# Time-domain Wavefield Reconstruction Inversion in a TTI medium.

Mathias Louboutin, Gabrio Rizzuti, and Felix J. Herrmann
School of Computational Science and Engineering, Georgia Institute of
Technology

##  1 Abstract

We introduce a generalization of time-domain wavefield reconstruction
inversion to anisotropic acoustic modeling. Wavefield reconstruction inversion
has been extensively researched in recent years for its ability to mitigate
cycle skipping. The original method was formulated in the frequency domain
with acoustic isotropic physics. However, frequency-domain modeling requires
sophisticated iterative solvers that are difficult to scale to industrial-size
problems and more realistic physical assumptions, such as tilted transverse
isotropy, object of this study. The work presented here is based on a recently
proposed dual formulation of wavefield reconstruction inversion, which allows
time-domain propagator that are suitable to both large scales and more
accurate physics.

##  2 Introduction

Wave-equation based seismic imaging has become increasingly popular due to its
ability to produce detailed and accurate subsurface models. In recent years,
however, the limitations of Full Waveform Inversion (FWI) have been widely
acknowledged due to the cycle skipping issue that arises with bandlimited data
and lack of long offsets (thus low frequencies). Simple geological settings,
such as shallow water sedimentary areas, have showcased the benefits of FWI,
but more challenging problems involve complex subsurface structures such as
salt bodies with strong anisotropy, which requires extensive manual
intervention for a consistently successful application. To address these
limitations, extended formulation have driven some of the most recent research
in seismic imaging. These methods rely on extra variables, usually a wavefield
[van Leeuwen and Herrmann, 2013b, a] as in Wavefield Reconstruction Inversion
(WRI) or time-space dependent source [Huang et al., 2018, Wang et al., 2016]
in the extended source method. Extra unknowns are designed to absorb
inaccuracies in the initial background model by relaxing the physics, while
additional constraints ensure that this relaxation is ultimately driven to a
physically consistent scenario. One of the main challenges associated with
extended formulations is the necessity to solve for an augmented least-squares
system that, in the frequency domain, is only feasible for small to mid-size
problems with simple representations of the physics or for industry-sized
problems but of limited type of acquisition geometries [Peters and Herrmann,
2019]. Conventional time-domain propagators, on the contrary, do not share the
same limitations.

This work is based on the time-domain formulation of WRI [Rizzuti et al.,
2019] which relies on conventional time-stepping [Louboutin et al., 2019,
Symes, 2015]. Therefore, it straightforwardly allows the adoption of a more
accurate representation of the physics. In this paper, we focus on the
transverse tilted isotropic (TTI) wave-equation [Zhang et al., 2011, Bube et
al., 2016, Louboutin et al., 2018]. The inclusion of anisotropic effects are
indeed crucial for the inversion of field data. Note that frequency-domain
methods do not enable TTI anisotropy in full generality, with the exception of
the limited vertical transverse anisotropy (VTI) [Aghamiry et al., 2019].
Regardless, VTI WRI still relies on iterative solvers for the augmented
system, and is therefore unfeasible for realistic sizes.

In this paper, we highlight the benefits of the time-domain WRI method [Peters
et al., 2014, van Leeuwen and Herrmann, 2013b, Rizzuti et al., 2019] for TTI
inversion and we demonstrate that TWRI can compensate for inaccuracies in the
anisotropic parameters in addition to compensating for a poor initial model.
We briefly introduce the dual formulation of WRI, then present two examples
that demonstrate that TTI WRI is feasible and more robust than FWI with
respect to modeling faulty assumptions.

##  3 Methodology

In previous work [Rizzuti et al., 2019], we introduced the dual formulation of
WRI, which only requires conventional forward and adjoint propagators in time
domain and do not need the solution of the extended wave equation. Time-domain
WRI starts from the constrained least-squares problem:

$\min_{\mathbf{m},\mathbf{u}}\dfrac{1}{2}\left\lVert\mathbf{q}-A(\mathbf{m})\mathbf{u}\right\rVert^{2}\quad\mathrm{s.t.}\quad\left\lVert\mathbf{d}-R\mathbf{u}\right\rVert\leq\varepsilon,$
(1)

where $\mathbf{m}$ represents the physical properties of interest and
$\mathbf{u}$ is the associated wavefield. The data is denoted by $\mathbf{d}$
and $\mathbf{q}$ is the (known) source term. The wave equation is denominated
by $A(\mathbf{m})$ and $R$ is receiver-restriction operator. We then derive
the reduced Lagrangian associated with Eq. 1 (c.f [Rockafellar, 1970, Rizzuti
et al., 2020]) to obtain the TWRI objective function:

$\begin{split}&\max_{\mathbf{y}}\min_{\mathbf{m}}\mathcal{L}_{\mathrm{TWRI}}(\mathbf{m},\mathbf{y}),\\\
&\mathcal{L}_{\mathrm{TWRI}}(\mathbf{m},\mathbf{y})=-\dfrac{1}{2}\left\lVert
F(\mathbf{m})^{{}^{*}}\mathbf{y}\right\rVert^{2}+\langle\mathbf{y},\mathbf{r}(\mathbf{m})\rangle-\varepsilon\left\lVert\mathbf{y}\right\rVert,\end{split}$
(2)

where the forward operator $F(\mathbf{m})=RA(\mathbf{m})^{-1}$ is the forward
modeling operator, and
$\mathbf{r}(\mathbf{m})=\mathbf{d}-F(\mathbf{m})\mathbf{q}$ the data residual
for the model $\mathbf{m}$. This dual problem has two unknowns, parameters
$\mathbf{m}$ (squared slowness), and variables $\mathbf{y}$(having the same
size of data). One of the advantages of the formulation in Eq. 2 is that,
unlike conventional WRI, the additional variable is of a manageable size
(compared to wavefields). To update the model $\mathbf{m}$, we calculate the
derivative:

$\nabla_{\mathbf{m}}\mathcal{L}_{\mathrm{TWRI}}=-J[\mathbf{m},\mathbf{q}+F(\mathbf{m})^{{}^{*}}\mathbf{y}]^{{}^{*}}\mathbf{y},$
(3)

where $J$ is the conventional FWI Jacobian operator for an extended source
$\mathbf{q}+F(\mathbf{m})^{{}^{*}}\mathbf{y}$. It is straightforward to extend
the previous acoustic implementation to the Transverse Tilted Isotropic case
[TTI, Zhang et al., 2011, Duveneck and Bakker, 2011, Louboutin et al., 2018],
just by simply replacing $F(\mathbf{m})$ with its TTI counterpart.

Previous work have demonstrated that TWRI behaves more robustly than FWI with
respect to local minimum issues in the purely acoustic case. Here we
illustrate that the edge of WRI over FWI still holds for TTI and, more
importantly, when the assumed physics do not match with the collected data.

##  4 Numerical experiments

These two examples aim to elucidate two main advantages of TWRI. First, since
we use a time-domain formulation that only necessitates the implementation of
standard forward and adjoint propagators, we are able to implement TTI TWRI by
simply replacing the acoustic time-stepper with an anisotropic version
thereof. Moreover, thanks to Devito [Louboutin et al., 2019, Luporini et al.,
2018] and JUDI[Witte et al., 2019], these propagators are implemented in a
simple high-level way and benefit from its highly efficient just-in-time
compiler. Finally, we show that TTI TWRI mitigates cycle skipping either with
true anisotropy parameters and a kinematically inaccurate initial model, or
with inaccurate anisotropic parameters and a fair initial model. Finally, we
verify a known secondary advantage of WRI that is its robustness to inaccurate
water layer velocity and ocean bottom position.

We concentrate on three increasingly difficult settings over these examples:

  1. 1.

Invert the TTI data with a TTI wave-equation and assuming the true anisotropy
parameters are known.

  2. 2.

Invert the TTI data with a TTI wave-equation with error introduced in the
anisotropic parameters.

  3. 3.

Invert the TTI data with an acoustic wave-equation.

The first two cases demonstrate the TWRI mitigates the sensitivity to cycle
skipping associated with both the velocity and anisotropy errors. The third
case demonstrates that TWRI can compensate for a numerical representation of
the physics that does not correspond to the physics of the observed data.

###  4.1 Gaussian lens

This first example follows [Huang et al., 2018] with a 2D singularity model
with a constant initial model. This model in known to be cycle skipped and
demonstrate the capability of TWRI to obtain a better update direction than
conventional FWI. To further highlight the robustness with respect to the
Thomsen parameters[Thomsen, 1986], we fix these to zero and compute the
gradient. This parameterization shows that TWRI can compensate for anisotropy
errors for the computation of the gradient with respect to the squared
slowness.

Figure 1: Squared slownes, Thomsen parameters and dip angle for the BG Compass
model.

We show the initial model (constant velocity) and true perturbation in Figure
2, and we expect to see the gradient update to have the same sign as the true
perturbation in Figure 2.

Figure 2: Initial models and true perturbation for the Gaussian lens model.
The initial model is a constant velocity and the true perturbation is the
velocity singularity.

The gradient obtained with each of the three cases are displayed on Figure 3.
We first see that, as expected from previous work, FWI does not produce a good
update direction as the sign is flipped compared to the true perturbation
(FIgure 2) and optimizations algorithms will not converge to a good solution.
On the other hand, we can see that in all three cases, the update direction
obtained with TWRI is consistent with the true perturbation and will lead to a
good model reconstruction. One interesting observation is that TWRI correctly
handles the errors in the anisotropy, including for the complete absence of
anisotropy in the modeling kernel. Such result is encouraging in light of more
realistic examples.

Figure 3: Gradients obtained with the anisotropic data for the three different
inversion settings.

###  4.2 BG Compass

The second example involves the BG Compass model. The Thomsen parameters are
synthesized from the velocity model, while dip angles are inferred from the
orientation of the layers. We show the TTI parameters in Figure 4. This model
is notoriously difficult due to the velocity kick back (situated at around the
one kilometer depth on Figure 4) that prevents turning waves from traveling
back to the surface.

Figure 4: Squared slowness, Thomsen parameters and dip angle for the BG
Compass model.

This experiment is divided in two settings. On one hand, we assume the water
velocity and depth of the water layer to be known and look at the first update
computed with TWRI and FWI. In the second case, we do not assume any knowledge
of the water layer, that is known to be delignated by WRI in the acoustic
case, and once again look at the first FWI and TWWRI update. These two initial
models and the true perturbation associated with it are on Figure 5, and a
good update is expected to correlate with the true perturbation well while a
cycle skipped update would have the wrong sign in most areas in particular for
the velocity kick-back part.

