion (BBD)
processes exhibit universal phases and phase transitions when the birth and
death rates depend on the instantaneous particle density and their time scales
are exponentially separated from the diffusion rates. In absence of birth,
these models exhibit non-unique QSSs and lead to an absorbing phase transition
(APT) at some critical nonzero death rate; the usual notion of universality is
broken as the critical exponents of APT here depend on the initial density
distribution.

Markovian dynamics is used extensively in Physics, Chemistry, Biology,
Economics, and many other branches of science to model real-world processes,
both in equilibrium and nonequilibrium conditions [1, 2]. Lattice models in
thermal equilibrium [3, 4, 5] are usually described by Markov processes (MPs)
known as Monte Carlo dynamics, a dynamics that obeys detailed balance
condition w.r.t. Gibbs measure, and they have found vast application in
polymer physics [6], active and granular media [7], traffic flows [8], protein
folding [9] etc.. In the absence of Gibbs measure, nonequilibrium systems are
primarily described by Markov jump processes [10] and they exhibit novel
nonequilibrium phases, and phase transitions [12, 11]. Ergodicity is crucial
for MPs to attain a unique steady state, which is trivially broken if the
system has absorbing configurations -the configurations where the system is
permanently trapped once reached. The stationary measure of models with
absorbing configurations are trivial: the steady-state probability for the
system to be any configuration except the absorbing one is zero. An important
question is to find a probability measure for the quasi-stationary state (QSS)
defined w.r.t. the survived (no absorbed) samples. Simple models like contact
process [13], directed percolation [14, 15], pair contact process [16], and
fixed energy sandpiles [17, 18, 19] are known to have unique QSSs and they
exhibit absorbing phase transitions (APT); some of them are realized
experimentally [20, 21].

In this letter, we study absorbing-state phase transition in a simple Birth-
Death-Diffusion (BDD) process where birth and death rates depend on the
instantanuous particle density. We find that irrespective of the nature of
diffusing(conserving) dynamics, these models exhibit APT when death rates
crosses a critical threshold. In absence of birth, the QSS becomes non-unique
in the sense that the probability measure there depends on the the initial
conditions. The nonuniqueness persists at the critical point resulting in
initial condition dependence of the standard critical exponents. We provide A
sufficient criteria for the system to have such nonunique QSSs.

The model: Let us associate particle deposition (birth) and evaporation
(death) dynamics

$1\xrightleftharpoons[B(\rho)]{D(\rho)}0,$ (1)

to an one dimensional system of size $L$ which otherwise follows a particle
conserving dynamics. The rates $B(.)$ and $D(.)$ are analytic functions of
instantaneous particle density $\rho=\frac{N}{L},$ where $N$ is the number of
particles. In particular, we are interested in rate functions of the form
$D(\rho)=e^{-L\theta(\rho)}$, and $B(\rho)=\rho e^{-L\phi(\rho)};$ $B(0)=0$
ensures that an empty lattice is the only absorbing configuration of the
system. Along with these birth-death dynamics (1), we assume the system to
have a natural conserving dynamics; the simplest one being the symmetric
exclusion process (SEP) on a one-dimensional periodic lattice labeled
$i=1,2...,L,$ with $s_{i}=0,1$ representing occupancy of the site $i.$

$10\xrightleftharpoons[1]{1}01.$ (2)

In absence of birth and death, the conserving dynamics leads the system having
$N$ particles to a stationary probability measure $g_{N}(\mathcal{C})$; for
SEP in Eq. (2), $g_{N}(\mathcal{C})=\binom{L}{N}^{-1}$ is a constant.

The birth-death rates in BBD model, being extremely slow
(${\mathcal{O}}(e^{-L})$) compared to the conserved dynamics
(${\mathcal{O}}(1)$), effectuates a separation of timescales between the
particle conserving and particle non-conserving dynamics. Between any two
density-altering events $N\to N\pm 1,$ the system gets enough time to
equilibriate to the stationary distribution $g_{N}(\mathcal{C}).$ This
separation of time scale decouples the conserved and nonconserved part of the
dynamics, making the nonconserved dynamics effectively a biased walk (RW) on
$N=0,1,2,\dots L,$ with an absorbing boundary at $N=0.$

$N-1\xleftarrow[]{D_{N}}N\xrightarrow[]{B_{N}}N+1,$ (3)

where $B_{N}=(L-N)B(\rho)$ and $D_{N}=ND(\rho)$ where $\rho\equiv\frac{N}{L}$.
The additional multiplicative factors in birth and death rates are natural to
expect as in any configuration having $N$ particles, death (of a particle) can
occur at any of the $N$ sites whereas birth can occur at $L-N$ sites. If the
QSS of this absorbing RW is $\\{f_{N}\\}$, $N=1,2,\dots L,$ then the same for
the whole system can be written assuming a good separation of time scale as,
$P(\mathcal{C})=f_{N}g_{N}(\mathcal{C}),{\rm where}~{}N\equiv N(\mathcal{C})$

Absorbing phase transition (APT): When the time-scales are well separated, the
steady state density of the system can be determined completely from the QSS
of the RW. Corresponding APT if any, and its critical behaviour, are the
governed solely by the birth-death dynamics, irrespective of the nature of the
diffusive (conserving) dynamics. But, will there be any APT? Consider a region
of parameter space where $B_{N}>D_{N}\ \forall\ N\in(0,L).$ Here the system is
drifted towards higher density. Since $B_{N}=e^{-L\phi(\rho)+\ln
L(1-\rho)}\simeq B(\rho)$ and similarly $D_{N}\simeq D(\rho),$ for any density
$0<\rho<1$ one can make $B(\rho)\gg D(\rho)$ by increasing $L$. Then, a super
fast drift towards higher density will pin the walker at $\rho=1.$ On the
other hand, for $D_{N}>B_{N}$ the system drifts towards $\rho=0$ and thus most
walkers are absorbed; those who survive will contribute to the QSS. Most of
the survived walker will be found at some $N^{*}=\rho^{*}L$ where the dynamics
is slowest or the waiting time is large, i.e. where $B_{N}+D_{N}$ has its
minimum. $\rho^{*}$ can be tuned continuously by birth and death rates to
obtain an absorbing transition in thermodynamically large systems when
$\rho^{*}$ falls below $\frac{1}{L}$ (i.e., $\rho\to 0$).

Figure 1: (Color online) Decay of $\rho(t)$, (a) for different $L=8,16,32,$
with initial density $\rho_{i}=0.7$ ($N=\lfloor 0.7L\rfloor$) and (b)
different initial conditions $\rho_{i}=\frac{11}{16},\frac{1}{2},\frac{1}{4}$
for $L=16.$ Three different overlapping curves corresponds to Monte Carlo
simulation of the BBD model for $b=1.4,d=0.6$ (solid red line), simulation of
an effective RW dynamics (dashed green line) and numerical integration of the
Master equation for the RW (dotted black line). The horizontal lines are the
value of $\rho_{s}$ obtained from a QSS simulation proposed in Ref. [22].
Inset of (b) shows that indeed $\rho_{s}\to d=0.6$ following Eq. (5) when
$L\to\infty;$ asymptotically, $\rho_{s}-\rho_{s}(L)\sim L^{-0.428}$ (dashed
line).

Let us consider a specific example. Along with the conserving dynamics SEP in
Eq. (2), we consider dynamics (1) with

$D(\rho)=e^{-L\rho(2d-\rho)};\quad B(\rho)=\rho e^{-L\rho(2/b-\rho)},$ (4)

which have two real parameters $b$ and $d.$ We take $b\geq 0$ as $B(\rho)$ can
not be analytically continued to $b<0.$ First let us check whether the time-
scales are well separated. In Fig. 1(a) we plot $\rho(t)$ for $b=1.4,d=0.6$
obtained from Monte Carlo simulations of the model for $L=8,16,32$ and
compared them with $\rho(t)$ obtained from simulations of the biases RW
dynamics (3). For small systems, the Master equation for the absorbing RW can
be integrated numerically to obtain $\rho(t),$ which is also plotted in Fig.
1. An excellent match of all three curves justifies the separation of time
scale and the decoupling of the non-conserved dynamics from the conserved one.

For rates (4), $B_{N}>D_{N}$ corresponds to $bd>1.$ In this regime we expect
the steady state density to be $\rho_{s}=1.$ For $bd<1,$
$\frac{1}{L}(D_{N}+B_{N})=(1-\rho)B(\rho)+\rho D(\rho),$ which has its minimum
at $\rho^{*}=d.$ Thus based on these phenomenological arguments,

$\rho_{s}=\begin{cases*}0&$d<0$\\\ d&$0\leq d<1$ \& $bd\leq 1$\\\
1&otherwise\end{cases*}.$ (5)

This result, of course, can be shown rigorously (see Supplemental Material).

With $\rho_{s}$ in hand, we can proceed to obtain the phase diagram in $b$-$d$
plane. But there is a problem. For $b=1.4,d=0.6,$ we have obtained the steady
state density $\langle\rho\rangle\equiv\rho_{s}$ for different $L$ in Fig. 1;
$\rho_{s}$ for the largest $L=32$ was found found to be $0.364$ which is far
way from the analytical result $\rho_{s}=0.6,$ in (5). The non-monotonic $L$
dependence of $\rho_{S}(L)$ adds more doubt on whether $\rho_{S}(L)\to d$ as
$L\to\infty.$ Moreover, simulation for larger $L$ takes astronomically large
time as the waiting times are ${\mathcal{O}}(e^{L}).$ Fortunately a recently
proposed simulation method for QSS [22] helps us to obtain $\rho_{s}$ for
reasonably large $L.$ The inset of Fig. 1, for $b=1.4,d=0.6,$ shows that
indeed $\rho_{s}$ approaches $d=0.6$ following an asymptotic power-law,
$\rho_{s}=0.6-L^{-0.428}.$

Figure 2: (Color online) Three phases of BBD model: $\rho=0$ (absorbing)
$0<\rho<1$ (active) and $\rho=1$ (maximal density). The APT occurs at $d=0.$
On PQR line $\bar{\rho}_{s}=1-\rho_{s}$ exhibits an ordinary phase transition,
continuous (discontinuous) along QR(PQ) segments with a tricritical point
$b=1=d$ (Q).

