# Error Analysis
of Domain Decomposition method for
4D Variational Data Assimilation (4DVAR DA): Consistence, Convergence and
Stability.

Rosalba Cacciapuoti, Luisa D’Amore

Abstract: We prove consistence, convergence and stability of the Domain
Decomposition in space and time method for 4DVAR Data Assimilation problem (DD
- 4D VAR). We give the condition number of the DD - 4D VAR method and
experimentally validate the analysis on the Shallow Water Equations.

##  0.1 4DVAR DA formulation

If $\Omega\subset\mathbb{R}^{n}$, $n\in\mathbb{N}$, is a spatial domain with a
Lipschitz boundary, let:

$\left\\{\begin{array}[]{ll}u^{\mathcal{M}}(t+h,x)=\mathcal{M}_{t,t+h}[u(t,x)]&\textrm{$\forall
x\in\Omega$, $t,t+h\in[0,T]$, $(h>0)$}\\\
u^{\mathcal{M}}(t_{0},x)=u_{0}(x)&\textrm{$t_{0}\equiv 0,\ \ x\in\Omega$}\\\
u^{\mathcal{M}}(t,x)=f(x)&\textrm{$x\in\partial\Omega$, $\forall
t\in[0,T]$}\\\ \end{array},\right.$ (1)

be a symbolic description of the model of interest where

$u^{\mathcal{M}}:(t,x)\in[0,T]\times\Omega\mapsto
u^{\mathcal{M}}(t,x)=[u^{\mathcal{M}}[1](t,x),u^{\mathcal{M}}[2](t,x),\ldots,u^{\mathcal{M}}[pv](t,x)],$
(2)

is the state function of $\mathcal{M}$ with $pv\in\mathbb{N}$ the number of
physical variables, $f$ is a known function defined on the boundary
$\partial\Omega$;let

$v:(t,x)\in[0,T]\times\Omega\mapsto v(t,x),$

be the observations function and

$\mathcal{H}:u^{\mathcal{M}}(t,x)\mapsto v(t,x),\ \ \ \ \
\forall(t,x)\in[0,T]\times\Omega,$

denote the non-linear observations mapping. To simplify future treatments we
assume $pv\equiv 1$.

###### Definition 1

(Discretization of $\Omega\times\Delta$) Let

$\Omega_{I}\equiv\\{x_{\tilde{i}}\\}_{\tilde{i}\in I}\subset\Omega$

be the discretization of $\Omega$ where

$I=\\{1,\ldots,N_{p}\\}\ \textrm{and}\ N_{p}=|I|$ (3)

are respectively the set of indices of nodes in $\Omega$ and its cardinality
i.e. the number of inner nodes in $\Omega$. Let

$\Delta_{K}\equiv\\{t_{\tilde{k}}\\}_{\tilde{k}\in K}\subset\Delta$

be the discretization of $\Delta$ where

$K=\\{1,\ldots,N\\}\ \textrm{and}\ N=|K|$ (4)

are respectively the set of indices of the time variable in $\Delta$ and its
cardinality i.e. number of time instants in $\Delta$. Consequently, we refer
to

$\Omega_{I}\times\Delta_{K}\equiv\\{(x_{\tilde{i}},t_{\tilde{k}})\\}_{\tilde{i}\in
I;\ \tilde{k}\in K}\subset\Omega\times\Delta$ (5)

as the discrete domain.

We introduce 4D-DA variational formulation, i.e. the 4D-VAR DA problem.

###### Definition 2

(The 4DVAR DA problem). Let $\Omega\subset\mathbb{R}^{n}$ and
$\Delta\subset\mathbb{R}$ be spatial domain and time interval. The 4DVAR DA
problem concerns the computation of:

$\mathbf{u}^{DA}=argmin_{u\in\mathbb{R}^{N_{p}\cdot N}}J(u),$ (6)

solution of 4DVar problem on $\Omega\times\Delta$ with

$J(u)=\alpha\|u-u^{M}\|_{\textbf{B}^{-1}}^{2}+\|Gu-y\|_{\textbf{R}^{-1}}^{2},$
(7)

where

  * •

$N_{p}$: is the number of nodes in $\Omega\subset\mathbb{R}^{n}$ defined in
(3);

  * •

$n_{obs}$: is the number of observations in $\Omega$; where $n_{obs}<<N_{p}$,

  * •

$N$: is the number of time instants in $\Delta$ defined in (4);

  * •

$\alpha$: is the regularization parameter

  * •

$u_{0}=\\{u_{0,j}\\}_{j=1,\ldots,N_{p}}\equiv\\{u_{0}(x_{j})\\}_{j=1,\ldots,N_{p}}\in\mathbb{R}^{N_{p}}$:
is the state at time $t_{0}$;

  * •

the operator

$M_{l-1,l}\in\mathbb{R}^{N_{p}\times N_{p}},\ \ \ l=1,\ldots,N,$ (8)

is the discretization of the linear approximation of
$\mathcal{M}_{t_{l-1},t_{l}}$ from $t_{l-1}$ to $t_{l}$;

  * •

the operator

$M\in\mathbb{R}^{N_{p}\times N_{p}}$ (9)

is the discretization of the linear approximation of $\mathcal{M}$ from
$t_{0}$ to $t_{N}$;

  * •

the matrix

$u^{M}:=\\{u_{j,l}^{{M}}\\}_{j=1,\ldots,N_{p};l=1,\ldots,N}\equiv\\{u^{M}(x_{j},t_{l})\\}_{j=1,\ldots,N_{p};l=0,1,\ldots,N-1}\in\mathbb{R}^{N_{p}\times
N}$ (10)

is the solution of $M$ i.e. the background;

  * •

$y:=\\{y(z_{j},t_{l})\\}_{j=1,\ldots,n_{obs};l=0,1,\ldots,N-1}\in\mathbb{R}^{n_{obs}\times
N}$: are the observations;

  * •

$H_{l}\in\mathbb{R}^{n_{obs}\times N_{p}},\ \ \ l=0,\ldots,N-1$ : is the
linear approximation of observation mapping $\mathcal{H}$;

  * •

$G\equiv G_{N-1}\in\mathbb{R}^{(N\times n_{obs})\times N_{p}}$:

$G_{l}=\left\\{\begin{array}[]{ll}\left[\begin{array}[]{ll}H_{0}\\\ H_{1}\\\
\vdots\\\ H_{l-1}\end{array}\right]&\textrm{$l>1$}\\\ \\\
H_{0}&\textrm{$l=1$}\end{array},\right.$

  * •

$\textbf{R}=diag(\textbf{R}_{0},\textbf{R}_{1},\ldots,\textbf{R}_{N-1})$ and
B$=\textbf{V}\textbf{V}^{T}$: are covariance matrices of the errors on
observations and background, respectively.

##  0.2 The space and time Domain Decomposition method: DD–4DVAR

The DD method consists in decomposing the domain of computation
$\Omega\times\Delta$ into subdomains and solving reduced forecast models and
local 4DVAR DA problems in [1]. The modules of DD method are: domain
decomposition of $\Omega\times\Delta$, Model Reduction, ASM based on Additive
Schwarz method [2], DD-4DVAR local solution and DD-4DVAR global solution. The
schematic description of DD-4DVAR algorithm is reported in Figure 1.