Figure 5: Initial models and true perturbation for the two test cases. The two
initial models are smoothed version of the true model with a 20 points
gaussian filter, and the right initial model has the true water layer speed
and ocean bottom position.

The results obtained with the correct water velocity are shown in Figure 6.
Similarly to the previous example, TWRI succeeds to capture most of the
features of the true perturbation while FWI displays major differences. The
most difficult part of the model to image is the middle section around 1 km
depth. The velocity kick-back makes the inversion very difficult as the
turning wave are diving deeper into the model rather than going back up
towards the receivers. We can see that TWRI perfectly captures this velocity
kick back while FWI only succeeds to match partially the shallower part of the
model.

Figure 6: Gradients obtained with the anisotropic data and an initial model
without the correct water layer.

In the second part of this experiment, the results obtained with the incorrect
water velocity are shown on Figure 6. As expected from previous results
[Peters et al., 2014], TWRI provides a correct update direction in the water
layer while still featuring the attributes necessary for inversion such as the
previously mentioned velocity kick-back.

Figure 7: Gradients obtained with the anisotropic data and an initial model
with the correct water layer.

These examples and related software can be found at TTIWRI in our
reproducibility repository https://github.com/slimgroup/Software.SEG2020.

##  5 Discussion & conclusions

In this work, we presented an application of TWRI to a realistic inversion
scenario, by including TTI physics. Because our work is based on time-domain
modeling, we can leverage on state of the art anisotropic propagators. The
extension to anisotropy is not trivial in the frequency domain due to the
current limitations of iterative solvers for large-scale, coupled PDEs. We
experimented that TWRI not only produces more qualitative results that
conventional FWI when data and modeling physics matches, but also fairs better
with respect to modeling inaccuracies, e.g. when the inverted data presents
anisotropic effects but an acoustic medium is postulated. While these
preliminary results are encouraging, future work will focus on a more thorough
validation of TWRI for TTI media and applications to 3D examples.

##  6 References

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The Kagan characterization theorem on Banach spaces

Margaryta Myronyuk

B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine,
Nauky Ave. 47, Kharkiv, Ukraine

Bielefeld University,
Universitätsstraße 25, Bielefeld, Germany

<EMAIL_ADDRESS>
###### Abstract

A. Kagan introduced classes of distributions $\mathcal{D}_{m,k}$ in
$m$-dimensional space $\mathbb{R}^{m}$. He proved that if the joint
distribution of $m$ linear forms of $n$ independent random variables belong to
the class $\mathcal{D}_{m,m-1}$ then the random variables are Gaussian. If
$m=2$ then the Kagan theorem implies the well-known Darmois-Skitovich theorem,
where the Gaussian distribution is characterized by the independence of two
linear forms of $n$ independent random variables. In the paper we describe
Banach spaces where the analogue of the Kagan theorem is valid.

Mathematics Subject Classification: 60B05, 62E10

Key words and phrases: Banach space, Characterization theorem, Gaussian
distribution, independent random variables, Q-independent random variables,
Kagan theorem, Darmois-Skitovich theorem

##  1 Introduction

In 1953 G. Darmois and V.P. Skitovich independently proved the following
theorem which characterizes the Gaussian distribution on the real line.

The Darmois-Skitovich theorem ([2], [28], see also [14, Sec. 3]) Let
$\xi_{1},\dots,\xi_{n}$, $n\geq 2,$ be independent random variables. Let
$a_{j},b_{j}$ be nonzero numbers. If linear forms
$L_{1}=a_{1}\xi_{1}+\dots+a_{n}\xi_{n}$ and
$L_{2}=b_{1}\xi_{1}+\dots+b_{n}\xi_{n}$ are independent then $\xi_{j}$ are
Gaussian random variables.

The generalizations of this characterization theorem to various algebraic
structures were considered in numerous works (see, e.g., [5] – [8], [19]–
[21], [23], for more references one can see [4]), in which the coefficients of
forms are topological automorphisms of the corresponding algebraic structures.
In 1962, S.G. Ghurye and I. Olkin generalized the Darmois-Skitovich theorem to
the case of random variables taking values in $\mathbb{R}^{n}$, where the
coefficients of forms are non-degenerate matrices ([8]). Later, A. Zinger
simplified their proof (see [14, Sec. 3.2]). The Darmois-Skitovich theorem was
also generalized to the case of random variables taking values in a Hilbert
space (see [16], [18]). In 1985, W. Krakowiak generalized the Darmois-
Skitovich theorem to the case of random variables taking values in an
arbitrary separable Banach space $X$, where the coefficients of forms are
invertible linear bounded operators $A_{j}$ and $B_{j}$ ([17]). In 2008,
M.Myronyuk gave another proof of the Darmois-Skitovich theorem on Banach
spaces ([20]). In this paper the Heyde theorem on Banach spaces was proved as
well. Later in [23] the analogues of these theorems were proved for
Q-independent random variables.

Consider a random vector $\boldsymbol{L}$= $(L_{1},\dots,L_{m})$. The
distribution $\mu_{\boldsymbol{L}}$ of the random vector $\boldsymbol{L}$
belongs to the class $\mathcal{D}_{m,k}$, $1\leq k\leq m$, if the
characteristic function $\hat{\mu}_{\boldsymbol{L}}(t_{1},\dots,t_{m})$ admits
the following factorization

$\hat{\mu}_{\boldsymbol{L}}(t_{1},\dots,t_{m})={\mathbb{E}}[e^{{\mathrm{i}}\left\langle\boldsymbol{L},(t_{1},\dots,t_{m})\right\rangle}]=\prod_{{i_{1},\dots,i_{k}}}R_{i_{1},\dots
i_{k}}(t_{i_{1}},\dots t_{i_{k}}),\quad t_{i}\in\mathbb{R},$

where $R_{i_{1},\dots,i_{k}}(t_{i_{1}},\dots,t_{i_{k}})$ are continuous
functions such that $R_{i_{1},\dots i_{k}}(0,\dots,0)=1$, and all indexes
$(i_{1},\dots,i_{k})$ in this product satisfy the conditions $1\leq
i_{1}<\dots<i_{k}\leq m$. A. Kagan proved the following generalization of the
Darmois-Skitovich theorem.

The Kagan theorem ([13]) Let $\xi_{1},\dots,\xi_{n}$, $n\geq 2,$ be
independent random variables. Let $A_{ji}$ be nonzero numbers. Put
$L_{j}=A_{j1}\xi_{1}+\cdots+A_{jn}\xi_{n}$, $j=1,2,\dots,m$. If
$\boldsymbol{L}$= $(L_{1},\dots,L_{m})\in\mathcal{D}_{m,m-1}$ then all
$\xi_{j}$ are Gaussian random variables.

The class $\mathcal{D}_{2,1}$ contains all distributions of $\mathbb{R}^{2}$
with independent components. Therefore the Darmois-Skitovich is a partial case
of the Kagan theorem. Note that the Kagan theorem was generalized for locally
compact Abelian groups ([6]). In this paper we prove analogues of the Kagan
theorem for Banach spaces.

##  2 Notation and definitions

We use standard facts of the theory of probability distributions in a Banach
space (see, e.g., [29]). Let $X$ be a separable real Banach space and let
$X^{*}$ be its dual space. Denote by ${\cal L}_{inv}(X)$ the set all linear
bounded invertible operators in $X$. Denote by $A^{*}$ the adjoint operator to
a linear bounded operator $A$. Denote by $I$ the identity operator. Denote by
$\left\langle x,f\right\rangle$ the value of a functional $f\in X^{*}$ at an
element $x\in X$. Denote by $\mathcal{M}^{1}(X)$ the convolution semigroup of
probability distributions on $X$. For $\mu\in\mathcal{M}^{1}(X)$ we denote by
$\overline{\mu}$ the distribution defined by the formula
$\overline{\mu}(E)=\mu(-E)$ for any Borel set $E\subset X$. Let $\xi$ be a
random variable with values in $X$ and with a distribution $\mu$. The
characteristic functional of $\mu$ is defined by the formula

$\widehat{\mu}(f)=\mathbf{E}[e^{\mathrm{i}\left\langle\xi,f\right\rangle}]=\int\limits_{X}e^{\mathrm{i}\left\langle
x,f\right\rangle}d\mu(x),\quad f\in X^{*}.$

Note that $\widehat{\overline{\mu}}(f)={\overline{\widehat{\mu}}}(f)$. It is
known that $\widehat{\mu}(f)$ is positive definite and continuous in the norm
topology ([29, Ch.4, §2]).

In the paper, unless otherwise specified, we work in a norm topology.

Let $\psi(f)$ be a complex-valued function on $X^{*}$, and let $h$ be an
arbitrary element of $X^{*}$. We denote by $\Delta_{h}$ an operator of the
finite difference

$\Delta_{h}\psi(f)=\psi(f+h)-\psi(f).$

A function $\psi(f)$ on $X^{*}$ is called a polynomial if

$\Delta_{h}^{n+1}\psi(f)=0$

for some $n$ and for all $f,h\in X^{*}$.

###### Definition 2.1

A random variable $\xi$ is called Gaussian if for any $f\in X^{*}$ the real-
valued random variable $\left\langle\xi,f\right\rangle$ is Gaussian. In other
words $\mu\in\mathcal{M}^{1}(X)$ is called Gaussian if its one-dimensional
image $\mu_{f}=\mu\circ f^{-1}$ is a Gaussian distribution for all $f\in
X^{*}$ and hence there exist real numbers $m_{f}$ and $\sigma_{f}\geq 0$ such
that $\hat{\mu}_{f}(t)=e^{{\mathrm{i}m_{f}t-{1\over 2}\sigma_{f}t^{2}}}$,
$t\in\mathbb{R}$.

The following proposition follows immediately from Definition 2.1.

###### Proposition 2.1

A distribution $\mu$ on $X$ is Gaussian if and only if for any $f\in X^{*}$
the characteristic function $\widehat{\mu}_{f}(t)$ of a random variable
$\left\langle\xi,f\right\rangle$ is a characteristic function of a Gaussian
distribution on $\mathbb{R}$.

Proposition 2.1 allows easily to obtain analogues of the well-known classical
theorems for a real separable Banach space such as the Kac-Bernstein theorem
([1], [10]) on characterization of a Gaussian distribution by the independence
of the sum of independent random variables and their difference, the Cramer
theorem on the decomposition of a Gaussian distribution ([27, §4.1]), and the
Marcinkiewicz theorem ([27, §3.13]). The proofs of these theorems reduces to
the description of solutions of the corresponding functional equations that
admit restrictions to each one-dimensional subspace.