The phase diagram, based on Eq. (5), is shown in Fig. 2 where $0<\rho_{s}<1$
region is separated from two other regions $\rho_{s}=0$ and $\rho_{s}=1.$ The
BBD model exhibits continuous absorbing-state phase transition at $d=0$ line.
For a fixed value of $b$, $\rho_{s}$ decreases continuously with decrease of
$d$ until a critical threshold $d_{c}=0$ is reached. Surprisingly, an ordinary
non-equilibrium phase transition also takes place along PQR line in Fig 2. The
order parameter $\bar{\rho}_{s}=1-\rho_{s}$ picks up a nonzero value when, for
a fixed $b,$ $d$ is decreased below

$\bar{d}_{c}=\begin{cases}1&0<b<1\\\ b^{-1}&b\geq 1.\end{cases}$ (6)

This transition is discontinuous when $b\in(1,\infty)$ (along PQ line in Fig.
2) as $\bar{\rho}_{s}$ jumps from $0$ to $1-\frac{1}{b}$ at $d=\bar{d}_{c}$,
whereas it is continuous $b\in(0,1],$ leading to the fact that Q
($(d,b)=(1,1)$) is a tri-critical point.

The line $b=0$ corresponds to no birth of particles. In this case, we can
solve the master equation of the system exactly. Let us assume that in absence
of birth and death, the diffusive dynamics follows a master equation
$\frac{d}{dt}|\phi_{N}\rangle=\mathbf{C}_{N}|\phi_{N}\rangle$ with
$N=0,1,\dots,L$ conserved. $|\phi_{N}\rangle$ is a $\binom{L}{N}$ dimensional
vector with components representing the probability of being in $\binom{L}{N}$
configurations of the conserved system. For $N=0,L$ the system has only one
configuration and evolution has no meaning; we set the scalars
$\mathbf{C}_{0}=0=\mathbf{C}_{L}$ for notational convenience. We also denote
the steady states as $|\bar{\phi}_{N}\rangle$ which satisfy
$\mathbf{C}_{N}|\bar{\phi}_{N}\rangle=0.$ Now the master equation on $b=0$
line can be written as $\frac{d}{dt}|P\rangle=\mathbf{M}|P\rangle$ with
$|P\rangle=(|\phi_{0}\rangle,|\phi_{1}\rangle,\dots,|\phi_{L}\rangle)^{T}$ and
$\mathbf{M}$ in block-form reads,

$\mathbf{M}=\begin{pmatrix}\mathbf{M}_{0}&D_{1}\mathbf{T}_{1}&\mathbf{0}&\mathbf{0}&\dots&\mathbf{0}\\\
\mathbf{0}&\mathbf{M}_{1}&D_{2}\mathbf{T}_{2}&\mathbf{0}&\dots&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{M}_{2}&D_{3}\mathbf{T}_{3}&\dots&\mathbf{0}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\dots&\mathbf{M}_{L}\end{pmatrix}.$
(7)

Here $\textbf{M}_{n}=\mathbf{C}_{n}-nD_{n}\mathbf{I}_{n}$ are $\binom{L}{n}$
dimensional square matrices with $D_{n}\equiv D(n/L)$ and $\mathbf{I}_{n}$ is
the identity matrix; obviously $\textbf{M}_{0}=0$ is a scalar. The
$\mathbf{T}_{n}$s are $\binom{L}{n-1}\times\binom{L}{n}$ matrices such that
each column has exactly $n$ number of $1$s corresponding to a surjective
mapping of $n$-particle configurations to $n-1$ particle configurations,
$\mathcal{C}_{n}\mapsto\mathcal{C}_{n-1}$, by operation of removal of a single
particle without disturbing any other particle. Clearly the eigenvalues of
this upper triangular block matrix $\mathbf{M}$ are
$\Lambda[\mathbf{M}]=\bigcup_{n}\Lambda[\mathbf{M_{n}}].$ In other words, the
set of $2^{L}$ eigenvalues of $\mathbf{M}$ are
$\lambda_{n\nu}=\Lambda[\mathbf{C}_{n}]_{\nu}-nD_{n},$ where
$\Lambda[\mathbf{C}_{n}]_{\nu}$ are $\nu^{th}$ eigenvalue of $\mathbf{C}_{n}$
in a decreasing order and $\nu=0,1,\dots\binom{L}{n}$ for any given $n.$ It is
now straight forward to calculate both the left and right eigenvectors of
$\lambda_{n\nu},$ respectively $\langle\Psi_{n\nu}|$ and
$|\Psi_{n\nu}\rangle.$ For our purpose calculating them for $\nu=0$ are enough
(see text below).

$\displaystyle\langle\Psi_{n0}|$ $\displaystyle=$
$\displaystyle\left(\langle\psi_{0}|,\langle\psi_{1}|,\dots\right);~{}~{}|\Psi_{n0}\rangle=\left(|\psi_{0}\rangle,|\psi_{1}\rangle,\dots\right)^{T}$
(8) $\displaystyle\langle\psi_{k}|$ $\displaystyle=$
$\displaystyle\begin{cases}\mathbf{0}&k<n\\\ \langle
1_{k}|\prod_{i=n+1}^{k}\frac{\lambda_{i0}}{\lambda_{i0}-\lambda_{n0}}&k\geq
n\end{cases}$ (9) $\displaystyle|\psi_{k}\rangle$ $\displaystyle=$
$\displaystyle\begin{cases}\prod_{i=1}^{n-1}\frac{\lambda_{i0}}{\lambda_{n0}-\lambda_{i0}}&k=0\\\
\prod_{i=k+1}^{n}\frac{\lambda_{i0}}{i}(\mathbf{M}_{i-1}-\lambda_{n0})^{-1}\mathbf{T}_{i}|\bar{\phi}_{n}\rangle&0<k\leq
n\\\ \mathbf{0}&k>n\end{cases}$

where ${\mathbf{0}}$ are null vectors and $\langle 1_{k}|\equiv(1,1,\dots)$ in
$\binom{L}{k}$ dimension.

The probability $Q_{mn}$ that system, starting from a configuration with $m$
number of particles at $t=0,$ is not absorbed until time $t$ and it has $n$
particles is

$Q_{mn}(t)=\frac{\langle n|e^{\mathbf{M}t}|m\rangle}{1-\langle
0|e^{\mathbf{M}t}|m\rangle}=\frac{\sum\limits_{k\neq 0}e^{-kD_{k}t}\langle
n|\Psi_{k0}\rangle\langle\Psi_{k0}|m\rangle}{-\sum\limits_{k\neq
0}e^{-kD_{k}t}\langle 0|\Psi_{k0}\rangle\langle\Psi_{k0}|m\rangle}$ (10)

where, in the spectral decomposition we kept only the dominant eigenvectors
(corresponding to $\lambda_{m0}$). We also denote $\langle
m|=\binom{L}{m}^{-1}({\mathbf{0},0},\dots,|1_{m}\rangle,\dots,{\mathbf{0}})$
and $|m\rangle=\langle m|^{T}.$ Clearly $|m\rangle$ represents a state of the
system where all configurations that has exactly $m$ particles are equally
likely. Note that, $\langle m|$ is a left eigenvector of any Markov matrix in
$\binom{L}{m}$ dimension, whereas $|m\rangle$ is the steady state for certain
specific dynamics, like the SEP defined in (2), where all configurations are
equally likely in steady state.

In the long time limit, one of the term
$k=\bar{k}=\arg\min\limits_{k>0}(kD_{k})~{}\forall~{}k$ is dominant in the
sums that appear in $Q_{mn}$ as long as $\langle\Psi_{k}|m\rangle\neq 0,$
leading to,

$\small Q_{mn}=\frac{\langle n|\Psi_{\bar{k}0}\rangle}{-\langle
0|\Psi_{\bar{k}0}\rangle}=\begin{cases}\frac{\bar{k}D_{\bar{k}}}{nD_{n}}\prod\limits_{k<\bar{k}}\left(1-\frac{\bar{k}D_{\bar{k}}}{kD_{k}}\right)&n\leq\bar{k}\\\
0&\bar{k}<n\end{cases}.$ (11)

It appears that $Q_{mn}$ for $b=0$ does not depend on the initial density
$m/L,$ however that is not true. Unlike $b>0$ case where
$\langle\Psi_{1}|m\rangle\neq 0~{}\forall~{}m$, here it is evident from Eq.
(9) that $\langle\Psi_{1}|m\rangle$ may become zero for some choice of $m$
leading to a possibility of a non-unique QSS that depends on the initial
condition.

Figure 3:  (Color online) $\rho_{s}$ and its fluctuation $\Delta\rho_{s}^{2}$
(in the inset) vs. $d$ for different initial density distributions
$\pi(\rho)=(1+a)\rho^{a}$, $a=0,1,2$. For $d_{c}\approx 0,$ $\rho_{s}\sim
d^{\beta}$ and $\Delta\rho_{s}^{2}\sim d^{-\gamma};$ the critical exponent
$\gamma=-(a+3)$ depends on initial condition, whereas $\beta$ does not.