Figure 1: Schematic description of DD–4DVAR algorithm. The modules of
DD–4DVAR: DD, Model Reduction, ASM, DD–4DVAR local solution and DD–4DVAR
global solution are identified and the Arabic numbers in parentheses refer to
the corresponding module in section 0.2. For each module we report the
corresponding solution.

  1. 1.

Domain Decomposition of $\Omega\times\Delta$.
We describe DD of $\Omega\times\Delta$ and $\Omega_{I}\times\Delta_{k}$.

     * •

DD of $\Omega\times\Delta$ consists of:
decomposition of $\Omega\subset\mathbb{R}^{n}$ into a sequence of subdomains
$\Omega_{i}$ such that:

$\Omega=\bigcup_{i=1}^{N_{sub}}\Omega_{i}$ (11)

and definition of set of indices of subdomains adjacent to $\Omega_{i}$ and
its cardinality, as follows

$J_{i}\subset\\{1,\ldots,N_{sub}\\}\quad\textrm{and}\quad ad_{i}=|J_{i}|$ (12)

in particular, $ad_{i}$ is the number of subdomains adjacent to $\Omega_{i}$.
For $i=1,\ldots,N_{sub}$, definition of overlap regions $\Omega_{ij}$ as
follows

$\Omega_{ij}:=\Omega_{i}\cap\Omega_{j}\neq\emptyset\quad\textrm{$\forall j\in
J_{i}$}.$ (13)

Definition of interfaces $\Omega_{i}$, for $i=1,\ldots,N_{sub}$:

$\Gamma_{ij}:=\partial\Omega_{i}\cap\Omega_{j}\quad\textrm{ $j\in J_{i}$}.$
(14)

Decomposition of time interval $\Delta\subset\mathbb{R}$ into a sequence of
time interval $\Delta_{k}$ such that:

$\Delta=\bigcup_{k=1}^{N_{t}}\Delta_{k}.$

Consequently. we define

$\\{\Omega_{i}\times\Delta_{k}\\}_{i=1,\ldots,N_{sub};k=1,\ldots,N_{t}}$ (15)

as local domains.

     * •

DD of $\Omega_{I}\times\Delta_{K}$ defined in (5) consists of:
identification of inner nodes of subdomains
$\\{\Omega_{i}\\}_{i=1,\ldots,N_{sub}}$: for $i=1,\ldots,N_{sub}$

$\\{x_{\tilde{i}}\\}_{\tilde{i}\in I_{i}}\subset\Omega_{i}$

are inner nodes of $\Omega_{i}$ where $I_{i}$ is set defined as follows

$I_{i}:=\left\\{(i-1)\times\frac{N_{p}}{N_{sub}}+1,\ldots,i\times\frac{N_{p}}{N_{sub}}+\frac{\delta}{2}\right\\}$

such that

$I=\bigcup_{i=1}^{N_{sub}}I_{i}.$ (16)

with $I$ set of indices of inner nodes in $\Omega$ defined in (3).

Identification of inner nodes of overlap regions
$\\{\Omega_{ij}\\}_{i=1,\ldots,N_{sub},j\in J_{i}}$: for $i=1,\ldots,N_{sub}$

$\Omega_{I_{i}}\equiv\\{x_{\tilde{j}}\\}_{\tilde{j}\in
I_{ij}}\subset\Omega_{ij}\quad\forall j\in J_{i}$

are inner nodes of $\Omega_{ij}$ where

$I_{ij}:=I_{i}\cap I_{j}\neq\emptyset\quad\forall j\in J_{i}$ (17)

and

$\delta:=|I_{i,j}|$ (18)

are respectively the set of indices of nodes in overlap region and its
cardinality i.e. the number of inner nodes in overlap region $\Omega_{ij}$
defined in (13). Consequently, for $i=1,\ldots,N_{sub}$ we define the
cardinality of $I_{i}$ as follows:

$N_{loc}:=|I_{i}|=\frac{N_{p}}{N_{sub}}+\frac{\delta}{2}$ (19)

i.e. number of inner nodes of spatial subdomain $\Omega_{i}$.

Identification of time instant in $\\{\Delta_{k}\\}_{k=1,\ldots,N_{t}}$: for
$k=1,\ldots,N_{t}$

$\Delta_{K_{k}}\equiv\\{t_{\tilde{k}}\\}_{\tilde{k}\in K_{k}}\subset\Delta.$

are the time instant in $\Delta_{k}$ where $K_{k}$ is defined as follows:

$K_{k}:=\left\\{(k-1)\times\frac{N}{N_{t}},\ldots,k\times\frac{N}{N_{t}}\right\\}$

where

$K_{k}\cap
K_{k+1}=\left\\{k\times\frac{N}{N_{t}}\right\\}\neq\emptyset\quad\forall
k=1,\ldots,N_{t}-1$

and

$N_{k}:=|K_{k}|=\frac{N}{N_{t}}$

is cardinality of $K_{k}$ i.e. the number of time instants in $K_{k}$, such
that

$K=\bigcup_{k=1}^{N_{t}}K_{k}.$ (20)

with $K$ set defined in (4).

Consistently with Definition 1, for $i=1,\ldots,N_{sub}$, $k=1,\dots,N_{t}$

$\Omega_{I_{i}}\times\Delta_{K_{k}}:=\\{(x_{\tilde{i}},t_{\tilde{k}})\\}_{\tilde{i}\in
I_{i};\tilde{k}\in K_{k}}\subset\Omega_{i}\times\Delta_{k}$

is local discrete domain.

In order to compute local estimates on local domains defined in (15) we
introduce the following notation. For $i=1,\ldots,N_{sub}$, $k=1,\dots,N_{t}$,
we pose

$z_{i,k}=\\{z({\tilde{i}},{\tilde{k}})\\}_{\tilde{i}\in I_{i};\ \tilde{k}\in
K_{k}}\in\mathbb{R}^{N_{loc}\times N_{k}}$

i.e. a vector defined on local domain $\Omega_{i}\times\Delta_{k}$.

We are able to define the restriction and extension operator.