For the convenience of references we formulate the analogues of the Cramer
theorem and the Marcinkiewicz theorem for Banach spaces as propositions.

###### Proposition 2.2

Let $\xi$ be a Gaussian random variable with values in a real separable Banach
space $X$. Let $\xi=\xi_{1}+\xi_{2}$, where $\xi_{1},\xi_{2}$ are independent
random variables with values in $X$. Then $\xi_{1},\xi_{2}$ are Gaussian
random variables.

###### Proposition 2.3

Let $X$ be a real separable Banach space, $\mu\in\mathcal{M}^{1}(X)$. Let
$\psi(f)$ be a polynomial on $X^{*}$. If

$\widehat{\mu}(f)=e^{\psi(f)}$ (1)

in a certain neighborhood of zero, then $\mu$ is a Gaussian distribution on
$X$.

Note that the anologue of the Darmois-Skitovich theorem for a Banach space can
not be proved in a such simple way.

###  2.1 The main results

Let $X$ be a separable real Banach space. Consider a random vector
$\boldsymbol{\zeta}$= $(\zeta_{1},\dots,\zeta_{m})$ in values in $X^{m}$.
Following A.M. Kagan ([13]) we say that the distribution of the random vector
$\boldsymbol{\zeta}$ belongs to the class $\mathcal{D}_{m,k}$, $1\leq k\leq
m$, if the characteristic functional
$\hat{\mu}_{\boldsymbol{\zeta}}(f_{1},\dots,f_{m})$ admits the following
factorization

$\hat{\mu}_{\boldsymbol{\zeta}}(f_{1},\dots,f_{m})={\mathbb{E}}[e^{{\mathrm{i}}\left\langle\boldsymbol{\zeta},(f_{1},\dots,f_{m})\right\rangle}]=\prod_{{i_{1},\dots,i_{k}}}R_{i_{1},\dots
i_{k}}(f_{i_{1}},\dots f_{i_{k}}),\quad f_{i}\in X^{*},$

where $R_{i_{1},\dots i_{k}}(f_{i_{1}},\dots,f_{i_{k}})$ are continuous
functions such that $R_{i_{1},\dots i_{k}}(0,\dots,0)=1$, and in the product
all indexes $(i_{1},\dots,i_{k})$ satisfy the conditions $1\leq
i_{1}<\dots<i_{k}\leq m$.

We formulate an analogue of this characterization for independent random
variables with values in a Banach space.

###### Theorem 2.1

Let $X$ be a real reflexive separable Banach space. Let $A_{pj}$,
$p=\overline{1,m}$, $i=\overline{1,n}$, be linear continuous operators of $X$
such that $KerA_{pj}=\\{0\\}$. Put

$G_{i}=\\{(A_{1i}x,...,A_{mi}x)\in X^{m}:x\in X\\},\quad i=\overline{1,n}.$
(2)

Suppose that the condition

$\overline{G_{i}}\cap\overline{G_{j}}=\\{0\\},\quad i,j=\overline{1,n},i\neq
j.$ (3)

holds. Let $\xi_{i}$, $i=1,2,\dots,n$, be independent random variables with
values in $X$. Consider the linear forms
$L_{j}=A_{j1}\xi_{1}+\cdots+A_{jn}\xi_{n}$, $j=1,2,\dots,m$. If the random
vector $\mathbf{L}$=$(L_{1},\dots,L_{m})$ $\in\mathcal{D}_{m,m-1}$ then all
$\xi_{i}$ are Gaussian random variables.

Theorem 2.1 is a spesial case of Theorem 2.2 which is formulated and is proved
further for $Q$-independent random variables.

Let $\xi_{1},\dots,\xi_{n}$ be random variables with values in separable
Banach space $X$. Following A. Kagan and G. Székely ([15]), we say that random
variables $\xi_{1},\dots,\xi_{n}$ are $Q$-independent if the characteristic
functional of the distribution of the vector $(\xi_{1},\dots,\xi_{n})$ can be
represented in the form

$\widehat{\mu}_{(\xi_{1},\dots,\xi_{n})}(f_{1},\dots,f_{n})={\bf
E}[e^{i\left\langle(\xi_{1},\dots,\xi_{n}),(f_{1},\dots,f_{n})\right\rangle}]=$
$=\left(\prod_{j=1}^{n}\widehat{\mu}_{\xi_{j}}(f_{j})\right)\exp\\{q(f_{1},\dots,f_{n})\\},\quad
f_{j}\in X^{*},$ (4)

where $q(f_{1},\dots,f_{n})$ is a continuous polynomial on $(X^{*})^{n}$ such
that $q(0,\dots,0)=0$.

The paper [15] has led to new researches of characterization problems on
locally compact Abelian groups ([5], [21], [22]) and Banach spaces ([23]).

It is obvious that if random variables $\xi_{1},\dots,\xi_{n}$ are independent
then they are $Q$-independent (in this case the function $q\equiv 0$).
Therefore if a statement is valid for Q-independent random variables then it
is valid for independent random variables (for example Theorem 2.2 implies
Theorem 2.1).

The following $Q$-analogue of the Proposition 2.2 (the Cramer theorem in a
Banach space) follows directly from Definition 2.1 and the $Q$-analogue of the
Cramer theorem on a real line (see [15]).

###### Proposition 2.4

Let $\xi$ be a Gaussian random variable with values in a real separable Banach
space $X$. Let $\xi=\xi_{1}+\xi_{2}$, where $\xi_{1},\xi_{2}$ are
$Q$-independent random variables with values in $X$. Then $\xi_{1},\xi_{2}$
are Gaussian random variables.

The following theorem is the $Q$-analogue of Theorem 2.1.

###### Theorem 2.2

Let $X$ be a real reflexive separable Banach space. Let $A_{pj}$,
$p=\overline{1,m}$, $i=\overline{1,n}$, be linear continuous operators of $X$
such that $KerA_{pj}=\\{0\\}$. Let $G_{i}$ be the sets defined by
$(\ref{Kag2})$. Suppose that the condition $(\ref{Kag1})$ holds. Let
$\xi_{i}$, $i=1,2,\dots,n$, be Q-independent random variables with values in
$X$. Consider the linear forms $L_{j}=A_{j1}\xi_{1}+\cdots+A_{jn}\xi_{n}$,
$j=1,2,\dots,m$. If the random vector $\mathbf{L}$=$(L_{1},\dots,L_{m})$
$\in\mathcal{D}_{m,m-1}$ then all $\xi_{i}$ are Gaussian random variables.

Proof. Consider the continuous linear functions $T_{i}:X\rightarrow X^{m}$ of
the form

$T_{i}(x)=(A_{1i}x,...,A_{mi}x)\in G_{i}$

and the factor mappings

$P_{i}:X^{m}\rightarrow X^{m}/\overline{G_{i}}.$

We will construct the compositions

$F_{ij}=P_{i}T_{j}$

and will check that $KerF_{ij}=\\{0\\}$ for $i\neq j$. Let $x\in KerF_{ij}$.
Then $F_{ij}(x)=P_{i}T_{j}(x)=P_{i}(A_{1j}x,...,A_{mj}x)\in KerP_{i}$ that is
$(A_{1j}x,...,A_{mj}x)\in\overline{G_{i}}$. It follows from condition (3) that
$x=0$ for $i\neq j$. Thus $KerF_{ij}=\\{0\\}$ for $i\neq j$. Since $X$ is a
reflexive space, it follows from this that

$\overline{F_{ij}^{*}(X^{*})}=X^{*}$ (5)

in the norm topology.

Note that $F_{ij}^{*}=T_{j}^{*}P_{i}^{*}$. Put $C_{pi}=A_{pi}^{*}$,
$p=\overline{1,m}$, $i=\overline{1,n}$. It is easy to verify that the mapping
$T_{i}^{*}:(X^{*})^{m}\rightarrow X^{*}$ has the form

$T_{i}^{*}(f_{1},...,f_{m})=C_{1i}f_{1}+\cdots+C_{mi}f_{m},\quad
i=\overline{1,n}.$

Put

$(\overline{G_{i}})^{\perp}=\\{(f_{1},...,f_{m})\in(X^{*})^{m}:\langle
x_{1},f_{1}\rangle+\ldots+\langle
x_{m},f_{m}\rangle=0\quad\forall(x_{1},...,x_{m})\in\overline{G_{i}}\\}.$

It is clear that

$(\overline{G_{i}})^{\perp}=\\{(f_{1},...,f_{m})\in(X^{*})^{m}:\langle
A_{1i}x,f_{1}\rangle+\ldots+\langle A_{mi}x,f_{m}\rangle=0\quad\forall x\in
X\\}=$ $=\\{(f_{1},...,f_{m})\in(X^{*})^{m}:\langle
x,C_{1i}f_{1}+\cdots+C_{mi}f_{m}\rangle=0\quad\forall x\in X\\}=KerT_{i}^{*}.$

Thus $P_{i}^{*}$ is a natural embedding of $KerT_{i}^{*}$ into $(X^{*})^{m}$.

Let $\mu_{i}$ be a distribution of the random variable $\xi_{i}$. We consider
the characteristic functional $\hat{\mu}_{\mathbf{L}}(f_{1},\dots,f_{m})$ of
the random vector $\mathbf{L}$. Since the random variables $\xi_{i}$ are
Q-independent, the functional $\hat{\mu}_{{\mathbf{L}}}(f_{1},\dots,f_{m})$
has the form

$\hat{\mu}_{{\mathbf{L}}}(f_{1},\dots,f_{m})={\bf
E}[e^{{{\mathrm{i}}\left\langle{\mathbf{L}},(f_{1},\dots,f_{m})\right\rangle}}]={\bf
E}[e^{{{\mathrm{i}}\left\langle(L_{1},\dots,L_{m}),(f_{1},\dots,f_{m})\right\rangle}}]=$
$={\bf E}\bigg{[}e^{{\mathrm{i}}\sum_{j=1}^{m}\left\langle
A_{j1}\xi_{1}+\cdots+A_{jn}\xi_{n},f_{j}\right\rangle}\bigg{]}={\bf
E}\bigg{[}e^{{\mathrm{i}}\sum_{i=1}^{n}\left\langle\xi_{i},C_{1i}f_{1}+\dots+C_{mi}f_{m}\right\rangle}\bigg{]}=$
$=\prod_{i=1}^{n}\hat{\mu}_{i}(C_{1i}f_{1}+\dots+C_{mi}f_{m})\exp\\{q(f_{1},\dots,f_{m})\\},\quad
f_{j}\in X^{*}.$ (6)

where $q(f_{1},\dots,f_{m})$ is a continuous polynomial on $(X^{*})^{n}$ such
that $q(0,\dots,0)=0$.,

Since ${\mathbf{L}}\in\mathcal{D}_{m,m-1}$, we have

$\hat{\mu}_{{\mathbf{L}}}(f_{1},\dots,f_{m})=\prod_{j=1}^{m}R_{j}(f_{1},\dots,f_{j-1},f_{j+1},\dots,f_{m}),\quad
f_{j}\in X^{*},$ (7)

where $R_{i_{1},\dots i_{k}}(f_{i_{1}},\dots,f_{i_{k}})$ are continuous
functions such that $R_{i_{1},\dots i_{k}}(0,\dots,0)=1$, and all indexes
$(i_{1},\dots,i_{k})$ satisfy the conditions $1\leq i_{1}<\dots<i_{k}\leq m$.