In continuum limit, we denote $(m,n,\bar{k})=(\rho_{i},\rho,\rho^{*})L$ and
thus the Matrix $Q_{mn}$ takes functional form

$Q(\rho_{i},\rho)=e^{2Ld(\rho-\rho^{*})}\Theta(\rho^{*}-\rho),$ (12)

which is the probability that a non-absorbed (or survived) system has steady
state density $\rho,$ when it is initialized at density $\rho_{i}.$ For an
arbitrary initial density distribution $\pi(\rho_{i})$ with $\rho_{i}>0$ the
QSS measure is ${\cal
Q}(\rho)=\int_{0^{+}}^{1}\mathrm{d}\rho_{i}Q(\rho_{i},\rho)\pi(\rho_{i})$ and
the steady state density, calculated using Laplace method [23], is

$\langle\rho\rangle=\int_{0}^{1}\mathrm{d}\rho\mathcal{Q}(\rho)\rho\approx\int_{0}^{1}\mathrm{d}\rho_{i}\pi(\rho_{i})\rho^{*};~{}\rho^{*}=\begin{cases}\rho_{i}&\rho_{i}\leq
d\\\ d&\rho_{i}>d\geq 0\\\ 0&d<0\end{cases}.$ (13)

Let us consider $\pi(\rho_{i})=(1+a)\rho_{i}^{a},a>-1.$ Then the moments of
the QSS $\langle\rho^{k}\rangle=d^{k}-k\frac{d^{k+1+a}}{k+1+a}~{}~{}\forall
k\geq 0$ depend on the initial condition. This features persists at the
critical point $d_{c}=0$ where the critical exponents too become non-unique.
For $d\gtrapprox 0,$ the order parameter $\langle\rho\rangle\sim d^{\beta}$
with $\beta=1$ whereas its variance
$\langle\rho^{2}\rangle-\langle\rho\rangle^{2}\sim d^{-\gamma}$ with a non-
unique exponent $\gamma=-(a+3).$ Continuous variation of critical exponents
[24] are not new, they have been observed theoretically [25, 26, 27] and
experimentally [28], but the exponents there vary w.r.t. certain marginal
parameters of the system under study. To the best of our knowledge, dependence
of the critical exponents on the initial conditions are new features of the
BBD class of models, which has not been observed earlier. What is so special
about BBD models at $b=0$?

In a Markov chain, two configurations belong to the same communicating class
(CC) if each one is accessible starting from the other. A “closed” CC is one
that is impossible to leave - CCs which are not ‘closed’ are ‘open’.
Irreducible Markov chains model many physical systems and they have only one
communicating class which is necessarily closed; corresponding Markov matrices
are irreducible and the uniqueness of its largest eigenvalue and the
uniqueness of the stationary state are well protected by the Perron–Frobenius
(PF) theorem [29]. Models like contact process [13], directed percolation [14,
15], and pair contact process[16] has only one open CC and as many closed CCs
as there are absorbing configurations. We have seen that the QSS is the
eigenvector corresponding to the largest eigenvalue of the Markov sub-matrix
defined on the open communicating class (see Supplemental Material for
details). For systems having one open CC this sub-matrix is irreducible and
corresponding QSS is strictly positive, following PF theorem. On the other
hand when a system has many open CCs, like a system described in Fig. 4(a),
the Markov sub-matrix becomes reducible, and might have a non-unique QSS.
Then, for any finite absorbing Markov chain the quasi-stationary density
$\langle\rho\rangle$ may depend on initial condition, except for the special
case when $\langle\rho\rangle=0.$ To have a non-unique $\langle\rho\rangle$ in
the $L\to\infty$ limit, we must choose the non-conserving dynamics such that
$\langle\rho\rangle$ does not vanish in the thermodynamic limit; birth and
death rates in the BDD model in $e^{-L\theta(\rho)}$ form serves the purpose.
Note that in BDD model for $b>0$ there is one closed and one open CC (as shown
in Fig 4(c)) whereas setting $b=0$ creates $L$ open CCs resulting in a
nonuinque QSS. If we consider another conserved dynamics instead SEP, say, the
Kawasaki dynamics of Ising model ($H=-J\sum_{i}\delta_{s_{i},s_{i+1}}$),

$\displaystyle
0100\xrightleftharpoons[1]{1}0010;~{}0101\xrightleftharpoons[\alpha]{1}0011;1100\xrightleftharpoons[1]{\alpha}1010;~{}1101\xrightleftharpoons[1]{1}1011,$

with $\alpha=e^{-2J}$, or when we extend SEP and Ising models with birth-death
dynamics (1) to higher dimensions, the structure of the communicating classes
remains same as shown in Fig. 4(b), (c) respectively for $b=0$ and $b\neq 0$.
Thus these models too exhibit an universal phase phase diagram Fig. 2 and a
non-unique APT at $b=0$ (as in Fig. 3).

Figure 4: (Color online) (a) Schematic representation of the open (discs) and
closed (squares) communicating classes (CCs) and possible transitions between
them. The CCs are arranged in decreasing order of their order parameter value
$\rho L.$ (b) and (c) represent CCs for $b=0$ and $b\neq 0$ cases of the BDD
model respectively.

In summary, we introduce and study a class of birth-death diffusion models
where the time-scale of birth and death rates are well separated from the
diffusion rate. These models exhibit ordinary and absorbing phase transitions
with universal critical lines that do not depend on the nature of diffusion
(conserved) dynamics. In the absence of birth, the QSS measure of these
systems become nonunique, leading to a novel APT with initial condition-
dependent critical exponents. Usually, the origin of varying critical
exponents is existence of an underlying marginal operator, which does not
scale under renormalization. It is impossible to extend these ideas to include
the dependence of critical exponents on initial conditions! In this letter we
provide sufficient criteria: A nonunique QSS is possible when the system has
multiple open communicating classes, and the transition rates between them
have a thermodynamically stable minimum at some nonzero value of the order
parameter. More theoretical understanding is required to design or access such
nonunique QSSs in experiments.

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# Wavelet-based regularization of the Galerkin truncated three-dimensional
incompressible Euler flows

Marie Farge  CNRS–INSMI, LMD–IPSL, Ecole Normale Supérieure–PSL, 24 rue
Lhomond, 75231 Paris Cedex 05, France  Naoya Okamoto  Center for Computational
Science, Nagoya University, Nagoya, 464-8603, Japan  Kai Schneider  I2M-CNRS,
Centre de Mathématiques et d’Informatique, Aix-Marseille Université, 39 rue F.
Joliot-Curie, 13453 Marseille Cedex 13, France  Katsunori Yoshimatsu
Institute of Materials and Systems for Sustainability, Nagoya University,
Nagoya, 464-8603, Japan

###### Abstract

We present numerical simulations of the three-dimensional Galerkin truncated
incompressible Euler equations that we integrate in time while regularizing
the solution by applying a wavelet-based denoising. For this, at each time
step, the vorticity filed is decomposed into wavelet coefficients, that are
split into strong and weak coefficients, before reconstructing them in
physical space to obtain the corresponding coherent and incoherent
vorticities. Both components are multiscale and orthogonal to each other.
Then, by using the Biot–Savart kernel, one obtains the coherent and incoherent
velocities. Advancing the coherent flow in time, while filtering out the
noise-like incoherent flow, models turbulent dissipation and corresponds to an
adaptive regularization. In order to track the flow evolution in both space
and scale, a safety zone is added in wavelet coefficient space to the coherent
wavelet coefficients. It is shown that the coherent flow indeed exhibits an
intermittent nonlinear dynamics and a $k^{-5/3}$ energy spectrum, where $k$ is
the wavenumber, characteristic of three-dimensional homogeneous isotropic
turbulence. Finally, we compare the dynamical and statistical properties of
Euler flows subjected to four kinds of regularizations: dissipative
(Navier–Stokes), hyperdissipative (iterated Laplacian), dispersive
(Euler–Voigt) and wavelet-based regularizations.