###### Definition 3

(Restriction Operator) For $i=1,\ldots,N_{sub}$ we define restriction matrices
$R_{i}$, $R_{ij}$ on $\Omega_{i}$, $\Omega_{ij}$, respectively. Given
$x\in\mathbb{R}^{N_{p}\times N}$ and $z\in\mathbb{R}^{N_{p}\times 1}$ we
define restriction of $x$ to $\Omega_{i}$ :

$x/\Omega_{{i}}:=R_{i}x=\\{x({\tilde{i}},\tilde{k})\\}_{\tilde{i}\in{I}_{i},\tilde{k}\in
K_{k}}\in\mathbb{R}^{N_{loc}\times 1}$ (21)
$x/\Omega_{{ij}}:=R_{ij}x=\\{x({\tilde{j}},\tilde{k})\\}_{\tilde{j}\in{I}_{ij},\tilde{k}\in
K_{k}}\in\mathbb{R}^{\delta\times 1}$

and restriction of $y$ to $\Omega_{i}\times\Delta_{k}$

$z/(\Omega_{{i}}\times\Delta_{k}):=\\{z(\cdot,\tilde{k})\\}_{\tilde{k}\in
K_{k}}/\Omega_{{i}}=R_{i}\cdot\\{z(\cdot,\tilde{k})\\}_{\tilde{k}\in
K_{k}}=\\{z({\tilde{i}},\tilde{k})\\}_{\tilde{i}\in{I}_{i},\tilde{k}\in
K_{k}}\in\mathbb{R}^{N_{loc}\times N_{k}}$
$z/(\Omega_{{ij}}\times\Delta_{k}):=\\{z(\cdot,\tilde{k})\\}_{\tilde{k}\in
K_{k}}/\Omega_{{ij}}=R_{ij}\cdot\\{z(\cdot,\tilde{k})\\}_{\tilde{k}\in
K_{k}}=\\{z({\tilde{i}},\tilde{k})\\}_{\tilde{i}\in{I}_{ij},\tilde{k}\in
K_{k}}\in\mathbb{R}^{N_{loc}\times N_{k}}$

where $I_{i}$ and $I_{ij}$ are respectively set of indices of inner nodes in
$\Omega_{i}$ and $\Omega_{ij}$, $\forall j\in J_{i}$.

###### Definition 4

(Extension operator) We define the Extension Operator (EO). If
$x\in\mathbb{R}^{N_{loc}\times N_{k}}$, it is

$EO(x):=R_{i}^{T}x=\left\\{\begin{array}[]{cc}x({\tilde{i}},\tilde{k})&\textit{if}\
({\tilde{i}},\tilde{k})\in I_{i}\times K_{k}\\\ 0&elsewhere\end{array}\right.$

where $R_{i}^{T}$ is the transpose of $R_{i}$ in (21) and
$EO(x)\equiv{x^{EO}}$.

At this point, we introduce the outer-loop solving the nonlinear aspects of
the assimilation problem which for DD-4DVAR, in general for 4DVAR techniques,
includes the integration of the nonlinear model.
For $n=0,1,\ldots,n_{stop}$, do

  2. 2.

Model Reduction: for $i=1,\ldots,N_{sub}$; $k=1,\ldots,N_{t}$, posed
$u_{i,k}^{0}:=\\{u^{{M}}(x_{\tilde{i}},t_{\tilde{k}})\\}_{\tilde{i}\in{I}_{ij},\tilde{k}\in
K_{k}}$ i.e. by using the background as local initial values, let
$u_{i,k}^{{M}_{i,k},n+1}$ be the solution of the local model
$(P_{i,k}^{{M}_{i,k},n})_{i=1\ldots,N_{sub},k=1,\ldots,N_{t}}$:

$(P_{i,k}^{{M}_{i,k},n})_{i=1\ldots,N_{sub},k=1,\ldots,N_{t}}\
:\quad\left\\{\begin{array}[]{ll}u_{i,k}^{{M}_{i,k},n}=M_{i,k}\cdot
u_{i,k-1}^{n}+b_{i,k}^{n},\\\ u_{i,k-1}^{n}=u_{i,k-1}^{{M}_{i,k},n}\\\
u_{i,k}^{n}/\Gamma_{ij}=u_{j,k}^{n}/\Gamma_{ij},\quad j\in
J_{i}\end{array}\right.$ (22)

where $u_{i,k}^{M}$, $b_{i,k}^{n}$ are respectively the background in
$\Omega_{i}\times\Delta_{k}$, the vector accounting boundary conditions of
$\Omega_{i}$ and

$M_{i,k}:=M_{k}/\Omega_{i}$ (23)

the restriction to $\Omega_{i}$ of the matrix

$M_{k}\equiv
M_{\bar{s}_{k-1},\bar{s}_{k}}:=M_{\bar{s}_{k-1},\bar{s}_{k-1}+1}\cdots
M_{\bar{s}_{k}-1,\bar{s}_{k}}.$ (24)

where

$\bar{s}_{k}:=\sum_{j=1}^{k-1}N_{j}-(N-1)\quad\textrm{and}\quad\bar{s}_{0}:=0$
(25)

are respectively the first index of $\Delta_{K_{K}}$ and $\Delta_{K_{1}}$ and
$b_{i,k}^{n}$ is vector computed from information on boundary of $\Omega_{i}$
at $\Delta_{k}$ that depends on the discretization scheme used.

Let:

$(P_{i,k}^{n})_{i=1,\ldots,N_{sub}\,,k=1,\ldots,N_{t}}\
:u_{i,k}^{ASM,n}=\arg\min_{u_{i,k}^{n}}\,\,\textbf{J}_{i,k}(u_{i,k}^{n})$ (26)

be the local 4DVAR DA model with

$\mathbf{J}_{i,k}(u_{i,k}^{n})=\textbf{J}(u_{i,k}^{n})/(\Omega_{i}\times\Delta_{k})+\mathcal{O}_{ij}.$
(27)

We let

$\mathcal{O}_{ij}=\sum_{j\in
J_{i}}\beta_{j}\cdot\|u_{i,k}^{n}/\Omega_{ij}-u_{j,k}^{n}/\Omega_{ij}\|_{\textbf{B}_{ij}^{-1}}^{2}$
(28)

be the overlapping operator on $\Omega_{ij}$, and

$\textbf{J}_{i,k}(u_{i,k}^{n})/(\Omega_{i}\times\Delta_{k})=\alpha_{i,k}\cdot\|u_{i,k}^{n}-u_{i,k}^{{M}_{i,k},n}\|_{\textbf{B}_{i}^{-1}}+\|{G}_{i,k}u_{i,k}^{n}-y_{i,k}\|_{\textbf{R}_{i}^{-1}}^{2}$

be the overlapping operator on $\Gamma_{ij}$, and

$\textbf{J}_{i,k}(u_{i,k}^{n})/(\Omega_{i}\times\Delta_{k})=\alpha_{i,k}\cdot\|u_{i,k}^{n}-u_{i,k}^{{M}_{i,k},n}\|_{\textbf{B}_{i}^{-1}}+\|{G}_{i,k}u_{i,k}^{n}-y_{i,k}\|_{\textbf{R}_{i}^{-1}}^{2}$

be the restriction of $\mathbf{J}$ to $\Omega_{i}\times\Delta_{k}$. Parameters
$\alpha_{i,k}$ and $\beta_{j}$ in (28) denotes the regularization parameters.
Following, we let $\alpha_{i,k}=\beta_{j}=1$, $j\in J_{i}$.

  3. 3.