It follows from (6) and (7) that

$\prod_{i=1}^{n}\hat{\mu}_{i}(C_{1i}f_{1}+\dots+C_{mi}f_{m})\exp\\{q(f_{1},\dots,f_{m})\\}=$
$=\prod_{j=1}^{m}R_{j}(f_{1},\dots,f_{j-1},f_{j+1},\dots,f_{m}),\quad f_{j}\in
X^{*}.$ (8)

Put $\nu_{i}=\mu_{i}*\bar{\mu}_{i}$. Then
$\hat{\nu}_{i}(f)=|\mu_{i}(f)|^{2}\geq 0$ and the characteristic functionals
$\hat{\nu}_{i}(f)$ also satisfy (8), in which all factors in the left and
right parts of the equation are non-negative. If we prove that all $\nu_{i}$
are Gaussian distributions then Proposition 2.4 implies that all $\mu_{i}$ are
Gaussian distributions. Thus, we may assume from the beginning that all
factors in the left and right parts of the equation (8) are non-negative.

Since $\hat{\mu}_{i}(0)=1$, it follows from the continuity of the
characteristic functionals $\hat{\mu}_{i}(f)$ in the norm topology that there
exists a ball $U_{R}$ around zero such that all $\hat{\mu}_{i}(f)>0$ for $f\in
U_{R}$. We choose in $U_{R}$ a ball $V_{r}$, where the radius $r={R\over
m\cdot\max\|C_{pi}\|}$ ($p=\overline{1,m}$, $i=\overline{1,n}$). Then

$C_{1i}(V_{r})+\dots+C_{mi}(V_{r})\subset U_{R},\quad i=\overline{1,n}.$ (9)

Put $\psi_{i}=\log\hat{\mu}_{i}$ in $V_{r}$, $i=\overline{1,n}$, $s_{j}=\log
R_{j}$, $j=1,2,\dots,m$. It follows from (8) that the functions $\psi_{i}$ and
$s_{j}$ satisfy equation

$\sum_{i=1}^{n}\psi_{i}(C_{1i}f_{1}+\cdots+C_{mi}f_{m})+q(f_{1},\dots,f_{m})=$
$=\sum_{j=1}^{m}s_{j}(f_{1},\ldots,f_{j-1},f_{j+1},\ldots,f_{m}),\quad
f_{j}\in V_{r},$ (10)

where $s_{j}$ are arbitrary functions.

We use the finite difference method to solve equation (10). Let $g_{nj}$,
$j=\overline{1,m}$, be arbitrary elements of $V_{r}$ such that

$C_{1n}g_{n1}+\cdots+C_{mn}g_{nm}=0.$

Note that the existence of such elements follows from (5) and the form of
$F_{ij}^{*}$. Substitute $f_{j}+g_{nj}$ for $f_{j}$ for all $j$ in equation
(10). We obtain

$\sum_{i=1}^{n-1}\Delta_{h_{in}}\psi_{i}(C_{1i}f_{1}+\cdots+C_{mi}f_{m})+\Delta_{\overline{{g}}_{n}}q(f_{1},\dots,f_{m})=$
$=\sum_{j=1}^{m}\Delta_{\overline{{v}}_{jn}}s_{j}(f_{1},\ldots,f_{j-1},f_{j+1},\ldots,f_{m}),\quad
f_{j}\in V_{r},$ (11)

where $h_{in}=C_{1i}g_{n1}+\cdots+C_{mi}g_{nm}$, $i=\overline{1,n-1}$,
$\overline{{v}}_{jn}=(g_{n1},\ldots,g_{n,j-1},g_{n,j+1},\ldots,g_{nm})$,
$j=\overline{1,m}$, $\overline{{g}}_{n}=(g_{n1},...,g_{nm})$. The left-hand
side of equation (11) no longer contains the function $\psi_{n}$.

Let $g_{n-1,j}$, $j=\overline{1,m}$, be arbitrary elements of $V_{r}$ such
that

$C_{1,n-1}g_{n-1,1}+\cdots+C_{m,n-1}g_{n-1,m}=0.$

Reasoning similarly we exclude the function $\psi_{n-1}$ from the left-hand
side of equation (11). We obtain

$\sum_{i=1}^{n-2}\Delta_{h_{in}}\Delta_{h_{i,n-1}}\psi_{i}(C_{1i}f_{1}+\cdots+C_{mi}f_{m})+\Delta_{\overline{{g}}_{n}}\Delta_{\overline{{g}}_{n-1}}q(f_{1},\dots,f_{m})=$
$=\sum_{j=1}^{m}\Delta_{\overline{{v}}_{jn}}\Delta_{\overline{{v}}_{j,n-1}}s_{j}(f_{1},\ldots,f_{j-1},f_{j+1},\ldots,f_{m}),\quad
f_{j}\in V_{r},$ (12)

where $h_{i,n-1}=C_{1i}g_{n-1,1}+\cdots+C_{mi}g_{n-1,m}$,
$i=\overline{1,n-2}$,
$\overline{{v}}_{j,n-1}=(g_{n-1,1},\ldots,g_{n-1,j-1},g_{n-1,j+1},\ldots,g_{n-1,m})$,
$j=\overline{1,m}$, $\overline{{g}}_{n-1}=(g_{n-1,1},...,g_{n-1,m})$.

By excluding the functions $\psi_{j}$ from the left-hand side of equation
(10), after $n-1$ steps we come to the equation of the form

$\Delta_{h_{12}}...\Delta_{h_{1n}}\psi_{1}(C_{11}f_{1}+\cdots+C_{m1}f_{m})+\Delta_{\overline{{g}}_{n}}...\Delta_{\overline{{g}}_{2}}q(f_{1},\dots,f_{m})=$

$=\sum_{j=1}^{m}r_{j}(f_{1},\ldots,f_{j-1},f_{j+1},\ldots,f_{m}),\quad
f_{j}\in V_{r},$ (13)

where

$h_{1j}=C_{11}g_{j1}+\cdots+C_{m1}g_{jm},\quad j=\overline{2,n},$ (14)

$\overline{{g}}_{j}=(g_{j1},...,g_{jm})$ and $r_{j}$ are arbitrary functions.
Note that we chose elements $g_{ij}$ at every step in such a manner that the
equlity

$C_{1i}g_{i1}+\cdots+C_{mi}g_{im}=0,\quad i=\overline{2,n},$ (15)

is fulfilled.

Let $k_{m}$ be an arbitrary element of $X^{*}$. Substitute $f_{m}+k_{m}$ for
$f_{m}$ in (13) and subtract equation (13) from the resulting equation. We get

$\displaystyle\Delta_{C_{mi}k_{m}}\Delta_{h_{12}}...\Delta_{h_{1n}}\psi_{1}(C_{11}f_{1}+\cdots+C_{m1}f_{m})+\Delta_{\overline{{k}}_{m}}\Delta_{\overline{{g}}_{n}}...\Delta_{\overline{{g}}_{2}}q(f_{1},\dots,f_{m})=$
$\displaystyle=\sum_{j=1}^{m-1}r_{j}(f_{1},\ldots,f_{j-1},f_{j+1},\ldots,f_{m}),\quad
f_{j}\in V_{r},$ (16)

where $\Delta_{\overline{{k}}_{m}}=(0,..,0,k_{m})$. Note that equation (2.1)
does not contain the function $r_{m}$.

By repeating this operation, we consistently exclude all functions $r_{j}$
from the right-hand side of the resulting equations. After $m$ steps we get

$\displaystyle\Delta_{C_{11}k_{1}}\dots\Delta_{C_{m1}k_{m}}\Delta_{h_{12}}...\Delta_{h_{1n}}\psi_{1}(C_{11}f_{1}+\cdots+C_{m1}f_{m})+$
$\displaystyle+\Delta_{\overline{{k}}_{1}}...\Delta_{\overline{{k}}_{m}}\Delta_{\overline{{g}}_{n}}...\Delta_{\overline{{g}}_{2}}q(f_{1},\dots,f_{m})=0,\quad
f_{j}\in V_{r}.$ (17)

Since $q(f_{1},\dots,f_{m})$ is a polynomial, we have

$\Delta^{l+1}_{(s_{1},...,s_{m})}q(f_{1},\dots,f_{m})=0,\quad f_{j}\in X^{*},$
(18)

for some $l$ and arbitrary elements $s_{1},...,s_{m}\in X^{*}$.

Taking into account (18), it follows from (2.1) that

$\displaystyle\Delta^{l+1}_{C_{11}s_{1}+\cdots+C_{m1}s_{m}}\Delta_{C_{11}k_{1}}\dots\Delta_{C_{m1}k_{m}}\Delta_{h_{12}}...\Delta_{h_{1n}}\psi_{1}(C_{11}f_{1}+\cdots+C_{m1}f_{m})=0,\quad
f_{j}\in V_{r}.$ (19)

Since $X$ is a reflexive space and $KerA_{pj}=\\{0\\}$, the subgroups
${C_{j1}(X^{*})}$ are dence in $X^{*}$ in the norm topology. Therefore we can
put $C_{11}k_{1}=\cdots=C_{m1}k_{m}=g$, where $g\in V_{r}$. Since elements
$h_{ij}$ are defined by (14) and belong to $F_{ij}^{*}(X^{*})$, it follows
from (5) that we can put $h_{12}=\cdots=h_{1n}=g$. Since
$C_{11}s_{1}+\cdots+C_{m1}s_{m}$ belong to $F_{ij}^{*}(X^{*})$, it follows
from (5) that we can put $C_{11}s_{1}+\cdots+C_{m1}s_{m}=g$. Thus, it follows
from (19) that the function $\psi_{1}(f)$ satisfies the equation

$\Delta^{l+m+n}_{g}\psi_{1}(f)=0,\quad f,g\in V_{r},$ (20)

i.e. $\psi_{1}(f)$ is a polynomial in a neighborhood of zero of $X^{*}$. In
the same way we can prove that all functions $\psi_{j}(f)$,
$j=\overline{2,n}$, are polynomials in a neighborhood of zero of $X^{*}$. It
follows from Proposition 2.3 that all $\mu_{i}$ are Gaussian distributions.
$\square$

It is known that a characteristic functional is continuous in the norm
topology and is sequentially continuous in the topology of pointwise
convergence ([29, P.4, §2]). We use only the continuity in the norm topology
to prove Theorem 2.2. An essential fact, which was used in the proof, is the
following assertion: the equality $KerA=\\{0\\}$ for some bounded operator $A$
implies that $\overline{ImA^{*}}=X^{*}$. This assertion is not valid for
arbitrary Banach spaces, but it is true for reflexive ones.