PACS numbers

47.27.E-, 47.27.Gs, 47.27.er

###### pacs:

Valid PACS appear here

##  I Introduction

A major challenge in computational fluid dynamics is the numerical simulation
of high Reynolds number turbulence and in particular the numerical solution of
the three-dimensional (3D) incompressible Euler equations. The nonlinearity of
Euler equations excites smaller and smaller scales and the same holds for
Navier–Stokes equations in the inviscid limit, which corresponds to very
strong turbulence when the Reynolds number tends to infinity. Since numerical
schemes are limited to a finite number of modes, or grid points, the numerical
integration of Euler equations requires to apply some kind of regularization
to obtain a physically relevant solution for a given resolution. Ideally such
techniques should preserve the flow’s nonlinear dynamics and the solution’s
properties. For example, vortex methods introduce a cut-off in the Biot–Savart
kernel. In the context of finite volume/difference methods, typically upwind
techniques are used which introduce numerical diffusion and also numerical
dispersion. Spectral methods have the advantage to avoid numerical diffusion
and dispersion, and furthermore they do preserve the conservation properties
of the governing equations. Truncated Fourier Galerkin approximations used to
solve Euler equations conserve kinetic energy and it was shown that the
solutions thus obtained tend in the limit of long time to energy equipartition
between all Fourier modes, which corresponds to an isotropic energy spectrum
with a $k^{2}$ behavior in three dimensions, where $k$ is the wavenumber Lee52
. For transient time numerical simulations of the 3D Euler equations
integrated with a truncated Fourier Galerkin method exhibit a $k^{-5/3}$
scaling, while at later time a $k^{2}$ spectrum builds up which corresponds to
the predicted energy equipartition CBD05 . The statistics of the velocity
field behave as a Gaussian white noise which satisfies the incompressibility
constraint. To obtain a physically relevant solution, typically
hyperdissipative (also known as hyperviscous) regularizations are applied,
which correspond to a Laplace operator which is iterated a certain number of
times, as introduced in Lady62 ; Lion69  and applied in, e.g., W84 ; FS89 .
Compared to viscous dissipation, which corresponds to the Laplace operator,
much wider inertial ranges can thus be obtained for a given numerical
resolution and therefore are frequently used to simulate geophysical and
astrophysical flows. Viscous and hyperviscous regularizations give rise to
bottlenecks in the compensated energy spectrum, $k^{5/3}E(k)$, which become
more pronounced as the order of the hyperdissipation, (corresponding to the
number of iterations of the Laplace operator) is increased LCP05 .
Hyperviscous regularizations for 3D homogeneous and isotropic turbulent flows
have been studied in SMC12 , and detailed analyses of bottleneck effects have
been published in FKP08 . An inviscid regularization called Euler–Voigt model
has been introduced by Oskolkov Osko82 ; CLT06 . This regularization is of
dispersive nature, which means that Fourier modes of different wavelength are
no more propagated with the same group and phase velocity. Dispersion thus
affects the phase of the Fourier modes, while diffusion modifies their
amplitude. The Euler–Voigt model can be obtained in the context of Navier-
Stokes alpha models, i.e., a Helmholtz filter is applied to the momentum
equation and the resulting equation is known as a simplified Bardina
turbulence model. Setting the viscosity equal to zero yields the Euler–Voigt
equations which formally correspond to adding the term
$\alpha^{2}\partial_{t}\Delta u$ to the momentum equation, where $\alpha>0$ is
a length scale that represents the width of the spatial filter, see, e.g., the
discussion in LaTi10 .

Wavelet techniques for simulating turbulent flows have been introduced in
FSK99 ; SF00 ; FS01 . For reviews we refer to Farge Fa92 , Schneider &
Vasilyev SV10  and Farge & Schneider FaSc15  . Wavelet-based regularization of
the one-dimensional Burgers equation and two-dimensional incompressible Euler
equations using Fourier Galerkin schemes has been presented in NFS09 ; PNFS13
. There it was shown that removing noise in the truncated Fourier Galerkin
simulations of the inviscid equations does yield results similar to the
viscous equations. Applying coherent vorticity extraction, introduced in FS01
; FSK99 , to high Reynolds number 3D turbulence shows that the incoherent
velocity field exhibits indeed an energy spectrum with a $k^{2}$ slope FPS01 ;
OYSFK07 . This wavelet-based extraction method presents the advantage over the
linear Fourier FSPWR03  and the nonlinear Fourier YOFS10  filtering method for
extracting coherent structures out of turbulent flows. These previous studies
motivate the present work.

The aim of this study is the application of wavelet-based regularization to
the truncated Fourier Galerkin approximation of Euler equations to examine if
wavelet-based denoising would yield the resulting flows which have similar
properties as Navier–Stokes flows in the fully-developed turbulent regime. The
idea is to remove the noise corresponding to the $k^{2}$ spectrum and to check
if this is equivalent to modeling turbulent dissipation as already suggested
in FPS01 ; SFPR05 . We also compare the results obtained using wavelet-based
denoising with several other kinds of regularization of Euler equations,
including hyperdissipative regularization by iterated Laplacian, and
dispersive regularization based on the Euler–Voigt model Osko82 ; CLT06 .

The outline of the paper is the following. In Section II, we describe the
governing equations, the different regularization methods of the Euler
equations used here, and the numerical schemes to implement them. In Section
III, we present the results of the numerical experiments we have performed and
analyze them using several statistical diagnostics and visualizations. In
Section IV, we draw some conclusions and propose perspectives for future work.

##  II Euler equations and regularization methods

First, we describe the Euler equations and the numerical methods used to solve
them. Then, we introduce a wavelet-based regularization of the Euler
equations, and present two more classical methods, one dissipative and one
dispersive, in order to compare the regularized Euler solutions thus obtained.

###  II.1 Euler equations and numerical method

We consider a velocity field $\bm{u}({\bm{x}},t)$ obeying the 3D
incompressible Euler equations,

$\displaystyle\partial_{t}{\bm{u}}+({\bm{u}}\cdot\nabla){\bm{u}}+\nabla p$
$\displaystyle=$ $\displaystyle{\bm{0}},$ (1)
$\displaystyle\quad\nabla\cdot{\bm{u}}$ $\displaystyle=$ $\displaystyle
0,\quad$ (2)

for ${\bm{x}}=(x^{1},x^{2},x^{3})$ in a periodic box $\Omega=[0,2\pi]^{3}$,
where unit density is assumed. The pressure is denoted by $p(\bm{x},t)$,
$\partial_{t}\equiv\partial/\partial t$ and
$\nabla\equiv(\partial/\partial{x^{1}},/\partial{x^{2}},\partial/\partial{x^{3}})$.
We omit the arguments $x$ and $t$, unless otherwise stated.

A truncated Fourier Galerkin approximation of the Euler equations (1) and (2)
is obtained by developing the velocity field and the pressure into truncated
Fourier series, e.g.,
${\bm{u}}({\bm{x}},t)=\sum_{\bm{k}}{\widehat{\bm{u}}}({\bm{k}},t)e^{i{\bm{k}}\cdot{\bm{x}}}$,
and requiring that the weighted residual vanishes with respect to test
functions, which are identical to the trial functions
$e^{i{\bm{k}}\cdot{\bm{x}}}$. Here ${\bm{k}}=(k^{1},k^{2},k^{3})$ is the wave
vector and $i=\sqrt{-1}$. The incompressibility constraint is taken into
account by eliminating pressure, which yields the Euler equations in Fourier
space; $\partial_{t}{\widehat{u}}^{\ell}({\bm{k}})=-P_{\ell
m}{\widehat{N}}^{m}({\bm{k}})$, where $P_{\ell m}=\delta_{\ell
m}-k^{\ell}k^{m}/k^{2}$ and ${\bm{N}}=({\bm{u}}\cdot\nabla){\bm{u}}$. Without
loss of generality, we set the mean velocity
$\langle\mbox{\boldmath$u$}\rangle=\mbox{\boldmath$0$}$, where
$\langle\cdot\rangle$ denotes spatial average over the periodic box. Then Eqs.
(1) and (2) are discretized with $N=2^{3J}=512^{3}\quad(J=9)$ grid points. The
nonlinear term is evaluated with a pseudo-spectral technique, i.e., in
physical space, and the aliasing errors are removed by means of the phase
shift method. Only modes with wavenumbers satisfying $k<k_{\rm
max}=2^{1/2}N^{1/3}/3$ are retained. For time integration we employ an
explicit Runge–Kutta scheme of fourth order. The dealiased pseudo-spectral
discretization is equivalent to the Galerkin approximation, which by
construction does conserve kinetic energy, i.e., $dE/dt=0$, where
$E=\int_{\Omega}|{\bm{u}}|^{2}d{\bm{x}}/2$.

###  II.2 Regularization methods

### Wavelet-based regularization

After a brief description of the orthogonal wavelet decomposition (i) and the
nonlinear wavelet filtering (ii), we describe the procedure of wavelet-based
regularization (iii). The choice of the threshold used in (iii) is described
(iv). The wavelet-based denoising regularization depends on the solution
projected onto an orthogonal wavelet basis and is therefore adaptive. Since
some wavelet coefficients are discarded, it has a dissipative effect. In order
to obtain statistically stationary states, a solenoidal forcing term $f$ is
imposed.

(i) Orthogonal wavelet decomposition
The 3D orthogonal wavelet transform unfolds a $2\pi$-periodic vector field
${\bm{v}}({\bm{x}},t)$ at a given instant $t$ into scale, positions and seven
directions ($\mu=1,...,7$) using a 3D mother wavelet $\psi_{\mu}({\bm{x}})$,
which is based on a tensor product construction. The wavelet $\psi$ is well-
localized in space $x$, oscillating, and smooth. The mother wavelet generates
a family of wavelets $\psi_{\mu,\lambda}(\mbox{\boldmath$x$})$ by dilation and
translation, which yields an orthogonal basis of $L^{2}({\mathbb{R}}^{3})$,
and also of $L^{2}({\mathbb{T}}^{3})$ with
$\mathbb{T}=2\pi\mathbb{R}/{\mathbb{Z}}$ being the torus through the
application of a periodization technique Mallat98 . The spatial average of
$\psi_{\mu,}(\mbox{\boldmath$x$})$, denoted by
$\langle\psi_{\mu,\lambda}\rangle$, vanishes for each index. The multi-index
$\lambda=(j,i_{1},i_{2},i_{3})$ denotes the scale $2^{-j}$ and position
$2\pi\times 2^{-j}{\bm{i}}=2\pi\times 2^{-j}(i_{1},i_{2},i_{3})$ of the
wavelets for each direction.