ASM: according to ASM [2] we compute solution of
$(P_{i,k}^{n})_{i=1,\ldots,N_{sub}\,,k=1,\ldots,N_{t}}$. Gradient of
$\textbf{J}_{i,k}$ is [9, 3]:

$\begin{array}[]{ll}\nabla\textbf{J}_{i,k}(\textbf{w}_{i,k}^{n})=&(\textbf{V}_{i}^{T}({G}_{i,k})^{T}(\textbf{R}_{i,k})^{-1}{G}_{i,k}\textbf{V}_{i}+I_{i}+ad_{i}\cdot\textbf{B}_{ij})\textbf{w}_{i,k}^{n}\\\
&-\textbf{c}_{i}+\sum_{j\in
J_{i}}\textbf{B}_{ij}\textbf{w}_{j,k}^{n},\end{array}$ (29)

where

$\textbf{w}_{i,k}^{n}=\textbf{V}_{i}^{-1}(u_{i,k}^{n}-u_{i,k}^{{M}_{i,k},n}),$
(30)
$\textbf{d}_{i}=(\textbf{v}_{i}-{G}_{i,k}{u}_{i,k}^{{M}_{i,k},n})\quad\textbf{c}_{i}=(\textbf{V}_{i}^{T}({G}_{i,k})^{T}(\textbf{R}_{i,k})^{-1}\textbf{d}_{i})$

and $I_{i}$ the identity matrix, $ad_{i}$ number of subdomains adjacent to
$\Omega_{i}$ defined in (12). Solution of
$(P_{i,k}^{n})_{i=1,\ldots,N_{sub}\,,k=1,\ldots,N_{t}}$ is obtained by
requiring $\nabla\textbf{J}_{i,k}(\textbf{w}_{i,k}^{n})=0$. This gives rise to
the linear system:

$A_{i,k}\textbf{w}_{i,k}^{n}=c_{i}-\sum_{j\in
J_{i}}\textbf{B}_{ij}\textbf{w}_{j,k}^{n},$ (31)

where

$A_{i,k}=(\textbf{V}_{i}^{T}({G}_{i,k})^{T}\textbf{R}_{i,k}^{-1}{G}_{i,k}\textbf{V}_{i}+I_{i}+ad_{i}\cdot\textbf{B}_{ij}).$
(32)

(7.1) For each iteration $n$, r.h.s. of (31) depends on unknown value
$\textbf{w}_{j,k}^{n}$ defined on sub domains $\Omega_{ij}$ adjacent to
$\Omega_{i}$,for $j\in J_{i}$. Hence, for solving the system in (31).
We introduce an inner-loops related to minimization of local 4DVAR model
defined in (26). For $r=0,1,\ldots,\bar{r}$

$A_{i,k}\textbf{w}_{i,k}^{r+1,n}=c_{i}-\sum_{j\in
J_{i}}\textbf{B}_{ij}\textbf{w}_{j,k}^{r,n},$ (33)

where at each step $r+1$, $\Omega_{i}$ receives $\textbf{w}_{i,k}^{r,n}$ from
$\Omega_{ij}$, $j\in J_{i}$ for computing the r.h.s. of (33) then it computes
$\textbf{w}_{i,k}^{r+1,n}$ by using Conjugate Gradient (CG) method and finally
it sends $\textbf{w}_{i,k}^{r+1,n}$ to $\Omega_{ij}$, $j\in J_{i}$ for
updating the r.h.s. of (33) needed to the next iteration.
$\textbf{w}_{i_{j},k}^{0,n}$ is an arbitrary initial value. Finally, we pose

$\textbf{w}_{i,k}^{n}\equiv\textbf{w}_{i,k}^{\bar{r},n},$

consequently

$u_{i,k}^{ASM,n}:=u_{i,k}^{ASM,\bar{r}}.$ (34)

  4. 4.

DD–4DVAR approximation in $\Omega_{i}\times\Delta_{k}$: final solution update,
using (22) and (30):

$\begin{array}[]{ll}u_{i,k}^{n+1}&=u_{i,k}^{{M}_{i,k},n+1}+\textbf{V}_{i}\textbf{w}_{i,k}^{n}=u_{i,k}^{{M}_{i,k},n+1}+[u_{i,k}^{ASM,n}-u_{i,k}^{{M}_{i,k},n}].\end{array}$
(35)

Endfor $n$

  5. 5.

DD–4DVAR solution in $\Omega\times\Delta$: computation of DD-4DVAR
approximation on $\Omega\times\Delta$: let

$\mathbf{\widetilde{u}^{DD-
DA,n}}:=\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}({{u}}_{i,k}^{{n}})^{EO}.$ (36)

be DD–4DVAR approximation at iteration $n$ where $({{u}}_{i,k}^{{n}})^{EO}$ is
extension to $\Omega\times\Delta$ of DD-4DVAR approximations in
$\Omega_{i}\times\Delta_{k}$, we define

$\mathbf{\widetilde{u}^{DD-DA}}:=\mathbf{\widetilde{u}^{DD-DA,{\bar{n}}}}$
(37)

as DD-4DVAR solution in $\Omega\times\Delta$.

Note that B${}_{i}=R_{i}\textbf{B}R_{i}^{T}$ and
$\textbf{B}_{ij}:=\textbf{B}/\Gamma_{ij}=R_{i}\textbf{B}R_{ij}^{T}$ are the
restrictions of covariance matrix B, respectively, to subdomain $\Omega_{i}$
and interface $\Gamma_{ij}$ in (14), ${G}_{i,k}$, $\textbf{R}_{i,k}$ are the
restriction of matrices ${G}_{k}:={G}_{\bar{s}_{k}}$ and
${\textbf{R}_{k}}:=diag(\textbf{R}_{0},\textbf{R}_{1},\ldots,\textbf{R}_{\bar{s}_{k}})$
to $\Omega_{i}$, and finally ${u}_{i,k}^{{M}}=R_{i}{u}_{k}^{{M}}$,
${u}_{i,k}^{n+1}/\Gamma_{ij}=R_{ij}{u}_{i,k}^{n+1}$,
${u}_{j,k}^{n}/\Gamma_{ij}=R_{ij}{u}_{k}^{n}$ are the restriction of vectors
${u}_{k}^{b}$, ${u}_{i,k}^{n+1}$, ${u}_{j,k}^{n}$ to $\Omega_{i}$ and
$\Gamma_{ij}$, for $i=1,2,\ldots,N_{sub}$, $j\in J_{i}$.

It is worth noting that each local functional $\textbf{J}_{i,k}$ is obtained
starting from a restriction of the global functional J and adding a local term
defined on the overlapping regions. In addition, regarding the decomposition
in time direction, we use DA as predictor operator for the local PDE-based
model, providing the approximations needed for solving the initial value
problem on each sub interval.

We prove that the minimum of $\mathbf{J}$ can be obtained by patching together
all the local solution obtained as minimum of local function
$\mathbf{J}_{i,k}$. We are able to prove the following result.