Suppose that $X$ is not a reflexive space. For which maximally wide class of
Banach spaces does Theorem 2.2 remain true?

Let us also assume at first that the random variables $\xi_{i}$,
$i=1,2,\dots,n$, have the distributions $\mu_{i}$ such that their
characteristic functions do not vanish. Then equation (19) is fulfilled on
$X^{*}$. Putting $f_{1}=f,f_{2}=\cdots=f_{m}=0$,
$s_{1}=s,s_{2}=\cdots=s_{m}=0$ in (19), we get

$\displaystyle\Delta^{l+1}_{C_{11}s}\Delta_{C_{11}g_{1}}\dots\Delta_{C_{m1}g_{m}}\Delta_{h_{12}}...\Delta_{h_{1n}}\psi_{1}(C_{11}f_{1})=0,\quad
f\in X^{*}.$ (21)

Elements $C_{11}f$, $C_{11}s$, $C_{11}g_{1},\ldots,C_{m1}g_{m}$,
$h_{12},\ldots,h_{1n}$ belong to images of the operators conjugate to
operators with zero kernel. Since $X$ is not a reflexive space, we can not
obtain from (21) the equation of type (20) on $X^{*}$.

Now we will use the fact that a characteristic functional is sequentially
continuous in the topology of pointwise convergence. It means that if a
sequence of elements $\\{f_{n}\\}$ of $X^{*}$ converges pointwise to $f\in
X^{*}$ ($\\{f_{n}(x)\rightarrow f(x)\\}$) for any $x\in X$ then
$\hat{\mu}(f_{n})\rightarrow\hat{\mu}(f)$. We need the following definition:

###### Definition 2.2

A Banach space $X$ is said to be quasireflexive if $X$ has finite codimension
in its second dual $X^{**}$.

We need the following proposition.

###### Proposition 2.5

Let $X$ be a quasireflexive separable Banach space. Let $A$ be a linear
bounded operator of $X$ such that $KerA=\\{0\\}$. Let $f$ be an arbitrary
element of $X^{*}\backslash Im(A^{*})(X^{*})$. Then there exists a sequence of
elements $\\{f_{n}\\}$ of the image $Im(A^{*})$ such that a sequence of
elements $\\{f_{n}\\}$ converges pointwise to $f$.

As V. Kadec informed the author, Proposition 2.5 holds (comments of V. Kadec
see in Appendix). Note that readers can also find useful information about
quasireflexive spaces and additional references in Appendix.

Using Proposition 2.5 we obtain from (21) the following equation:

$\Delta^{l+m+n}_{g}\psi_{1}(f)=0,\quad f,g\in X^{*}.$ (22)

Now we can formulate the following theorem.

###### Theorem 2.3

Let $X$ be a real quasireflexive separable Banach space. Let $A_{pj}$,
$p=\overline{1,m}$, $i=\overline{1,n}$, be linear continuous operators of $X$
such that $KerA_{pj}=\\{0\\}$. Let $G_{i}$ be the sets defined by
$(\ref{Kag2})$. Suppose that the condition $(\ref{Kag1})$ holds. Let
$\xi_{i}$, $i=1,2,\dots,n$, be Q-independent random variables with values in
$X$ and with non vanishing characteristic functionals. Consider the linear
forms $L_{j}=A_{j1}\xi_{1}+\cdots+A_{jn}\xi_{n}$, $j=1,2,\dots,m$. If the
random vector $\mathbf{L}$=$(L_{1},\dots,L_{m})$ belongs to the class
$\mathcal{D}_{m,m-1}$ then all $\xi_{i}$ are Gaussian random variables.

Note that we cannot omit the condition that the characteristic functionals of
the distributions $\mu_{i}$ do not vanish because otherwise we cannot claim
that the equation (22) is fulfilled in some ball.

Theorem 2.3 implies the following theorem.

###### Theorem 2.4

Let $X$ be a real quasireflexive separable Banach space. Let $A_{pj}$,
$p=\overline{1,m}$, $i=\overline{1,n}$, be linear continuous operators of $X$
such that $KerA_{pj}=\\{0\\}$. Let $G_{i}$ be the sets defined by
$(\ref{Kag2})$. Suppose that the condition $(\ref{Kag1})$ holds. Let
$\xi_{i}$, $i=1,2,\dots,n$, be independent random variables with values in $X$
and with non vanishing characteristic functionals. Consider the linear forms
$L_{j}=A_{j1}\xi_{1}+\cdots+A_{jn}\xi_{n}$, $j=1,2,\dots,m$. If the random
vector $\mathbf{L}$=$(L_{1},\dots,L_{m})$ belongs to the class
$\mathcal{D}_{m,m-1}$ then all $\xi_{i}$ are Gaussian random variables.

##  3 Appendix

The content of this section belongs to V. Kadec and is published with his
consent.

###  3.1 Density and sequential density in weak-star topology

Let $X$ and $Y$ be normed spaces over the field $\mathbb{K}$
($\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$). We denote by
$\mathcal{L}(X,Y)$ the space of all bounded linear operators from $X$ into
$Y$. By $B_{X}$ and $S_{X}$ we denote the closed unit ball and the unit sphere
of $X$ respectively.

###### Definition 3.1

Let $X$ be a Banach space. A set $F\subset{X}^{*}$ is called _total over_ $X$,
if for any $y\in X\setminus\\{0\\}$ there exists an $f\in F$ such that
$f(y)\neq 0$.

Evidently, $F$ is total over $X$ if and only if its linear span ${\rm Lin\ }F$
is total, and if and only if the norm-closure $\overline{\rm Lin\ }F$ of the
linear span is total. The bipolar theorem applied to the dual pair $(X^{*},X)$
implies that $F\subset{X}^{*}$ is total over $X$ if and only if ${\rm Lin\ }F$
is $w^{*}$-dense in $X^{*}$.

###### Definition 3.2

Let $X$ be a Banach space and $\theta\in(0,1]$. A set $F\subset X^{*}$ is said
to be $\theta$-_norming over_ $X$ if

$\sup_{f\in F\backslash\\{0\\}}\dfrac{|f(x)|}{\|f\|}\geq\theta\|x\|$

for all $x\in X$. The set $F\subset X^{*}$ is said to be _norming_ if there
exists a $\theta\in(0,1]$ such that $F$ is $\theta$-norming over $X$.

The following well-known result can be deduced from the bipolar theorem,
applied to the dual pair $(X^{*},X)$.

######  Proposition 3.1 ([12, Section 17.2.4, Exercise 2])

Let $X$ be a Banach space, and $\theta\in(0,1]$. A set $F\subset S_{X^{*}}$ is
$\theta$-norming for $X$ if and only if the $w^{*}$-closure of the absolute
convex hull of $F$ contains $\theta{B}_{X^{*}}$.

######  Proposition 3.2 ([12, Section 17.2.4, Corollary 4])

Let $X$ be a separable Banach space. Then the $w^{*}$-topology
$\sigma(X^{*},X)$ is metrizable on the bounded subsets of the space $X^{*}$.

The very first example of quasireflexive nonreflexive space was constructed by
R.C.James in [9]. Since then, quasireflexive spaces form an important part of
Banach space theory.

The following result by W.J. Davis and J. Lindenstrauss ([3]) is highly non-
trivial. Two years later it was rediscovered by A. Plichko ([26]). The proof
in both papers develops ideas by Yuri Petunin ([24]). One can find a detailed
exposition with all necessary preliminaries in the book [25].

###### Proposition 3.3

A Banach space $X$ has a total nonnorming subspace in its dual if and only if
$X$ is not quasireflexive.

###### Definition 3.3

Let $X$ be a Banach space. A set $F\subset{X}^{*}$ is called _w*-sequentially
dense_ in $X^{*}$, if for every $g\in X^{*}$ there exists a sequence $f_{n}\in
F$ such that $\lim_{n\to\infty}f_{n}(x)=g(x)$ for all $x\in X$.

###### Lemma 3.1

Let $X$ be a separable Banach space, $F\subset X^{*}$ be a linear subspace.
Then the following assertions are equivalent: $(i)$ $F$ is w*-sequentially
dense in $X^{*}$, and $(ii)$ $F$ is norming.

Proof. $(i)\Rightarrow(ii)$. Assume $F$ is w*-sequentially dense in $X^{*}$.
Denote $U$ the w*-closure of the unit ball $B_{F}$ of $F$. Thanks to Banach-
Steinhaus theorem [12, Section 10.4.2, Theorem 1], the sequence $f_{n}$ from
Definition 3.3 is bounded, hence it belongs to a set of the form $nB_{F}$.
Consequently, every $g\in X^{*}$ belongs to some set of the form $nU$, that is

$X^{*}=\bigcup_{n\in\mathbb{N}}nU.$ (23)

The application of Baire theorem gives that $U$ contains some ball $\theta
B_{X^{*}}$. Since $U$ is the $w^{*}$-closure of the absolute convex hull of
$S_{F}$, Proposition 3.1 says that $S_{F}$ is norming, so $F$ is norming as
well.

$(ii)\Rightarrow(i)$. Evidently, a linear subspace $F$ is norming if and only
if its unit sphere is norming, so in notations of the previous implication
Proposition 3.1 says that $U$ contains some ball $\theta B_{X^{*}}$. So, (23)
holds true. Consequently, every $g\in X^{*}$ belongs to the w*-closure of some
ball $nB_{F}$. But, thanks to metrizability (Proposition 3.2), this means that
$g$ is a w*-limit of some sequence from $nB_{F}\subset F$. $\square$

After this, Proposition 3.3 reformulates as follows:

###### Theorem 3.1

For a separable Banach space $X$ the following assertions are equivalent:
$(i)$ every $w^{*}$-dense subspace of $X^{*}$ is $w^{*}$-sequentially dense,
and $(ii)$ $X$ is quasireflexive.