A vector field $\bm{v}(\bm{x})=(v^{1},v^{2},v^{3})$ sampled on $N=2^{3J}$
equidistant grid points, having zero mean value, can be decomposed into an
orthogonal wavelet series:

${\bm{v}}({\bm{x}})=\sum_{j=0}^{J-1}{\bm{v}}_{j}({\bm{x}}),$ (3)

where ${\bm{v}}_{j}$ is the contribution of ${\bm{v}}$ at scale $2^{-j}$
defined by

${\bm{v}}_{j}({\bm{x}})=\sum_{\mu=1}^{7}\sum_{i_{1},i_{2},i_{3}=0}^{2^{j}-1}{\widetilde{\bm{v}}}_{\mu,\lambda}\psi_{\mu,\lambda}(\mbox{\boldmath$x$}).$
(4)

Due to orthogonality of the wavelets, the coefficients are given by
${\mbox{\boldmath$\widetilde{v}$}}_{\mu,\lambda}=\left<{\mbox{\boldmath$v$}},\psi_{\mu,\lambda}\right>$,
where $\left<\cdot,\cdot\right>$ denotes the $L^{2}$-inner product defined by
$\left<f,g\right>=\int_{\Omega}f(\mbox{\boldmath$x$})\,g(\mbox{\boldmath$x$})d\mbox{\boldmath$x$}$.
At scale $2^{-j}$ we have $N_{j}=7\times 2^{3j}$ wavelet coefficients for each
component of $v$. Thus, in total we have $N$ coefficients for each component
of the vector field corresponding to $N-1$ wavelet coefficients and the
vanishing mean value. These coefficients are efficiently computed from the $N$
grid point values for each component of $v$ using the fast wavelet transform,
which has linear computational complexity. In the present work, the compactly
supported Coiflet wavelets with filter width 12 are used. For more details on
wavelets, we refer the reader to text books, e.g., Mallat Mallat98 .

(ii)  Wavelet-based denoising
Thresholding the wavelet coefficients
${\widetilde{\mbox{\boldmath$v$}}}_{\mu,\lambda}$ at a given time instant, we
can define the coherent subset of the wavelet coefficients
${\widetilde{\mbox{\boldmath$v$}}}_{\mu,\lambda}^{\rm c}$ by

${\widetilde{\mbox{\boldmath$v$}}}_{\mu,\lambda}^{\rm
c}=\left\\{\begin{array}[]{lll}{\widetilde{\mbox{\boldmath$v$}}}_{\mu,\lambda}&{\mathrm{for}}&|{\widetilde{\mbox{\boldmath$v$}}}_{\mu,\lambda}|>{T},\\\
0&{\mathrm{for}}&|{\widetilde{\mbox{\boldmath$v$}}}_{\mu,\lambda}|\leq{T},\end{array}\right.$
(5)

where $T$ is a given threshold value. The choice of the threshold value is
discussed in (iv). The coherent field $\mbox{\boldmath$v$}_{c}$ is then
reconstructed by inverse wavelet transform. The remaining incoherent field
$\mbox{\boldmath$v$}_{i}$ is given as
$\mbox{\boldmath$v$}_{i}=\mbox{\boldmath$v$}-\mbox{\boldmath$v$}_{c}$.

(iii) Wavelet-based regularization of Euler equations
The numerical simulation of the Euler equations with wavelet-based
regularization is also called Coherent Vorticity Simulation (CVS). The
procedure of CVS, starting from the Fourier coefficients of the velocity field
$\mbox{\boldmath${\hat{u}}$}(\mbox{\boldmath$k$},t)$ at $t=t_{n}$, is as
follows.
(a) Time integration in spectral space: The velocity
$\mbox{\boldmath${\hat{u}}$}(\mbox{\boldmath$k$},t)$ is advanced in time in
$k$–space up to $t=t_{n+1}$ using the fourth-order Runge–Kutta method.
(b) Reconstruction of vorticity in physical space:
The vorticity field $\mbox{\boldmath$\omega$}=\nabla\times\mbox{\boldmath$u$}$
at $t=t_{n+1}$ is reconstructed by applying the inverse Fourier transform to
$\mbox{\boldmath${\widehat{\omega}}$}=i\mbox{\boldmath$k$}\times\mbox{\boldmath${\hat{u}}$}$.
(c) Extraction of coherent vorticity and addition of safety zone in wavelet
space:
The set of wavelet coefficients of vorticity is obtained by applying the fast
wavelet transform to $\omega$. In order to track the evolution of coherent
vorticity in space, scale and direction we have to keep, not only the coherent
wavelet coefficients ${\widetilde{\mbox{\boldmath$\omega$}}}_{c}$, but also
the neighboring wavelet coefficients in space, scale and directions. For this
we first define the index set $\Lambda$ that is the union of all
$(\mu,\lambda)$ corresponding to the coherent wavelet coefficients kept in Eq.
(5). We then define an expanded index set $\Lambda_{*}$ which adds to
$\Lambda$ the indices $(\mu,\lambda)$ of the neighboring coefficients in
position, scale and direction. For details on the definition of the safety
zone, we refer to OYSFK11 . Finally, all the coefficients, which do not belong
to $\Lambda_{*}$ are set to zero. The expanded wavelet coefficients indexed by
$\Lambda_{*}$ correspond to the coherent ones plus those of the safety zone,
and are denoted by ${\widetilde{\mbox{\boldmath$\omega$}}}_{c*}$.
(d) Reconstruction in physical space of the expanded coherent vorticity:
Applying the inverse wavelet transform to
${\widetilde{\mbox{\boldmath$\omega$}}}_{c*}$ yields the coherent vorticity
including the safety zone $\mbox{\boldmath$\omega$}_{c*}$.
(e) Calculation of the expanded coherent velocity:
The induced velocity $\mbox{\boldmath$u$}_{c*}$, which is divergence free, is
computed using the Biot–Savart relation
$\mbox{\boldmath$u$}_{c*}=-\Delta^{-1}(\nabla\times\mbox{\boldmath$\omega$}_{c*})$
in wavenumber space. The steps (a)–(e) are applied in each time step.

(iv) Choice of the threshold
The choice of the threshold is motivated by the fact that CVS of Euler
equations does not work for threshold values $T=0$ or $T=\infty$. This is
because the former corresponds to the simulation for Euler equations (all
wavelet coefficients are kept) and thus the $k^{2}$ range grows with time,
while no coefficients are retained in the latter CVS. It is anticipated that
there are appropriate values such that CVS can simulate flows without a
$k^{2}$ range in the energy spectrum. After some trial and error to avoid the
appearance of the $k^{2}$ range, we selected the value $T=2T_{0}$, where
$T_{0}=\\{(4/3)Z\ln N\\}^{1/2}$ and the enstrophy
$Z=\langle|\bm{\omega}_{c*}|^{2}\rangle/2$. The value of $T_{0}$ is based on
the Donoho threshold without iteration used in OYSFK11  to perform CVS of
Navier–Stokes equations.

### Other regularizations

For the sake of comparison, we will consider other kinds of regularization of
the Euler equations. We will thus add a term having a dissipative effect on
the solution, and another one having a dispersive effect, to obtain the
regularized Euler equations

$\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}+\nabla
p=\nu_{h}(-1)^{h+1}\nabla^{2h}\bm{u}+\alpha^{2}\nabla^{2}\partial_{t}\bm{u}+\bm{f},\quad\quad\nabla\cdot\bm{u}=0,$
(6)

where $\nu_{h}(-1)^{h+1}\nabla^{2h}\bm{u}$ is a dissipative term,
$\alpha^{2}\nabla^{2}\partial_{t}\bm{u}$ is a dispersive term, and $\bm{f}$ is
a solenoidal forcing term.

Dissipative regularization
We study two kinds of dissipative regularizations. The dissipative term in Eq.
(6) has a non-zero positive coefficient, i.e., $\nu_{h}>0$, while the
dispersive term vanishes, i.e., $\alpha=0$. Since energy is then dissipated we
need to add a forcing term $\bm{f}$ in Eq. (6) to keep the flow statistically
steady.
(a) Viscous regularization (Navier–Stokes equations)
The choice $h=1$ results in the regular Newtonian viscosity term and Eq. (6)
then corresponds to the Navier–Stokes equations.
(b) Hyperviscous regularization
Higher integer values of $h$ correspond to different kinds of
hyperdissipation, for which the energy dissipation becomes more and more
localized in a narrower and narrower range of high wavenumbers in Fourier
space, see, e.g., SMC12 . This implies a longer inertial range at the expense
of the dissipation range which is thus reduced.

Dispersive regularization
We also perform a simulation where we apply the Euler–Voigt regularization
(EV) to the Euler equations CLT06 ; Osko82 . In this case, $\nu_{h}$ and $f$
are set to zero in Eq. (6). Since it is an inviscid regularization, whose
effect is dispersive rather than dissipative, the modified energy, defined as
$E_{m}=E+\alpha^{2}Z$, is conserved in time. The Euler–Voigt regularization
with parameter $\alpha=0$ corresponds to the Euler equations and, since we
solve them using a Fourier Galerkin scheme, energy cascades and piles up at
the cutoff wavenumber during the flow evolution. To avoid such a pile-up, the
value of $\alpha$ has to be sufficiently large and here we choose $\alpha$ =
2/5.

###  II.3 Numerical methods used for the regularizations

In total we have performed five flow simulations:

  * i)

Euler equations with wavelet–based denoising (CVS),

  * ii)

Navier–Stokes equations (NS),

  * iii)

Euler equations with hyperviscous regularization (HV),

  * iv)

Euler equations with dispersive regularization (EV),

  * v)

Euler equations without any regularization (Euler).