###### Theorem 0.2.1

If $\mathbf{J}$ is convex:

$\mathbf{J}(\mathbf{u}^{DA})=\mathbf{J}(\mathbf{\widetilde{u}^{DD-DA}}).$ (38)

Proof: As $\mathbf{u}^{DA}$ is the global minimum of $\mathbf{J}$ it follows
that:

$\mathbf{J}(\mathbf{{u}^{DA}})\leq\mathbf{J}\left(\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}{(\mathbf{u}}_{i,k}^{\bar{n}})^{EO}\right),\quad\forall\,i,k$
(39)

then, from the (37) it follows that

$\mathbf{J}(\mathbf{u^{DA}})\leq\mathbf{J}\left(\mathbf{\widetilde{u}^{DD-
DA}}\right)\quad.$ (40)

Now we prove that if $\mathbf{J}$ is convex, then

$\mathbf{J}(\mathbf{u^{DA}})=\mathbf{J}(\mathbf{\widetilde{u}^{DD-DA}})$

by reduction to the absurd. Assume that

$\mathbf{J}(\mathbf{u^{DA}})<\mathbf{J}(\mathbf{\widetilde{u}^{DD-DA}}).$ (41)

In particular,

$\mathbf{J}(\mathbf{u^{DA}})<\mathbf{J}\left(\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\mathbf{\widetilde{u}_{i,k}}\right)\leq\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\mathbf{\mathbf{J}_{i,k}}({{u}_{i,k}^{\bar{n}}})\quad.$
(42)

This means that

$\mathbf{J}(\mathbf{u^{DA}})<\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\mathbf{\mathbf{J}_{i,k}}({{u}_{i,k}^{\bar{n}}})\quad.$
(43)

From the (43) and the (42), it is:

$\nabla\mathbf{J}(\mathbf{u}^{DA})<\nabla\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\mathbf{\mathbf{J}_{i,k}}({{u}_{i,k}^{\bar{n}}})=\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\nabla\mathbf{\mathbf{J}_{i,k}}({{u}_{i,k}^{\bar{n}}})$
(44)

$\mathbf{u}^{DA}$ is global minimum of $J$ i.e.
$\nabla\mathbf{J}(\mathbf{u}^{DA})=0$, from (44) it is

$\nabla\mathbf{\mathbf{J}_{i,k}}({u}_{i,k})>0\quad\forall(i,k).$ (45)

The (45) is an absurd as the values of ${{u}_{i,k}^{\bar{n}}}$ are the minimum
for $\mathbf{\mathbf{J}_{i,k}}$ for $\forall(i,k)$. Hence, the (38) is proved.

##  0.3 Consistency, convergence and stability of DD-4DVAR method

We first prove DD–4DVAR convergence (see Figure 1).

###### Theorem 0.3.1

(Convergence of outer loop) The DD-4DVAR method described in section 0.2 is a
convergent method i.e.

$lim_{n\rightarrow\infty}\|\mathbf{\tilde{u}^{DD-
DA,n+1}}-\mathbf{\tilde{u}^{DD-DA,n}}\|=0$ (46)

where $\mathbf{u^{DA,n}}$ is the DD–4DVAR approximation at iteration $n$
defined in (36).

Proof. From (38) and (37), it is

$\displaystyle\begin{split}\mathbf{J}(\mathbf{u^{DA}})=J(\mathbf{\widetilde{u}^{DD-
DA}})=\mathbf{J}(\mathbf{\widetilde{u}^{DD-DA,\bar{n}}})\end{split}$ (47)

where $\bar{n}$ is the latest iteration of DD-4DVAR outer loop (see Figure 1).
By minimum properties of $\mathbf{\widetilde{u}^{DD-DA,\bar{n}}}$, we get

$\displaystyle\begin{split}\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,\bar{n}}})\leq\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,\bar{n}-1}}).\end{split}$ (48)

From (48), by recurring we obtain

$\displaystyle\begin{split}0\leq\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,\bar{n}}})\leq\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,\bar{n-1}}})\leq\ldots\leq\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,{1}}})\leq\mathbf{J}(\mathbf{\widetilde{u}^{DD-DA,{0}}})\end{split}$

then $\\{\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,{n}}})\\}_{\mathbf{n}\in\mathbb{N}}$ is monotonically decreasing and
bounded from below by 0 then it is convergent i.e.

$lim_{n\rightarrow\infty}\left[\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,{n+1}}})-\mathbf{J}(\mathbf{\widetilde{u}^{DD-DA,{n}}})\right]=0.$ (49)

Concerning iteration $n$ of the outer loop, we get

$\displaystyle\begin{split}\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,{n}}})=\|\mathbf{\widetilde{u}^{DD-
DA,{n}}}-u^{{M}}\|_{\mathbf{B}^{-1}}^{2}+\|G\mathbf{\widetilde{u}^{DD-
DA,{n}}}-v\|_{\mathbf{R}^{-1}}^{2}\geq\|\mathbf{\widetilde{u}^{DD-
DA,{n}}}-u^{{M}}\|_{\mathbf{B}^{-1}}\end{split}$ (50)

from the triangle inequality we get

$\displaystyle\begin{split}\mathbf{J}(\mathbf{\widetilde{u}^{DD-
DA,{n}}})\geq|\|\mathbf{\widetilde{u}^{DD-
DA,{n}}}\|_{\mathbf{B}^{-1}}-\|u^{{M}}\|_{\mathbf{B}^{-1}}|\end{split}$ (51)

where $u^{{M}}$ is defined in (10) and does not depend on $n$ then
$\\{\mathbf{\widetilde{u}^{DD-DA,{n}}}\\}_{\mathbf{n}\in\mathbb{N}}$ is
convergent respect to $\|\cdot\|_{\mathbf{B}^{-1}}$, consequently respect to
$\|\cdot\|$. Then we get the thesis in (46).

###### Theorem 0.3.2

(Convergence of inner loop) The ASM module is convergent.

Proof. The convergence of ASM is proved in [2].

We also analyse the convergence of DD-4DVAR modules in terms of local
truncation errors $E_{i,k}^{{M}_{i,k}}$, $E_{i,k}^{ASM}$, $E_{i,k}$ and
$E_{g}$ reported in Figure 2. Similar to the differential equations case [17],
the local truncation errors $E_{i,k}^{{M}_{i,k}}$, $E_{i,k}^{ASM}$, $E_{i,k}$
in $\Omega_{i}\times\Delta_{k}$ and $E_{g}$ in $\Omega\times\Delta$ are
defined as the remainder after solutions $u^{\mathcal{M}}$ of model (1) and
$\mathbf{u^{DA}}$ of 4D-DA problem (6) are substituted into the discrete
models. To this aim, we give the following definitions.

###### Definition 5

We define

$u^{\mathcal{M}\rightarrow
M}:=M\cdot\\{u^{\mathcal{M}}(x_{i},t_{k})\\}_{\tilde{i}\in I;\ \tilde{k}\in
K}$ (52)

approximation of $u^{\mathcal{M}}$ in (1) in $\Omega\times\Delta$ obtained by
replacing $u^{\mathcal{M}}$ evaluated in $\Omega_{I}\times\Delta_{K}$ defined
in (1), into $M$ defined in (9).