###  3.2 Applications to the range of adjoint operator

Recall that for arbitrary Banach spaces $X,Y$ and for arbitrary
$T\in\mathcal{L}(X,Y)$ the $w^{*}$-density of $T^{*}(Y^{*})$ in $X^{*}$ is
equivalent to the injectivity of $T$ ([12, Section 17.1.3, Theorem 5]). This
and Theorem 3.1 give the following corollary:

###### Theorem 3.2

Let $X$ be a separable quasireflexive Banach space, $Y$ be a Banach space and
$T\in\mathcal{L}(X,Y)$ be injective. Then $T^{*}(Y^{*})$ is
$w^{*}$-sequentially dense in $X^{*}$.

It is not clear for us whether for every norm-closed total subspace $F\subset
X^{*}$ there exist a Banach space $Y$ and an injective operator
$T\in\mathcal{L}(X,Y)$ such that $T^{*}(Y^{*})\subset F$. In the case of
positive answer one would have the following inverse to Theorem 3.2 result:

Hypothesis. Let $X$ be a separable Banach space that is not quasireflexive.
Then there exist a Banach space $Y$ and an injective operator
$T\in\mathcal{L}(X,Y)$ such that $T^{*}(Y^{*})$ is not $w^{*}$-sequentially
dense in $X^{*}$.

We can demonstrate a partial result. At first, a technical tool.

###### Lemma 3.2

Let $X$, $Y$ be infinite-dimensional Banach spaces, and let $F\subset X^{*}$
be a $w^{*}$-separable closed linear subspace. Then there exists a
$T\in\mathcal{L}(X,Y)$ such that $T^{*}(Y^{*})\subset F$ and $T^{*}(Y^{*})$ is
$w^{*}$-dense in $F$.

Proof. Fix a $w^{*}$-dense in $S_{F}$ countable set
$\\{f_{n}\\}_{n\in{\mathbb{N}}}\subset S_{F}$ and a normalized basic sequence
$\\{e_{n}\\}_{n\in{\mathbb{N}}}\subset S_{Y}$ (the existence of the latter was
demonstrated by S. Masur, see a detailed exposition in [11, Theorem 6.3.3]).
Denote $\\{e_{n}^{*}\\}_{n\in{\mathbb{N}}}\subset Y^{*}$ there corresponding
coordinate functionals extended to the whole $Y$ by the Hahn-Banach theorem.
Define $T\in\mathcal{L}(X,Y)$ by formula

$Tx=\sum_{n=1}^{\infty}2^{-n}f_{n}(x)e_{n}.$

Then, the adjoint operators $T^{*}\colon Y^{*}\to X^{*}$ acts on every $g\in
Y^{*}$ as

$T^{*}g=\sum_{n=1}^{\infty}2^{-n}g(e_{n})f_{n}.$

Evidently, $T^{*}(Y^{*})\subset\overline{{\rm Lin\
}}\\{f_{n}\\}_{n\in\mathbb{N}}\subset F$. On the other hand, for every
$k\in\mathbb{N}$ we have $T^{*}e_{k}^{*}=2^{-k}f_{k}$, consequently
$T^{*}(Y^{*})$ contains the $w^{*}$-dense in $F$ set ${\rm Lin\
}\\{f_{n}\\}_{n\in\mathbb{N}}$. $\square$

###### Corollary 3.1

$X$, $Y$ be infinite-dimensional Banach spaces, $X$ and $X^{*}$ are both
separable, and $X$ is not quasireflexive. Then there exist an injective
operator $T\in\mathcal{L}(X,Y)$ such that $T^{*}(Y^{*})$ is not
$w^{*}$-sequentially dense in $X^{*}$.

Proof. Indeed, Theorem 3.1 gives the existence of a $w^{*}$-dense norm-closed
linear subspace $F\subset X^{*}$ which is not $w^{*}$-sequentially dense. By
separability of $X^{*}$ the subspace $F$ is separable, and consequently $F$ is
$w^{*}$-separable. Applying Lemma 3.2 we obtain a $T\in\mathcal{L}(X,Y)$ such
that $T^{*}(Y^{*})\subset F$ and $T^{*}(Y^{*})$ is $w^{*}$-dense in $F$. Then
$T^{*}(Y^{*})$ is $w^{*}$-dense in $X^{*}$, so $T$ is injective. But the
inclusion $T^{*}(Y^{*})\subset F$ implies that $T^{*}(Y^{*})$ is not
$w^{*}$-sequentially dense in $X^{*}$. $\square$

Remark, that for separable $X$ the dual space $X^{*}$ is $w^{*}$-separable
([12, Section 17.2.4, Corollary 2]). Unfortunately, this does not guaranty the
$w^{*}$-separability of subspaces of $X^{*}$. This is the reason, why in the
Corollary 3.1 we make the assumption of separability of $X^{*}$ in norm
topology. We don’t know if in the case of separable non-quasiraflexive $X$ the
corresponding $w^{*}$-dense but not $w^{*}$-sequentially dense $F\subset
X^{*}$ can be always selected to be $w^{*}$-separable. We cannot extract this
additional property neither from the demonstration by Davis-Lindenstrauss
([3]), nor from Plichko’s demonstration, but such a possibility does not
contradict our intuition.

## Acknowledgements

The author would like to thank the Volkswagen Foundation (VolkswagenStiftung),
the Bielefeld University and Prof. Dr. Friedrich Götze for the support and
warm reception.

## Funding

This research was supported by VolkswagenStiftung - Az. 9C108.

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1]Universität Göttingen, Göttingen, Germany

2]GWDG, Göttingen, Germany

3]E4 Computer Engineering S.p.A., Scandiano, Italy

4]KTH Royal Institute of Technology, Stockholm, Sweden

5]HLRS, Stuttgart, Germany

6]Huawei Technologies Düsseldorf GmbH, Düsseldorf, Germany

7]Forschung Burgenland GmbH, Eisenstadt, Austria

8]SYNYO GmbH, Wien, Austria

9]Marmara University, Istanbul, Turkey

10]Consorzio TOP-IX, Torino, Italy

11]Alma Mater Studiorum - Università di Bologna, Bologna, Italy

12]Numerical Algorithms Group Limited, Oxford, United Kingdom

13]BigTRI Bilisim A.S., Istanbul, Turkey

# DECICE: Device-Edge-Cloud Intelligent Collaboration Framework

Julian Kunkel<EMAIL_ADDRESS>Christian Boehme
<EMAIL_ADDRESS>Jonathan Decker<EMAIL_ADDRESS>Fabrizio Magugliani<EMAIL_ADDRESS>Dirk Pleiter
<EMAIL_ADDRESS>Bastian Koller<EMAIL_ADDRESS>Karthee Sivalingam
<EMAIL_ADDRESS>Sabri Pllana
<EMAIL_ADDRESS>Alexander Nikolov
<EMAIL_ADDRESS>Mujdat Soyturk<EMAIL_ADDRESS>Christian Racca<EMAIL_ADDRESS>Andrea Bartolini
<EMAIL_ADDRESS>Adrian Tate<EMAIL_ADDRESS>Berkay Yaman
<EMAIL_ADDRESS>[  [  [  [  [  [  [  [  [  [  [  [  [

###### Abstract

DECICE is a Horizon Europe project that is developing an AI-enabled open and
portable management framework for automatic and adaptive optimization and
deployment of applications in computing continuum encompassing from IoT
sensors on the Edge to large-scale Cloud / HPC computing infrastructures. In
this paper, we describe the DECICE framework and architecture. Furthermore, we
highlight use-cases for framework evaluation: intelligent traffic
intersection, magnetic resonance imaging, and emergency response.

###### keywords:

AI-enabled Computing Continuum, Cloud-Edge Orchestration, Cognitive Cloud,
Digital Twin, KubeEdge

##  1 Introduction

In the past decade, cloud computing has revolutionized computing by providing
services over the Internet that can scale elastically for user requirements.
The economics of scale have resulted in powerful computing and storage
resources being provided at reduced costs. Furthermore, fault tolerance
mechanism including replication allow to significantly enhance availability of
services deployed in the cloud. Moreover, various application areas including
smart home, smart city, or industrial automation come with needs that require
computing services to be deployed closer to the end-user. Edge computing
complements cloud computing by placing compute resources at the edge of the
network [8]. It also allows for connecting Internet of things devices that
utilize alternative wireless protocols such as LoRaWAN.

While cloud and edge computing do share some characteristics, there are also
characteristics and features that make edge computing unique. Placing
computational resources in the proximity of users or data sources reduces
latency and data transport and can even prevent shipping of sensitive data.
The trend of placing computing resources outside of the data center has
resulted in the emergence of the term compute continuum (or cloud continuum)
[14]. Having a continuum of resources available from the network edge (and
beyond) to cloud and HPC data centers has attracted increased interest, also
in the European context [5]. The emergence of a compute continuum is of
particular interest for workflows that extend beyond the data center and have
to leverage services that provide access to distributed edge, cloud, and HPC
resources across federated infrastructures.

Due to inter-dependencies between distributed components and the potentially
high level of heterogeneity of the compute, storage, and network resources,
the scheduling of these resources becomes a huge challenge [3]. The problem of
finding the optimal scheduling decisions may be NP-complete. Moreover, dynamic
factors such as faults, changing network capabilities and system topologies
require constant monitoring and automatic adjustments to the placement of
processes and data such that the manual creation of good static assignments of
processes to hardware becomes infeasible. To overcome these challenges, we
believe that new approaches are necessary that include the utilization of AI
methods to facilitate timely resource allocation decisions for complex systems
with thousands of nodes.

In this paper, we describe DECICE project [9] that is developing an AI-based,
open, and portable cloud management framework for automatic and adaptive
optimization and deployment of applications in a compute continuum. The
framework relies on holistic monitoring to construct a digital twin of the
system that reflects all components and characteristics of the system. This
digital twin will supply schedulers with information such that scheduling
decisions for dynamic load balancing and data placements can be reached. This
will lead to improved throughput and reduced resource consumption. The
framework’s prototype will be integrated into Kubernetes, which is the most
popular solution for orchestrating containers in the cloud. Kubernetes itself
has no built-in support for edge computing. For this, DECICE relies on an
extension of Kubernetes that is called KubeEdge [15] and allows for
transparently integrating edge nodes in the main cluster. Furthermore, to
support HPC-like batch job execution in Kubernetes [10] the prototype will
explore the utilization of technology such as Volcano [4]. The DECICE project
is working towards the following:

  * •

Development of a cloud management framework based on open source and
standards, which can seamlessly connect and deploy applications to devices
across HPC, cloud and edge continuums.