We apply the same Fourier Galerkin method to discretize in space all those
governing equations. For the time integration we use a fourth order
Runge–Kutta method, with the time increment chosen as $1.0\times 10^{-3}$ to
insure that the Courant–Friedrichs–Lewy number remains below $0.4$ for all the
computations. For the Euler equations, with and without Euler–Voigt
regularization, we set $\nu_{h}=0$. Therefore we do not need to add any
forcing (${\bm{f}}={\bm{0}}$), since either kinetic energy $E$ or modified
energy $E+\alpha^{2}Z$ is conserved. For the remaining computations, we add a
solenoidal random forcing $f$, whose time correlation is $1.0\times 10^{-3}$
and with magnitude $1.0\times 10^{-3}$, to compensate the dissipated energy
and obtain a statistically stationary state. The forcing ${\bm{f}}$ is applied
only in the low wavenumber range $1\leq k<2.5$. Readers interested in details
on generating such a random force are referred to OYSFK11 . Note that we use
the same realization of the random forcing in all simulations. For the
coefficients of the dissipative regularizations, we set $\nu_{1}=4.0\times
10^{-4}$ for the Navier–Stokes equations, and $\nu_{4}=1.5\times 10^{-14}$ for
the hyperviscous regularization with $h=4$. The choice of these coefficients
is determined in such a way that the enstrophy, obtained when the flow
evolution has become quasi-steady, has about the same value for the different
simulations made with either dissipative regularizations or CVS
regularization. For the Euler–Voigt regularization, we set $\nu_{h}=0$ and
$\alpha=2/5$.

For all simulations, except the one with hyperviscous regularization (HV), the
number of grid points $N$ is $512^{3}$. For HV we use $N_{\rm HV}=256^{3}$ and
only those Fourier modes with wavenumbers smaller than the cutoff wavenumber
$k_{c}$ are retained. This choice is motivated by the fact that the number of
retained modes matches the number of retained wavelet coefficients in CVS.
Thus $k_{c}$ is set to $105$ using $4\pi k_{c}^{3}/3\sim N_{\rm CVS}$, where
$N_{\rm CVS}=0.036\times 512^{3}$, as we shall see later.

Type | $\nu_{h}$ | $\alpha$ | wavelet |  forcing term
---|---|---|---|---
CVS | 0 | 0 | Yes |  Yes
NS | $\nu_{1}=4.0\times 10^{-4}$ | 0 | – |  Yes
HV | $\nu_{4}=1.5\times 10^{-14}$ | 0 | – |  Yes
EV | 0 | $\alpha=2/5$ | – |  –
Euler | 0 | 0 | – |  –
Table 1:  Summary of the different regularization methods presented in this
article, mentioning the dissipative parameter $\nu_{h}$ and the dispersive
parameter $\alpha$ which have been used.

A summary of the different regularizations compared here, indicating the
parameters used for each simulation, is given in Table 1. The initial
condition of all simulations, except HV, corresponds to a fully developed
turbulent flow at Taylor microscale Reynolds number $R_{\lambda}=257$, which
was obtained by a direct numerical simulation forced with a negative viscosity
as explained in IKYIU07 . For the simulation with hyperviscous regularization
(HV), we use an initial velocity field which retains only the modes whose
wavenumber is below $k_{c}$.

Concerning the CPU time, CVS is about 60 % more expensive than NS, EV, HV and
EE, which are about the same. In the current implementation the wavelet
transform is not optimized and its parallelization is based on a transposition
technique, which requires global data communication and hence slows down the
computation. For CVS the ultimate goal is to perform Euler simulations
directly in an adaptive wavelet basis, thus reducing memory and CPU time
requirements. Viscous dissipation is then absent and dissipation is only due
to filtering out the incoherent part. In Roussel and Schneider RoSc10
computations of a slightly compressible turbulent mixing layer showed a speed-
up of the computation and memory reduction for CVS of about a factor $3$ in
comparison to DNS of Navier–Stokes. The fully adaptive version advances in
time only the coherent flow (represented by few wavelet coefficients), and
adds a safety zone at each time step to account for translation of vortices
and the generation of finer scales. We anticipate that similar performance
will be obtained in a fully adaptive version of the CVS Euler code.

##  III Numerical Results

In the following we discuss the results obtained for the five flow
simulations.

###  III.1 Time evolution of statistics

Figure 1:  Time evolution of the percentage of the retained wavelet
coefficients, $100N_{c}/N$ (top), and the percentage of the retained wavelet
coefficients at each scale $j$, $100N_{c,j}/N_{j}$ (bottom).

All computations are integrated in time for about four initial eddy turnover
times $\tau=L/u_{0}$, where $u_{0}=\sqrt{2E/3}$ and $L$ is the integral length
scale defined by $L=\pi/(2u_{0}^{2})\int_{0}^{k_{\rm max}}e(k)/kdk$, with
$e(k)$ being the isotropic energy spectrum, defined as
$e(k)=\frac{1}{2}\sum_{k-1/2\leq|\bm{p}|<k+1/2}|\bm{{\widehat{u}}}(\bm{p})|^{2}$.

Figure 1 (top) shows the time evolution of the percentage of the wavelet
coefficients retained by CVS, namely $100N_{c}/N$, where $N_{c}$ is the number
of the wavelet coefficients which correspond to the coherent flow, including
the safety zone. We see that the percentage of retained coefficients does not
vary much, around $3.5\%$, after a transient decay for $t\lesssim 0.2\tau$.
The computations presented here do not benefit from this compression in terms
of computational cost, since the flow field is reconstructed in Fourier space
or in physical space on the full grid $N=512^{3}$ at each time step and a
spectral method is used for space discretization. Nevertheless, the percentage
of retained wavelet coefficients remains a good indicator of the potential
gain which can be achieved by adaptive wavelet simulations FS01 ; SV10 . In
Fig. 1 (bottom), we plot the number of wavelet coefficients that CVS retains
at each scale, namely $100N_{c,j}/N_{j}$, where $N_{c,j}$ is the number of
wavelet coefficients which correspond to the coherent flow including the
safety zone, at a given scale indexed by $j$. Note that
$\sum_{j=0}^{J-1}N_{c,j}=N_{c}$.

Figure 2: Time evolution of energy $E$ (top), and enstrophy $Z$ (bottom).
Figure 3: Time evolution of the dissipation rate $D$.

From the largest scale $j=0$ to scale $j=5$, we observe that 100% of the
wavelet coefficients are retained as coherent flow, while at scale $j=6$ the
percentage drops to $80\%$, then to $15\%$ at scale $j=7$, and finally to less
than $1\%$ at the smallest scale $j=8$. Therefore it is the compression
obtained at the smallest scales that dominates, since the number of
coefficients, $N_{j}$, drastically increases with the scale index $j$, as
$N_{j}=7\times 2^{3j}$.

Figure 2 plots the time evolution of turbulent kinetic energy,
$E=\langle|\bm{u}|^{2}\rangle/2$, and of enstrophy,
$Z=\langle|\mbox{\boldmath$\omega$}|^{2}\rangle/2$, for the Euler equations
and for the different regularized Euler equations, namely CVS, NS, HV, and EV.
Figure 2 (top) shows that both CVS and HV present the same time evolution of
the energy as NS. It also confirms that the numerical scheme used to solve the
Euler equations is sufficiently conservative, since only 0.024 % of the
initial energy is lost after four eddy turnover times $\tau$ which is due to
the time discretization. In contrast, for EV energy decreases significantly in
time, because only the modified energy, $E+\alpha^{2}Z$, is conserved (this
within $4.3\times 10^{-6}\%$ of its initial value after $t=4\tau$). Figure 2
(bottom) shows how, after a transient period up to $t=2.5\tau$, enstrophy
reaches almost the same value for CVS, HV and NS, as expected (since we have
adjusted the parameter $\nu_{h}$ in HV and NS to match the level of enstrophy
of CVS for the steady state). For the Euler case, enstrophy grows rapidly in
time due to energy piling up at high wavenumbers in absence of regularization.
For the dispersive regularization EV, Fig. 2 (bottom) shows that the enstrophy
is almost conserved, which suggests that the nonlinear transfer of energy
towards smaller scales is inhibited. The values of energy $E$ and enstrophy
$Z$ at $t=3.4\tau$ are summarized in Table 2 for all computations.

Let us recall that the nonlinear wavelet filtering of CVS regularization
removes the noise-like incoherent part from the flow at each time step, and
thus CVS is dissipative. In order to estimate the energy dissipation rate $D$
in this case, we use

$\displaystyle D=\langle\bm{u}\cdot\bm{f}\rangle-\frac{dE}{dt},$ (7)

where $dE/dt$ is estimated by the first order forward finite difference in
time. Since NS and HV have dissipative terms, energy dissipation can be
directly estimated by
$\langle\epsilon_{h}\rangle=\nu_{h}(-1)^{h}\langle\mbox{\boldmath$u$}\cdot\nabla^{2h}\mbox{\boldmath$u$}\rangle$.
We verified that the difference between the values estimated by the two
methods, i.e., either $D$ or $\langle\epsilon_{h}\rangle$, is negligibly small
for NS and HV (the differences are less than $0.17\%$ for $t>0.1\tau$). It can
be noted that Eq. (7) is similar to what was used to estimate numerical
viscosity in DXS03 .

Figure 3 plots the energy dissipation rate $D$ for CVS, NS, HV, and EV, but
not for the Euler equations which conserve energy. We observe that the energy
dissipation of CVS is close to those for NS and HV for $t>2.5\tau$. This
suggests that the mean energy dissipation rate $D$ is insensitive to the
detailed structure of the vorticity field. This shows that using CVS removes
the incoherent noise-like contribution to the flow which corresponds to energy
dissipation. Indeed, the incoherent enstrophy is a measure of turbulent
dissipation. The insensitivity of the energy dissipation rate is consistent
with the observation in YAK15  that the scrambling the high wavenumber
contribution of the flow field does not modify the mean energy dissipation
rate. The value of $D$ for EV is determined by $D=-dE/dt$, because
$\bm{f}=\bm{0}$.