###### Definition 6

(Local truncation errors in $\Omega_{i}\times\Delta_{k}$) Let us consider

  * •

$(\Omega_{1},\Omega_{2},\ldots,\Omega_{N_{sub}})$ and
$(\Delta_{1},\Delta_{2},\ldots,\Delta_{N_{t}})$: be decomposition of $\Omega$
and $\Delta$ (DD in Figure 2);

  * •

$u^{\mathcal{M}\rightarrow M}/({\Omega_{i}}\times\Delta_{k})$: be restriction
of approximation of $u^{\mathcal{M}}$ defined in (52) to
$\Omega_{i}\times\Delta_{k}$;

  * •

$u_{i,k}^{{M}_{i,k},\bar{n}}$: be solution of $P_{i,k}^{{M}_{i,k},\bar{n}}$ in
$\Omega_{i}\times\Delta_{k}$ defined in (22) at iteration $\bar{n}$ (Model
Reduction in Figure 2);

  * •

$\mathbf{{u}^{DA}}$: be solution of 4DVAR DA problem in (6);

  * •

$\mathbf{\widetilde{u}^{DD-DA}}$: be DD-4DVAR solution in $\Omega\times\Delta$
of $\mathbf{{u}^{DA}}$ in (6);

  * •

$\mathbf{u^{DA}}/({\Omega}_{i}\times\Delta_{k})$: be the restriction to
${\Omega_{i}}\times\Delta_{k}$ of $\mathbf{{u}^{DA}}$;

  * •

$u_{i,k}^{ASM,\bar{n}}$: be solution of $P_{i,k}^{\bar{n}}$ on
$\Omega_{i}\times\Delta_{k}$ defined in (26) (ASM in Figure 2);

  * •

$u_{i,k}^{\bar{n}}$: be local DD-4DVAR solution in
$\Omega_{i}\times\Delta_{k}$ defined in (35) (DD–4DVAR local solution).

Then, $\forall i=1,\ldots,N_{sub}$ and $k=1,\ldots,N_{t}$, we define

$E_{i,k}^{{M}_{i,k},\bar{n}}:=\left\|u^{\mathcal{M}\rightarrow
M}/({\Omega_{i}}\times\Delta_{k})-u_{i,k}^{{M}_{i,k},\bar{n}}\right\|$ (53)

as local truncation error of numerical scheme restricted to
$\Omega_{i}\times\Delta_{k}$ at iteration $\bar{n}$;

$E_{i,k}^{ASM,\bar{n}}:=\left\|\mathbf{u^{DA}}/({\Omega}_{i}\times\Delta_{k})-u_{i,k}^{ASM,\bar{n}}\right\|$
(54)

as local truncation error of ASM restricted to $\Omega_{i}\times\Delta_{k}$ at
iteration $\bar{n}$;

$E_{i,k}^{\bar{n}}:=\left\|\mathbf{u^{DA}}/({\Omega}_{i}\times\Delta_{k})-u_{i,k}^{\bar{n}}\right\|$
(55)

as local truncation error of DD–4DVAR method restricted to
$\Omega_{i}\times\Delta_{k}$ at iteration $\bar{n}$;

$E_{g}:=\left\|\mathbf{u^{DA}}-\mathbf{\widetilde{u}^{DD-DA}}\right\|$ (56)

as local truncation error of DD–4DVAR method in $\Omega\times\Delta$.

DD-4DVAR method needs few iterations of DD–4DVAR outer loop on $n$ to update
DD-4DVAR approximation in (35). Consequently, we neglect the dependency on
outer loop iteration $\bar{n}$ of $u_{i,k}^{M_{i,k},\bar{n}}$,
$u_{i,k}^{ASM,\bar{n}}$ and $u_{i,k}^{n}$ defined respectively in (22), (26)
and (35) and $E_{i,k}^{{M}_{i,k},\bar{n}}$, $E_{i,k}^{ASM,\bar{n}}$ and
$E_{i,k}^{\bar{n}}$ in $\Omega_{i}\times\Delta_{k}$ defined respectively in
(53), (54) and (55).

Figure 2: Local truncation errors related to each module of DD–4DVAR method.

For $i=1,\ldots,N_{sub}$ and $k=1,\ldots,N_{t}$, we pose

$\mathbf{\widetilde{u}_{i,k}}:={{u}}_{i,k}^{\bar{n}}.$ (57)

From (57), the DD–4DVAR approximation in $\Omega\times\Delta$ defined in (37)
becomes

$\mathbf{\widetilde{u}^{DD-
DA}}=\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\mathbf{\widetilde{u}_{i,k}}^{EO}.$
(58)

For $i=1,\ldots,N_{sub}$ and $k=1,\ldots,N_{t}$, we pose

$\begin{split}u_{i,k}^{ASM}&=u_{i,k}^{ASM,\bar{n}};\\\
u_{i,k}^{{M}_{i,k}}&=u_{i,k}^{{M}_{i,k},\bar{n}};\\\
u_{i,k}&=u_{i,k}^{\bar{n}}.\end{split}$ (59)

Consequently, from (59) we pose

$E_{i,k}^{{M}_{i,k}}=E_{i,k}^{{M}_{i,k},\bar{n}},$ (60)

as local model truncation error on $\Omega_{i}\times\Delta_{k}$,

$E_{i,k}^{ASM}=E_{i,k}^{ASM,\bar{n}},$ (61)

as local ASM truncation error on $\Omega_{i}\times\Delta_{k}$ in (54),

$E_{i,k}=E_{i,k}^{\bar{n}}.$ (62)

as local truncation error $\Omega_{i}\times\Delta_{k}$ in (55).

We introduce the definition of consistency of DD–4DVAR method. We pose
$\|\cdot\|=\|\cdot\|_{2}$222We refer to
$\|z_{i,k}\|_{2}=\|\\{z_{i,k}(\bar{i},\bar{k})\\}_{\bar{i}\in{I}_{i},\bar{k}\in
K_{k}}\|_{2}$ where $z\in R^{N_{p}\times N}$ and
$z_{i,k}:=z/(\Omega_{i}\times\Delta_{k})$ and ${I}_{i}$ and $K_{k}$ are
defined in (16) and (16), respectively..

###### Definition 7

(Consistency of DD–4DVAR method) Let $E_{g}$ be local truncation error of
DD–4DVAR in $\Omega\times\Delta$ defined in (56). The DD–4DVAR method is said
to be consistent, if $E_{g}\rightarrow 0$ as $\Delta x,\ \Delta t\rightarrow
0$, where

  * •

$\Delta x:=max_{i=1,\ldots,N_{sub}}(\Delta x)_{i}$, where $\\{(\Delta
x)_{i}\\}_{i=1,\ldots,N_{sub}}$ are spatial step sizes of $M_{i,k}$ defined in
(23);

  * •

$\Delta t:=max_{k=1,\ldots,N_{t}}(\Delta t)_{k}$, where $\\{(\Delta
t)_{k}\\}_{k=1,\ldots,N_{t}}$ are temporal step sizes of $M_{i,k}$ defined in
(23).

In order to prove the consistency of DD–4DVAR method, we perform the analysis
of local truncation errors $E_{i,k}^{{M}_{i,k}}$, $E_{i,k}^{ASM}$, $E_{i,k}$
and $E_{g}$ defined respectively in (60), (61), (62) and (56) related to
Module Reduction, ASM, DD–4DVAR local solution and DD–4DVAR solution modules
as described in Figure 3.