  * •

Providing system administration and DevOps tools to access, control and manage
the continuous service environment that includes the deployment of network,
computing, data infrastructures and services.

  * •

Creating a dynamic digital twin for heterogeneous infrastructure by supplying
relevant data via monitoring systems such that it can enable simulations and
AI-based forecasting.

  * •

Enhance application performance, reliability, throughput, energy efficiency
and cost by automating decisions for application scheduling with the help of
AI models.

The rest of this paper is structured as follows. Section 2 provides context
information. An overview of DECICE framework is provided in Section 3. Section
4 describes the architecture. The strategy for DECICE evaluation via use-cases
is described in Section 5. Section 6 concludes the paper.

##  2 Context

For the rise of cloud computing, the ability to provide virtualized resources
has been critical. Beyond virtual machines, containers have become a widely
used technologies with Kubernetes being the de facto orchestration platform
for managing execution of containers at scale. Kubernetes is an open source
container orchestration platform that is able to dynamically handle
containerized workloads across thousands of nodes by assigning resources and
virtualizing networks. One of the major reasons for the widespread usage of
Kubernetes is its adaptability, which enables anyone to extend, replace and
remove various parts of these systems.

Before the times of cloud computing and container orchestration solutions,
scaling resources available to a service required proprietary solutions for
integrating load balancing capabilities and adding additional servers. Such
approaches had various short-comings as it was hard to roll out application
updates or to dynamically react to sudden bursts in service requests.

The virtualization offered by containers simplifies orchestration systems like
Kubernetes the choice of the physical machine for running the service as long
as it provides enough resources. This works well for data centers with
homogeneous hardware and fast and stable networks. This situation changes in
an environment with different types of hardware as it is typically the case
when including edge devices.

Another challenge is that Kubernetes has been designed for cloud computing
data centers and not for distributed environments involving edge devices. This
challenge has been addressed by KubeEdge [15], which extends Kubernetes such
that it is able to use edge devices. The key features of KubeEdge are its
lightweight Kubernetes installation for less powerful edge devices and a pull-
based network infrastructure. Via this network infrastructure, edge devices
can appear as regular nodes on Kubernetes. A key feature of KubeEdge is an
implementation of OSI network protocol layers that enables connecting edge
devices and cloud servers within one virtual network.

With edge devices being added, the underlying set of hardware resources
becomes heterogeneous and, therefore, scheduling of resources significantly
more complex. Scheduling refers here to algorithms that allow – based on the
current state of available resource utilization – to decide on where to place
a new container. To the best of our knowledge there are currently no solutions
available to address the resulting scheduling challenges and to, e.g., decide
on whether to execute containers on cloud or edge nodes. For this, the
specific hardware characteristics of the available edge nodes as well as the
current network connectivity need to be taken into account.

Container scheduling is an active research field with various techniques being
explored. Ahmad et al. [1] classifies scheduling techniques into one of four
categories: (1) Mathematical modeling techniques, (2) heuristic techniques,
(3) meta-heuristics techniques and (4) machine-learning techniques.

Mathematical techniques provide a mathematical description of an optimization
problem that must be solved taking a set of constraints under account. Integer
linear programming is a widely used technique. A typical challenge with such
techniques is that they become too complex for large-scale systems and take
too long to solve. Heuristic techniques are generally scalable and fast,
however, there is no guarantee for good scheduling decisions.

Meta-heuristic and machine-learning techniques have generally become generally
popular for optimization problems related to parallel computing systems (see
[13] for a recent survey). Meta-heuristic techniques are based on search
strategies for finding good but not necessary optimal solutions for a given
optimization problem. Typically nature-inspired algorithms are used [6].

Machine-learning techniques train a model using data collected in a given
context. These models later allow to make scheduling decisions based on what
has been learned earlier, for instance by using the model for making
predictions about the number of needed containers [11].

For DECICE it is planned to use machine-learning techniques for anomaly
detection. Such techniques have already been successfully used for HPC systems
(see, e.g., [2]). Furthermore, such techniques have been used for reducing
energy-to-solution (see, e.g., [12]). Within DECICE extending such approaches
to the compute continuum will be explored.

Recently, a very significant increase of popularity of the digital twin
concept can be observed. It is based on the realization of one or more virtual
entities that describe a physical entity. A key feature of digital twins is
the twinning of the physical and virtual entities, i.e., the implementation of
physical-to-virtual twinning and virtual-to-physical twinning processes [7].
To the best of our knowledge, the concept of digital twinning has not been
applied to the case of cloud-edge systems.

##  3 Framework Overview

DECICE aims to develop an AI-based, open, and portable cloud management
framework for automatic and adaptive optimization and deployment of
applications in a federated infrastructure, including computing from the very
large (e.g., HPC systems) to the very small (e.g., IoT sensors connected on
the edge).

Working at such vastly different scales requires an intelligent management
plane with advanced capabilities that allow it to proactively adjust workloads
within the system based on their needs, such as latency, compute power, and
power consumption. Therefore, we envision an AI-model, which can use a digital
twin of the resources available, to make real-time scheduling decisions based
on telemetry data from the resources.

The DECICE framework will be able to dynamically balance different workloads,
optimize the throughput and latency of the system resources (compute, storage,
and network) regarding performance and energy efficiency and quickly adapt to
changing conditions. The framework also provides the necessary tools and
interfaces for the administrators and deployment experts to interface with all
the infrastructure components and control them to achieve the desired result.

The integration of the DECICE framework with orchestration systems will be
done through open APIs to make it portable, modular, and extensible. The
DECICE framework will be co-designed and evaluated through concrete use cases.

Figure 1: An example of computing continuum

The system depicted in Figure 1 can be seen as an example of a computing
continuum executing a non-trivial application. Consider an application that
requires programs running on IoT devices to collect data, which is transferred
via two edge nodes to the cloud nodes. Due to heavy computation and privacy
requirements, a data center might contribute compute resources and data
volumes in a private cloud. The results are then transferred back to the cloud
where they are served to the end users. Finally, the cloud offers frontends
that allow the end users to access the application and the results of the
computation, for example, via a webpage. The distribution of data across the
continuum is another open challenge that is scheduled to tackle during the
performance test phase.

##  4 DECICE Architecture

Figure 2: High-level DECICE architecture

The DECICE framework is built on top of existing open-source technologies that
are already well-established in the community.

Kubernetes, KubeEdge, Prometheus, SEDNA and Volcano are part of the _Cloud
Native Computing Foundation (CNCF)_ , which is a hub for many open source
projects from the cloud native ecosystem as well as over 250 members including
many public cloud and enterprise cloud vendors.

Application users deploying to cloud-edge face multiple challenges and are
usually not experts in cloud management frameworks or heterogeneous hardware
resources. DECICE aims to help such users by minimizing the effort required to
efficiently utilize diverse hardware devices for their application needs. For
example, users of the DECICE framework will be able to build hybrid
edge/cloud/HPC workloads using standard containers, knowing that the DECICE
scheduler will allow them to specify data locality, network latency or other
constraints and that the AI-based scheduler will place their jobs according to
the requirements – and move them if needed. Primarily these capabilities will
be provided via the careful selection of base technologies and use of
standardized interfaces for interacting with DECICE. More importantly, via
properly mapping and modeling the entire heterogeneous continuum into
respective DECICE model.

Users interact with DECICE via the standard Kubernetes interface, which is
extended with DECICE-specific extensions. This allows them to use an existing,
well-supported and familiar interface. Furthermore, it allows the immediate
movement of existing workloads into DECICE and minimizes the “learning curve”
required for users to begin using DECICE.

The compute plane, the DECICE Manager and the DECICE Model as illustrated in
Figure 2 will be outlined in the following sections.

###  4.1 Compute Plane

The compute plane exists to run the user workloads. It is built on existing
open source technology, which is then integrated with the DECICE components in
the control plane. The primary component in the compute plane is Kubernetes.
It provides the basic building blocks for running containerized jobs over
multiple physical systems. Kubernetes is further extended with KubeEdge to
allow the use and management of autonomous edge devices. Additionally,
Prometheus is used to gather metrics from devices via Kubernetes/KubeEdge
interfaces. Finally, SEDNA provides AI model training capabilities, while
Volcano adds the ability to run HPC jobs.

###  4.2 DECICE Manager

The DECICE Manager integrates the Digital twin and AI models with the
orchestration system (e.g., Kubernetes) through its standard API. This
integration can be adapted to make the DECICE framework portable across
orchestration frameworks allowing integration with any commercial or open
source cloud solutions. Furthermore, the DECICE manager also includes a data
repository for storing metadata of infrastructure and applications as well as
monitoring data. This data is used to update the Digital twin and train the
AI-models. Moreover, the manager also provides a synthetic training and test
environment to simulate different infrastructure and application deployment
scenarios for the training of AI-models.

###  4.3 DECICE Model

The Dynamic Digital Twin is a virtual representation of an actual or potential
system at micro and macro level. It models the complete life-cycle of the
system using simulation, real-time monitoring and enables forecasting, which
is then used to make decisions. Creating a digital twin of the heterogeneous
cloud-edge infrastructure and the deployed applications is one of the main
goals of the project. The DECICE twin models the compute, memory, storage and
network resources as well as the application tasks. Simulation and real time
monitoring are used to continuously update the model. AI models are then used
to forecast the system behavior in case of new deployment or a failure. AI
modeling is also used to continuously predict the application performance
based on the current state of the system and makes suggestions of optimization
or adaptation to increase performance, energy efficiency, reliability and
throughput. These suggestions are then acted upon by the DECICE manager.

###  4.4 Virtual Training Environment

In order to train, improve and test AI models for scheduling, we employ a
Virtual Training Environment (VTE), which simulates a Digital Twin (DT) based
on a pre-configured scenario. The training process involves running the AI
model under training against training scenarios, updating the scenario and
evaluating the performance of the AI scheduler based on a set of metrics. This
process is depicted in Figure 3 with the VTE loading a given training
scenario, which in turn is used to configure the DT. Within the main loop of
the training, the VTE advances the scenario by progressing through the
scenario, which in turn submits additional updates in the form of metrics and
jobs to the DT. Afterward, the scheduler accesses the DT to perform scheduling
decisions.

Figure 3: Sequence diagram: Virtualized Training; VTE: Virtual Training
Environment; DT: Digital Twin

As the VTE is responsible for advancing the scenario through the simulation
and the AI scheduler itself has no concept of past states, it is possible to
perform faster than real-time training. This enables us to quickly train many
instances of the AI scheduler in parallel and keep the best performing
instances for further training.