Run | $E$ |  $Z$ | $\quad A$ | $\quad L$ |  $\lambda$ |  $R_{\lambda}^{S}$
---|---|---|---|---|---|---
CVS | 0.448 |  85.53 |  0.46 |  1.07 |  0.162 |  217
NS | 0.453 |  86.22 |  0.46 |  1.11 |  0.162 |  226
HV | 0.450 |  86.57 |  0.43 |  1.11 |  0.161 |  228
EV | 0.303 |  162 |  0.10 |  0.657 |  0.097 |  –
Euler | 0.500 |  1.41$\times$$10^{4}$ |  0 |  0.260 |  0.013 |  –
Table 2:  Energy $E$, enstrophy $Z$, normalized energy dissipation rate
$A=DL/u_{0}^{3}$, integral scale $L$, Taylor microscale $\lambda$ and the
Taylor microscale Reynolds number $R_{\lambda}^{S}$ at $t=3.4\tau$. The
statistics for CVS, NS and HV are statistically stationary, while the non-
conservative statistics of EV and Euler are time-dependent.

The normalized mean energy dissipation rate $A=DL/{u_{0}}^{3}$ is a key
quantity to study the phenomenology of turbulence. Our results listed in Table
2 show that the values of $A$ are slightly smaller than 0.5 for CVS, NS, and
HV. These values agree excellently with asymptotic values for isotropic
turbulence at high Reynolds number ($R_{\lambda}\gtrsim 200$) obtained by DNS
of the Navier-Stokes equations KIY03  and hyperviscous computations HB04 . In
SMC12 , the Taylor-microscale Reynolds number as a function of $L/\lambda$,
$R_{\lambda}^{S}=36.4L/\lambda-23.1$, was introduced using data fitting and
applied for hyperviscous computations, where $\lambda$ is the Taylor-
microscale $\lambda=\sqrt{5E/Z}$. The values of $R_{\lambda}^{S}$ for our
dissipative regularizations (HV and NS) are also summarized in Table 2. The
value of $R_{\lambda}^{S}$ for NS is 226 at $t=3.4\tau$, which is close to the
value of 223 estimated by the classical definition of
$R_{\lambda}=u_{0}\lambda/\nu$ for NS. The value of $R_{\lambda}^{S}$ for CVS
is $R_{\lambda}^{S}=217$, which is close to the values of $R_{\lambda}^{S}$
for NS and HV. The values of the corresponding Taylor-microscale $\lambda$ are
also listed in Table 2 and we find very similar values for CVS, NS and HV.

###  III.2 Energy spectra and fluxes

Figure 4:  Compensated energy spectra $k^{5/3}e(k)$ in log-log plot (top), and
$D^{-2/3}k^{5/3}e(k)$ in semi-log plot (bottom) at $t=3.4\tau$.

To get insight into the spectral distribution of turbulent kinetic energy, we
plot in Fig. 4 the compensated energy spectrum $k^{5/3}e(k)$ for the five
flows as a function of wavenumber $k$ at time $t=3.4\tau$. In both the energy
containing and the inertial range ($k\lesssim 10$) we observe that CVS, NS and
HV yield similar compensated energy spectra. In contrast for EV it
substantially differs from the others due to the absence of large scale
forcing. At moderate wavenumbers ($10\lesssim k\lesssim 60$) we find for CVS
and NS similar spectral behaviors. We also notice that the compensated spectra
of all regularizations, including NS, exhibit bottlenecks with different peak
wavenumbers $k_{p}$ ($k_{p}=20$ for CVS and NS, $k_{p}=35$ for HV and EV). For
large wavenumbers ($k>60$) the energy spectrum is significantly damped for HV
compared to NS due to the hyper-dissipative term. Moreover, CVS retains much
more energy than HV and a little less than NS as the noise removed by CVS is
predominant at high wavenumbers, due to its $k^{2}$ behavior. Now considering
the Euler case, we find that for all wavenumbers the energy spectrum differs
from the four other cases and, in particular, we observe that $e(k)\propto
k^{2}$ for $k>20$, which corresponds to energy equipartition. Notice, to
compute the 1D energy spectrum $e(k)$ we have integrated the 3D energy
spectrum over spherical shells of radius $k$, i.e., the shell surface scales
as $k^{2}$. Hence, energy equipartition in 3D Fourier space corresponds to
$e(k)\propto k^{2}$. Figure 4 (bottom) shows the compensated energy spectrum
nondimensionalized by the energy dissipation rate $D$, namely
$D^{-2/3}k^{5/3}e(k)$. In the inertial range ($3\lesssim k\lesssim 10$) we
observe that both CVS and HV keep almost constant values, similar to NS, which
are close to the value $1.62$ assumed for the Kolmogorov constant. The value
$1.62$ for the 3D energy spectrum is obtained by applying the correction
factor $55/18$ MoYa75  to the value $0.530$ of the Kolmogorov constant
estimated from a large set of experimental data in Sr95  for the 1D
longitudinal energy spectrum.

Figure 5 (top) plots the energy fluxes $\Pi(k)$ for CVS, NS, HV, EV and Euler.
Here, $\Pi(k)$ and $T(k)$ are defined by $\Pi(k)=-\int_{0}^{k}T(x)dx$ and
$T(k)=-\sum_{k-{1}/{2}\leq|{\bm{p}}|<k+{1}/{2}}\mbox{\boldmath${\hat{u}}$}(-{{\bm{p}}})\cdot\mbox{\boldmath${\widehat{N}}$}({\bm{p}})$,
respectively, where
$\mbox{\boldmath$N$}=(\mbox{\boldmath$u$}\cdot\nabla)\mbox{\boldmath$u$}$. We
observe that in the inertial range ($k\lesssim 10$) the energy flux $\Pi(k)$
of CVS is close to those of NS and HV, while for moderate wavenumbers
($10<k\lesssim 60$) $\Pi(k)$ of CVS and HV are more pronounced than the energy
flux of NS. This suggests that CVS and HV well preserve the nonlinear dynamics
of turbulence in the inertial range. The flux $\Pi(k)$ in EV and Euler is
significantly reduced for $k\lesssim 10$ compared to the fluxes of CVS, NS and
HV, because forcing is absent in EV and Euler. Note that $\Pi(k_{\rm max})=0$
due to the solenoidal constraint of the velocity and the skew symmetry of the
nonlinear term $T(k)$.

The energy fluxes normalized by the energy dissipation rate $D$ are shown in
Fig. 5 (bottom) excluding the Euler case. We observe that NS, CVS and HV
exhibit a plateau range, where $\Pi(k)/D\sim 1$, for $k>2$. For NS the plateau
ends at $k\sim 10$, corresponding to the end of the intertial range. As
expected, HV exhibits the longest plateau up to $k\sim 40$, extending the
inertial range at the expense of the dissipative range, which is thus reduced.
The plateau of CVS ends at $k\sim 20$ and the corresponding energy flux
remains in between NS and HV. Indeed, CVS offers a kind of interpolation
between NS and HV. In contrast, for EV no plateau is observed and maximum
energy flux is found at $k=50$ being three times larger than for the other
cases, showing that EV is very different from NS.

Figure 5:  Energy flux $\Pi(k)$ (top), and $\Pi(k)/D$ (bottom) at $t=3.4\tau$.

###  III.3  Visualizations and $Q-R$ diagrams

Figure 6:  Visualization of intense vorticity regions for NS (green), CVS
(red), HV (magenta), EV (purple) and Euler (gray) at $t=3.4\tau$. Isosurfaces
of vorticity are shown for $|\bm{\omega}|=M+4\sigma$, where $M$ and $\sigma$
denote respectively the mean value and standard deviation of the modulus of
the vorticity field of NS. The values of $M$ and $\sigma$ are 10.2 and 8.27,
respectively. Only $1/8$ subcubes are shown to enlarge the structures.

Figure 6 shows the most intense structures of the vorticity field for CVS, NS,
HV, EV and Euler, visualized by the isosurface $|\bm{\omega}|=M+4\sigma$, $M$
being the mean value and $\sigma$ the standard deviation of the modulus of
vorticity for the Navier–Stokes simulation. The isosurface value of
$|\bm{\omega}|$ is the same for all computations. We observe that vorticity
structures are tube-like for CVS, NS, HV and EV. The structures of HV are more
sparsely distributed compared to NS which is consistent with SMC12 . In
contrast, we do not see any coherent structures in the Euler solution, which
behaves as a Gaussian white noise since the $k^{2}$ scaling of the energy
spectrum (Fig. 4) corresponds to decorrelation in physical space and the PDF
of the longitudianl velocity derivative is Gaussian (Fig. 8).

We also analyzed the velocity gradient tensor $\partial u^{i}/\partial x^{j}$
(for a review we refer to Wall09 ) of the five different flows at $t=3.4\tau$.
We study the second and and third invariants, $Q=-\frac{1}{2}\partial
u^{i}/\partial x^{j}\partial u^{j}/\partial x^{i}$ and $R=-\frac{1}{3}\partial
u^{i}/\partial x^{j}\partial u^{j}/\partial x^{\ell}\partial u^{\ell}/\partial
x^{i}$, respectively, as proposed in ChPC90 . The joint PDFs of the
dimensionless invariants, called $Q-R$ diagram, in Fig. 7 present a very
similar teardrop shape for CVS, HV and EV, close to the shape found for NS. In
contrast, for EE we observe a symmetric joint PDF with respect to the line
$R=0$, which exhibits a keyhole shape. These observations illustrate that the
small scale properties of the CVS, HV and EV flows agree well with those
observed for NS, which is not the case for what we find for EE.