###### Assumption 0.3.1

(Local truncation error of Model Reduction in $\Omega_{i}\times\Delta_{k}$)
Let

$E_{i,k}^{{M}_{i,k}}=\mathcal{O}((\Delta x)_{i}^{p}+(\Delta
t)_{k}^{q}),\quad\forall(i,k)\in\\{1,\ldots,N_{sub}\\}\times\\{1,\ldots,N_{t}\\}$
(63)

be local truncation error defined in (60) where $(\Delta x)_{i}$ and $(\Delta
t)_{k}$ are spatial and temporal step sizes of ${M}_{i,k}$ defined in (23) and
$p$ and $q$ are the order of convergence in space and in time.

In experimental results (see section 0.4), in order to discretize the Shallow
Water Equations (SWEs) model we consider Lax–Wendroff scheme [4]. Hence, in
that case, $p=q=2$.

###### Lemma 0.3.1

(Local truncation error of ASM in $\Omega_{i}\times\Delta_{k}$) Let us
consider

  * •

$(\Omega_{1},\Omega_{2},\ldots,\Omega_{N_{sub}})$ and
$(\Delta_{1},\Delta_{2},\ldots,\Delta_{N_{t}})$: be decomposition of $\Omega$
and $\Delta$ (DD in Figure 3);

  * •

$E_{i,k}^{ASM}$: be local truncation error in $\Omega_{i}\times\Delta_{k}$ of
ASM defined in (61);

  * •

$u_{i,k}^{ASM}$: be solution of $P_{i,k}^{\bar{n}}$ on
$\Omega_{i}\times\Delta_{k}$ defined in (59) (ASM in Figure 3);

  * •

$\sigma_{0}^{2}$: be observational error variance;

  * •

$\mathbf{B}_{i}=\mathbf{V}_{i}\mathbf{V}_{i}^{T}$: be restriction of
covariance matrices of the error on background to $\Omega_{i}$;

  * •

$G_{i,k}$: be restriction to $\Omega_{i}\times\Delta_{k}$ of matrix $G$
defined in (• ‣ 2);

  * •

$ad_{i}$: be number of subdomains adjacent to $\Omega_{i}$, defined in (12);

  * •

$\mathbf{B}_{ij}=\mathbf{V}_{ij}\mathbf{V}_{ij}^{T}$: be restriction to
$\Omega_{ij}$ of variance matrix of the error on background;

  * •

${M}_{i,k}$: be matrix defined in (24), restricted to
$\Omega_{i}\times\Delta_{k}$;

  * •

$\mu(\mathbf{V}_{i})$, $\mu(G_{i,k})$, $\mu(M_{i,k})$ and
$\mu(\mathbf{V}_{ij})$: be condition number of matrices
$\mathbf{V}_{i},G_{i,k}$, $M_{i,k}$ and $\mathbf{V}_{ij}$, respectively.

Then, $\forall i=1,\ldots,N_{sub};k=1,\ldots,N_{t}$, it holds that:

$E_{i,k}^{ASM}\leq\mu_{i,k}^{DD-DA}\cdot E_{i,1}^{ASM}$ (64)

where

$\displaystyle\begin{split}\mu_{i,k}^{DD-
DA}&:=\left[1+\frac{1}{\sigma_{0}^{2}}\mu^{2}(\mathbf{V}_{i})\mu^{2}(G_{i,k})+ad_{i}\cdot\mu^{2}(\mathbf{V}_{ij})\right]\mu({M}_{i,k}).\end{split}$
(65)

Proof: By applying error approximation in [9], we have

$\|\mathbf{u}^{DA}/(\Omega_{i}\times\Delta_{k})-u_{i,k}^{ASM}\|\leq\mu(\mathbf{J}_{i,k})\mu({M}_{i,k})\cdot\|\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{1})-u_{i,1}^{ASM}\|$
(66)

where $\|\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{1})-u_{i,1}^{ASM}\|$ is
error on $\Omega_{i}\times\Delta_{1}$. As proved in [9], it is

$\mu(\mathbf{J}_{i,k})=\mu(A_{i,k})$ (67)

where $\mathbf{J}_{i,k}$ and $A_{i,k}$ are respectively defined in (27) and
(32) and form the triangle inequality, it is

$\displaystyle\mu(A_{i,k})$ $\displaystyle\leq
1+\frac{1}{\sigma_{0}^{2}}\mu^{2}(\mathbf{V}_{i})\mu^{2}(G_{i,k})+ad_{i}\cdot\mu(\mathbf{B}_{ij})$
(68) $\displaystyle\leq
1+\frac{1}{\sigma_{0}^{2}}\mu^{2}(\mathbf{V}_{i})\mu^{2}(G_{i,k})+ad_{i}\cdot\mu^{2}(\mathbf{V}_{ij}),$
(69)

where $\sigma_{0}^{2}$ is observational error variance, $ad_{i}$ is number of
adjacent subdomains to $\Omega_{i}$ defined in (12), $\mu(\mathbf{V}_{i})$,
$\mu(G_{i,k})$, $\mu(M_{i,k})$ and $\mu(\mathbf{V}_{ij})$ are condition
numbers of matrices $\mathbf{V}_{i},G_{i,k}$, $M_{i,k}$ and $\mathbf{V}_{ij}$,
respectively. From (66) and (68), the (64) follows.

Figure 3: Assumption, Lemmas, Theorems related to each module of DD-4DVAR
method.

In the following we approximation the local truncation error of DD-4DVAR
method in $\Omega_{i}\times\Delta_{k}$ defined in (62).

###### Lemma 0.3.2

(Local truncation error of DD-4DVAR method in $\Omega_{i}\times\Delta_{k}$)
Let us consider

  * •

$(\Omega_{1},\Omega_{2},\ldots,\Omega_{N_{sub}})$ and
$(\Delta_{1},\Delta_{2},\ldots,\Delta_{N_{t}})$: be decomposition of $\Omega$
and $\Delta$ (DD in Figure 3);

  * •

$\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{k})$: be the restriction to
$\Omega_{i}\times\Delta_{k}$ of 4DVAR DA problem’s solution in (6);

  * •

$u_{i,k}$: be DD-4DVAR local solution on $\Omega_{i}\times\Delta_{k}$ at
iteration ${n}$ defined in (35) (DD-4DVAR local solution in Figure 3);

  * •

$u^{\mathcal{M}\rightarrow M}/({\Omega_{i}}\times\Delta_{k})$: be restriction
of $u^{\mathcal{M}}$ defined in (52) to $\Omega_{i}\times\Delta_{k}$;

  * •

$E_{i,k}^{{M}_{i,k}}$: be local truncation error in
$\Omega_{i}\times\Delta_{k}$ defined in (60);

  * •

$E_{i,1}^{ASM}$: be local truncation error in $\Omega_{i}\times\Delta_{1}$
defined in (61);

Then $\forall i=1,\ldots,N_{sub};k=1,\ldots,N_{t}$, it holds that:

$\displaystyle\begin{split}E_{i,k}\leq\mu_{i,k}^{DD-DA}\cdot
E_{i,1}^{ASM}+2\cdot E_{i,k}^{{M}_{i,k}}\end{split}$ (70)

where $\mu_{i,k}^{DD-DA}$ is defined in (65).