The training loop used by the VTE in Figure 3 is similar to the scheduling
loop that will be deployed on real systems. Instead of a VTE, the real system
is controlled by the Cloud Manager (CM) and updates of the DT are pulled in
from the Metrics System (MS) and the CM. Furthermore, the scheduling decisions
requested by the CM are then forwarded to the underlying Kubernetes (K8s)
platform and realized.

Notably, the AI scheduler always views the entire DT, which also includes the
entire job queue and state of all running jobs. The AI scheduler has no
obligation to immediately schedule a newly arrived job but its goal is to
optimally utilize the underlying cluster for all active and queued jobs while
honoring the user specifications. This also includes rescheduling running jobs
if a better placement becomes available by regularly reevaluating the current
status of the system.

Overall, the AI scheduler is able to view and optimize the entire cluster
through the DT. A common heuristics-based scheduler operates by individually
evaluating each queued job but could be expanded to view all jobs at once.
However, from our understanding, this would highly increase the complexity of
the used heuristics without gaining value from a deeper understanding of the
interplay of the components of the system as a machine learning-based
scheduler would do. We will also explore various deep learning strategies.

###  4.5 CI/CD-Test Strategy

The objective of our CI/CD systems is to verify that all components are
functional individually and in composition. Components developed in the
context of the DECICE project, such as the Cloud Manager, the Digital Twin and
the AI scheduler have their logic verified through individual component and
mock testing.

As the DECICE systems integrate with an underlying orchestration platform that
also evolves together with its ecosystem independently of DECICE, it becomes
necessary to regularly perform integration tests. For now, the prototype of
the DECICE systems only targets Kubernetes as an orchestration platform.

By definition, the DECICE systems should be usable by administrators of
heterogeneous compute systems that include cloud, edge and HPC resources.
Administrators may have additional requirements such as using a specific
version of Kubernetes on their systems, which might be distinct from the
version used to develop DECICE. Therefore, the integration tests of DECICE
cover a range of versions that are commonly used to ensure compatibility.

The testing of individual components is done via unit and mock testing.
Integration testing, however, requires spinning up a temporary Kubernetes
cluster and deploying the components into it before running tests on them.
This allows using various configurations, versions and additional elements
(e.g., edge and HPC nodes) over multiple test runs.

In practice, we are looking at GitLab runners for handling our CI/CD
pipelines. They can be integrated with Terraform and OpenStack to
automatically roll out virtual machines, install Kubernetes on them and deploy
our DECICE systems into it. Such a system enables us to perform complete end-
to-end tests of our software against multiple versions of Kubernetes, KubeEdge
and Volcano.

##  5 Evaluation Strategy

In this section we describe our approach for evaluation of DECICE framework.
The DECICE project adopts a co-design process in order to ensure that
functionality and performance of the DECICE solutions do meet demands for
real-life challenges. Figure 4 depicts use-cases that are used for
demonstrating the capabilities of technology developed in this project,

  * •

Intelligent intersection

  * •

Magnetic resonance imaging (MRI)

  * •

In-the-field intelligence supporting emergency response

Figure 4: Use-cases for DECICE evaluation: intelligent intersection, MRI, and
emergency response

###  5.1 Intelligent intersection

This use case focuses on real-time processing aspects in Connected &
Autonomous Driving and Cooperative Intelligent Transportation Systems (C-ITS).
Connected and Autonomous Vehicles (CAVs) use their onboard sensors for object
detection and situation awareness. Intersections introduce additional
difficulties, challenges, and safety issues due to varying road users (cars,
trucks, buses, pedestrians, bicycles, motorcycles, wheelchaired users, etc.)
in varying numbers moving in different directions. Road user including
Vulnerable Road User (VRU) detection is crucial and critical to ensuring
safety and traffic efficiency. Road users which might be behind other objects
or corners may not be detected by onboard sensors in vehicles. This poses
risks to both road and driving safety and traffic efficiency. In addition to
sharing the status information of the vehicles, it is aimed to share the
information obtained from the sensors (e.g. camera) in the vehicles and at the
intersections with the road users (vehicles, VRUs, etc.) in real-time. This
approach is included by C2C-CC in Day 2 and Day3+ applications.

Cameras positioned at an intersection continuously monitor the Region of
Interest (RoI) and the captured images/videos are analyzed for object
(vehicle, VRU, etc.) detection, which thereafter has to be shared with the
road users (vehicles, VRUs). Processing of images and communication has to be
in real-time (at low latencies) to provide safety and avoid possible
accidents.

The question arises on the level of hierarchy where the images will be
processed. It could be done in the cloud, or edge, depending on the
requirements and cost of doing that. Accuracy, latency, efficiency are the key
technical performance metrics to measure. Data communication between the
camera, and the central cloud introduces additional (communication) delay
which is not acceptable. Therefore, there are general aims of carrying some
fast processing, low latency, fast response applications at the edge, while
learning models can work at the edge and cloud. On the other hand, resource
utilization, cost, and traffic safety are the business-related performance
metrics. DECICE provides an efficient, fast processing, low-latency, energy-
efficient edge and cloud computing continuum that also considers the load at
edge nodes due to the spatiotemporal variation of the users (vehicles, VRUs)
and environmental conditions (light, illumination, weather conditions, etc).

###  5.2 Magnetic resonance imaging (MRI)

MRI scans are a common tool for medical examination due to the noninvasive way
to get a view of the inner body parts. Typically, MRI scans are conducted on
edge nodes; they only cover individual body parts of a patient and create a
high-resolution 3D-image of the body part using magnetic fields. Advances in
technology lead to higher resolutions and, consequently, larger data sizes.
Like other medical data, MRI scans often contain sensitive personal
information about the patient. Therefore, special care with respect to
storage, transfer and access is necessary. We propose the analysis of MRI
scans as a use case for the DECICE framework to validate restrictions on
storage and access rights for data. Furthermore, a fast and accurate analysis
of the MRI scans is important, adding fast compute times to the list of
requirements.

The analysis of the MRI scans poses difficulties, both from a compute and
storage perspective. Each MRI scan consists of multiple slices of data, thus
creating a complex object for analysis. Complex models for analysis can be
challenging for the compute capabilities of edge devices. Thus, moving
analysis jobs to the cloud or HPC system can be necessary. Managing access
rights to the data, the storage location of data and preventing data loss are
key components from the storage perspective. Additionally, the network
transfer can be a limiting factor creating a complex decision environment.
Instead of having users manually figure out a solution, the DECICE schedule
will map the data and tasks to suitable locations and also adapt to changes in
system topology.

###  5.3 In-the-field intelligence supporting emergency response

In an emergency response, compute infrastructure and connectivity in the
mission area can be seriously disturbed and unreliable. To enable continuous
monitoring of the affected areas and an effective response after a disaster
struck, drones can be of help, data from multiple sensors in the drones and
also additional data sources such as satellite images can be integrated to
define mission areas or extend the area of interest. Quadcopters are an
interesting opportunity but they have limited mission time. Also, image
processing on board can be used but needs to be adapted to conditions, we need
flexibility of computation. Recharging stations can be used to increase drones
autonomy by also supporting upload of mission plans, data gathering, and
communication. Missions can be made adaptive to weather. Coordination between
operators and drone surveys to cover areas that are not covered by operators.
The processing must be optimized, the integration of external sources such as
satellite data to upgrade missions is possible.

We aim at developing an open digital platform to support emergency response
operators exploiting data from drones and satellites. The objective is to
bring intelligence in the operational activities in the field, by providing
the computation support for embedding ML algorithms to support drone
autonomous flight as well as mission operations. To do so, the platform will
implement an efficient and adaptive computation of AI algorithms in-the-field,
orchestrating the processing and the communication between cloud/edge/drone in
a fault-tolerant fashion, allowing the operator to control the data flows
between the computation entities and providing the adaptivity in the system
platform necessary to handle the emergency situation, such as size of the area
to be explored, recognition tasks to be performed, communication link
bandwidth, energy availability and weather conditions.

##  6 Conclusions

The cloud computing industry has grown massively over the last decade and with
that new areas of application have arisen. Modern cloud applications have also
become more complex as they usually run on a distributed computer system,
split up into components that must run with high availability.

In the DECICE project, our goal is to develop an open and portable cloud-edge
management framework for automatic and adaptive optimization of applications
by mapping jobs to the most suitable resources in a heterogeneous system
landscape and compute continuum. By utilizing holistic monitoring, we
construct a digital twin of the system that reflects on the original system.
An AI-scheduler makes decisions on placement of jobs and data as well as
conducting job rescheduling to adjust to system changes dynamically. A virtual
training environment is developed that generates test data for training of ML-
models and the exploration of what-if scenarios. The portable framework is
integrated into the Kubernetes ecosystem, co-designed and evaluated using
relevant use cases on real-world heterogeneous systems.

## Acknowledgments

This research is performed in the context of DECICE project that is funded by
the European Union (grant agreement no. 101092582).

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$$f=\sum\limits_{\bj \in \Z_0^d} f_{\bj}({\vec x}) \text{ with } f_{\bj}({\vec x})=\sum\limits_{\bk \in J_{\bj}} c_{\bk}\e^{2\pi \im\langle\bk,{\vec x}\rangle}$$

in case that $\bj\in \N_0^d$ and $f_{\bj}:=0$ otherwise. This immediately gives

\begin{equation}\label{dyadic}
\|f\|_{H^s_{\mix}}^2 \asymp \sum\limits_{\bj \in \N_0^d} 2^{2|\bj|_1s} \|f_{\bj}\|_{L_2(\T^d)}^2.
\end{equation}

Interestingly, there is also a Fourier analytic characterization of the above defined Besov-Nikolskij spaces $\bB^s_{p,\infty}(\T^d)$ which even works for $1<p<\infty$. Instead of taking the $\ell_2$-norm of the weighted sequence $(2^{|\bj|_1s}\|f_{\bj}\|_{L_p(\T^d)})_{\bj \in \N_0^d}$ we take the $\ell_{\infty}$-norm. We have

\begin{equation*}\label{dyadic_besov}
\|f\|_{\bB^s_{p,\infty}} \asymp \sup\limits_{\bj \in \N_0^d} 2^{|\bj|_1 s }\|f_j\|_{L_p(\T^d)}\,.
\end{equation*}

§ ACKNOWLEDGMENTS

Laura Lippert and Daniel Potts acknowledge funding by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 416228727 - SFB 1410. Tino Ullrich would like to acknowledge support by the DFG Ul-403/2-1.

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