Figure 7: Joint PDFs of the dimensionless invariants $Q/\langle
S_{ij}S_{ij}\rangle$ and $R/\langle S_{ij}S_{ij}\rangle^{3/2}$ where
$S_{ij}=\frac{1}{2}\left(\partial u^{i}/\partial x^{j}+\partial u^{j}/\partial
x^{i}\right)$ for CVS, HV, EV and EE in comparison with NS (dotted lines).
Shown are the isolines $1,10^{-1}$ and $10^{-2}$ together with the so-called
Vieillefosse curve Vieil84  given by $27R^{2}/4+Q^{3}=0$ which distinguishes
the four domains corresponding to different local flow patterns (A: stable
focus-stetching, B: unstable focus-compressing, C: stable node-saddle-saddle,
D: unstable node-saddle-saddle) and the vertical line $R=0$ (dashed lines).

###  III.4 Probability density functions and scale-dependent flatness

Now we show in Fig. 8 the probability density functions (PDFs) of velocity,
$P[u^{\ell}]$, and of the longitudinal velocity derivative, $P[\partial
u^{1}/\partial x^{1}]$, estimated using their histograms computed with 200
bins. Each PDF is normalized by its standard deviation. We observe that the
shape of the velocity PDF for each case, except for EV, remains close to the
shape of the normal distribution. Table 3 summarizes skewness and flatness
factors for $u^{\ell}$: The skewness values $S[u^{\ell}]$ are small and
negative (of order $10^{-2}\sim 10^{-1}$) for all computations. The flatness
values $F[u^{\ell}]$ for CVS, NS, HV and Euler are close to $3$, the flatness
of the normal distribution, while for EV it is $3.52$, which confirms a slight
departure from Gaussianity.

The longitudinal velocity derivative $\partial u^{1}/\partial x^{1}$ is a
quantity well suited to characterize small scale intermittency. In the
following we study its PDF, $P[\partial u^{1}/\partial x^{1}]$, for the five
simulations as shown in Fig. 8 (bottom). Each PDF is again normalized by the
corresponding standard deviation. First we find that the skewness of the
longitudinal velocity derivative $S[\partial u^{1}/\partial x^{1}]$ is
negative for all cases, and for CVS its value is closer to the one of NS
compared to the other cases, as shown in Table 3. For the Euler case this
skewness almost vanishes, which confirms its Gaussian behavior. While for the
Euler case the PDF is indeed Gaussian (with flatness $F[\partial
u^{1}/\partial x^{1}]=3.02$), the other cases progressively depart from
Gaussianity, reflected in heavier tails, this in the order: HV ($F[\partial
u^{1}/\partial x^{1}]=4.30$), EV ($6.32$), NS ($6.90$) and CVS ($11.2$), as
given in Table 3. These findings confirm that CVS is more intermittent than
NS, while HV is less intermittent, the latter being consistent with previous
work SMC12 . For the Euler case the normal distribution proves that the flow
is non intermittent.

Run | $S[u^{\ell}]$ |  $S[\partial u^{1}/\partial x^{1}]$ |  $F[u^{\ell}]$ |  $F[\partial u^{1}/\partial x^{1}]$
---|---|---|---|---
CVS | $-5.4\times 10^{-2}$ | -0.60 | 2.82 | 11.2
NS | -0.12 | -0.54 | 2.84 | 6.90
HV | $-5.8\times 10^{-2}$ | -0.43 | 2.86 | 4.30
EV | $-3.1\times 10^{-2}$ | -0.38 | 3.52 | 6.32
Euler | $-1.1\times 10^{-2}$ | $-3.7\times 10^{-4}$ | 3.02 | 3.02
Table 3: Skewness and flatness factors of velocity and longitudinal velocity
derivative at $t=3.4\tau$.  Figure 8: PDFs of velocity (top), and longitudinal
velocity derivative $\partial u^{1}/\partial x^{1}$ (bottom) at $t=3.4\tau$.
The Gaussian distribution is plotted as a reference.

Wavelet coefficients allow us to further quantify the flow intermittency BLS09
; YOSKF09 , since wavelets are well-localized functions in space, which are
contracted and dilated to explore a large range of scales. The scale-dependent
flatness at scale $j$ is defined by the flatness of a wavelet-filtered
quantity. The scale index $j$ corresponds to the wavenumber
$k_{j}=k_{\psi}2^{j}$, where $k_{\psi}$ is the centroid wavenumber of the
chosen wavelet ($k_{\psi}=0.77$ for the Coiflet 12 used here). For a wavelet-
filtered quantity at $k_{j}$, $v^{\ell}_{j}(\mbox{\boldmath$x$})$, given by
Eq. (4), we define the scale-dependent flatness of
$v^{\ell}_{j}(\mbox{\boldmath$x$})$ by

$F[v^{\ell}_{j}]=\langle(v^{\ell}_{j})^{4}\rangle/\langle(v^{\ell}_{j})^{2}\rangle.$
(8)

(Note that $\langle v^{\ell}_{j}\rangle=0$.) Figure 9 plots the scale-
dependent flatness for the $x^{1}$-component of velocity, $F[u^{1}_{j}]$, and
for the longitudinal velocity derivative, $F[(\partial u^{1}/\partial
x^{1})_{j}]$. For CVS, NS and EV we observe that both $F[u^{1}_{j}]$ and
$F[(\partial u^{1}/\partial x^{1})_{j}]$ increase with $k_{j}$, and this is
more significant for CVS after $k\gtrsim 50$, than for NS, and less
significant for EV compared to NS for the same scales. The flatness values
$F[u^{1}_{j}]$ and $F[(\partial u^{1}/\partial x^{1})_{j}]$ of Euler and HV
hardly depend on scale, which shows that the two flows are not, or much less
intermittent, respectively. In contrast, the wavelet-based regularization
leads to a stronger intermittency for CVS than for NS, since CVS extracts
coherent structures by denoising the vorticity field at each time step.
Reversely for EV regularization intermittency of the flow is reduced with
respect to NS. The HV regularization suppresses the flow intermittency
significantly, resulting in reduced flatness values similar to those observed
for Euler. This is also reflected in reduced tails of the PDF of the
longitudinal velocity derivative, a result which is consistent with SMC12 .

Figure 9: Scale-dependent flatness factors for $u^{1}$ and $\partial
u^{1}/\partial x^{1}$ at time $t=3.4\tau$.

##  IV Conclusions and perspectives

We have proposed a wavelet-based approach to adaptively regularize the
solution of three-dimensional incompressible Euler equations computed with a
classical Fourier Galerkin spectral method. We compared the wavelet-based
method with three regularizations: Navier–Stokes, hyperviscous and
Euler–Voigt. In addition we performed computations for the Euler equations
without regularization. The main findings can be summarized as follows: First,
wavelet-based regularization (CVS), as well as hyperviscous regularization
(HV), preserve the Navier–Stokes (NS) dynamics in the inertial range selecting
in both cases only a reduced set of the total number of modes used for NS. For
the wavelet regularization the flow is more intermittent than for NS, since it
extracts coherent structures by removing Gaussian decorrelated noise at each
time step. In contrast, the flow obtained by hyperviscous regularization is
less intermittent than for NS. CVS offers a significant compression rate
reducing the number of active degrees of freedom to only about 3.5% for the
turbulent flows studied here, i.e., $R_{\lambda}\sim 200$. For higher
$R_{\lambda}$ flows the compression rate will even be more efficient, as shown
in OYSFK07  for high resolution DNS of Navier–Stokes. From the time evolution
of energy and enstrophy of EV, it is speculated that the large-scale flow
decays in time with the small-scale flow being almost frozen. Further studies
of the flow structure and its dynamics for different values of $\alpha$ at
higher resolution for longer time computations would lead to an improved
understanding of the effect of Euler-Voigt regalization GKB15 . In conclusion,
the comparison of different regularization methods of the Euler equations
shows the potential of CVS for simulating fully developed turbulence using a
reduced number of degrees of freedom, while preserving the intermittency of
the flow. Perspectives for future work are adaptive simulations of turbulent
flows solving the Euler equations at a reduced computational cost and applying
CVS filtering for modeling turbulent dissipation.

###### Acknowledgements.

The computations were carried out on the FX100 system at the Information
Technology Center of Nagoya University. This work was partially supported by
JSPS KAKENHI Grant Number (S)16H06339, (A)2524701 and (C)17K05139. MF and KS
acknowledge support by the French Research Federation for Fusion Studies
within the framework of the European Fusion Development Agreement (EFDA).

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Denoising Diffusion Probabilistic Models for Robust Image
Super-Resolution in the Wild

Hshmat Sahakbrain,intern
Daniel Watsonbrain
Chitwan Sahariabrain
David Fleetbrain

brainGoogle Research, Brain Team.
internWork done as a student researcher

Hshmat<EMAIL_ADDRESS>
Diffusion models have shown promising results on single-image super-resolution and other image-to-image translation tasks.
Despite this success, they have not outperformed state-of-the-art GAN models on the more challenging blind super-resolution task, where the input images are out of distribution, with unknown degradations.
This paper introduces SR3+, a diffusion-based model for blind super-resolution, establishing a new state-of-the-art.
To this end, we advocate self-supervised training with a combination of composite, parameterized degradations for self-supervised training, and noise-conditioing augmentation during training and testing.
With these innovations, a large-scale convolutional architecture, and large-scale datasets, SR3+ greatly outperforms SR3.
It outperforms Real-ESRGAN when trained on the same data, with a DRealSR FID score of 36.82 vs. 37.22, which further improves to FID  of 32.37 with larger models, and further still with larger training sets.

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