Proof: From (59) and (35), $E_{i,k}$ defined in (62) can be rewritten as
follows

$\begin{array}[]{ll}E_{i,k}&:=\|\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{k})-u_{i,k}\|\\\
&=\|\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{k})-u_{i,k}^{{M}_{i,k},\bar{n}}-(u_{i,k}^{ASM,\bar{n}-1}-u_{i,k}^{{M}_{i,k},\bar{n}-1})\|\end{array}$

from the triangle inequality

$\begin{array}[]{ll}E_{i,k}&\leq\|\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{k})-u_{i,k}^{ASM,\bar{n}}\|+\|u_{i,k}^{{M}_{i,k},\bar{n}-1}-u_{i,k}^{{M}_{i,k},\bar{n}}\|\end{array}$
(71)

as consequence of Lemma 0.3.1 and (61) we have

$E_{i,k}\leq\mu_{i,k}^{DD-DA}\cdot
E_{i,1}^{ASM}+\|u_{i,k}^{{M}_{i,k},\bar{n}-1}-u_{i,k}^{{M}_{i,k},\bar{n}}\|$
(72)

where $E_{i,1}^{ASM}$ is defined in (61) and $\mu_{i,k}^{DD-DA}$ is defined in
(65).
In particular, by adding and subtracting $u^{\mathcal{M}\rightarrow
M}/({\Omega_{i}}\times\Delta_{k})$ in
$\|u_{i,k}^{{M}_{i,k},\bar{n}-1}-u_{i,k}^{{M}_{i,k},\bar{n}}\|$ we get:

$\displaystyle\begin{split}\|u_{i,k}^{{M}_{i,k},\bar{n}-1}-u_{i,k}^{{M}_{i,k},\bar{n}}\|&=\|(u_{i,k}^{{M}_{i,k},\bar{n}-1}-u^{\mathcal{M}\rightarrow
M}/({\Omega_{i}}\times\Delta_{k}))\\\ &+(u^{\mathcal{M}\rightarrow
M}/({\Omega_{i}}\times\Delta_{k})-u_{i,k}^{{M}_{i,k},\bar{n}})\|\end{split}$

and from the triangle inequality

$\displaystyle\begin{split}\|u_{i,k}^{{M}_{i,k},\bar{n}-1}-u_{i,k}^{{M}_{i,k},\bar{n}}\|&\leq\|u_{i,k}^{{M}_{i,k},\bar{n}-1}-u^{\mathcal{M}\rightarrow
M}/({\Omega_{i}}\times\Delta_{k})\|\\\ &+\|u^{\mathcal{M}\rightarrow
M}/({\Omega_{i}}\times\Delta_{k})-u_{i,k}^{{M}_{i,k},\bar{n}}\|\\\
&=E_{i,k}^{{M}_{i,k},\bar{n}}+E_{i,k}^{{M}_{i,k},\bar{n}-1}\end{split}$ (73)

From Theorem 0.3.1, we get that $\\{\mathbf{\widetilde{u}^{DD-
DA,n}}\\}_{n\in\mathbb{N}}$ is a convergent sequence, then it is a Cauchy
sequence. From (36), we get that $\\{u_{i,k}^{M_{i,k},n}\\}_{n\in\mathbb{N}}$
is also a Cauchy sequence, i.e.

$\forall\ \epsilon>0\ \exists\ \mathbf{N}>0\ :\ \
\|u_{i,k}^{M_{i,k},{n}}-u_{i,k}^{M_{i,k},m}\|\leq\epsilon\quad\forall
n,m>\mathbf{N}.$ (74)

In particular, (74) is true for $n=\bar{n}$ and $m=\bar{n}-1$, assuming that
$\bar{n}$ is large enough. Consequently, we can neglect the dependency on
outer loop in $E_{i,k}^{{M}_{i,k},\bar{n}}$ and
$E_{i,k}^{{M}_{i,k},\bar{n}-1}$ in (73) i.e.

$\displaystyle\|u_{i,k}^{{M}_{i,k},\bar{n}-1}-u_{i,k}^{{M}_{i,k},\bar{n}}\|\leq
2\cdot E_{i,k}^{{M}_{i,k}}$ (75)

where $E_{i,k}^{{M}_{i,k}}$ is defined in (60). From (72), (73) and (75) we
get the thesis in (70).

###### Lemma 0.3.3

If $e_{0}$, the error on initial condition of ${M}$ in (9) is equal to zero
i.e.

$e_{0}=0$ (76)

then

$E_{i,k}\leq c\cdot((\Delta x)_{i}^{p}+\cdot(\Delta t)_{k}^{q})$ (77)

where $E_{i,k}$ is the local truncation error defined in (62) and $c$ is
positive constant independent on DD.

Proof. By applying the Lemma 61 to local ASM truncation error $E_{i,1}^{ASM}$
on $\Omega_{i}\times\Delta_{1}$, we get

$E_{i,1}^{ASM}\leq F_{i,1}\cdot e_{0}/\Omega_{i}.$ (78)

where $F_{i,1}$ is defined in (65) and $e_{0}/\Omega_{i}$ is the restriction
of $e_{0}$ to $\Omega_{i}$. From the assumptions it is $e_{0}/\Omega_{i}=0$
and by replacing it in (78) we have

$E_{i,1}^{ASM}=0$ (79)

consequently local ASM truncation error on $\Omega_{i}\times\Delta_{k}$ is

$E_{i,k}^{ASM}=0.$ (80)

From (80), (63) and (70) we get the thesis in (77).

Now, we are able to prove the following result on global truncation error.

###### Theorem 0.3.3

(Local truncation error in $\Omega\times\Delta$) Let assume the assumption of
Lemma 0.3.3 in (76). The local truncation error in $\Omega\times\Delta$ is

$E_{g}\leq c\cdot(N_{sub}\cdot N_{t})\cdot[(\Delta x)^{p}+(\Delta t)^{q}],$
(81)

where

  * •

$\mathbf{u^{DA}}$: solution of 4DVAR DA problem in (6);

  * •

$\mathbf{\widetilde{u}^{DD-DA}}$: DD–4DVAR solution in $\Omega\times\Delta$
defined in (37);

  * •

$\Delta x:=max_{i=1,\ldots,N_{sub}}(\Delta x)_{i}$: where $\\{(\Delta
x)_{i}\\}_{i=1,\ldots,N_{sub}}$ are spatial step sizes of $M_{i,k}$ defined in
(23);

  * •

$\Delta t:=max_{k=1,\ldots,N_{t}}(\Delta t)_{k}$: where $\\{(\Delta
t)_{k}\\}_{k=1,\ldots,N_{t}}$ are temporal step sizes of $M_{i,k}$ defined in
(23);

  * •

$c>0$: is positive constant independent of DD.

Proof From (58) the error $E_{g}$ defined in (56) can be rewritten as follows

$\displaystyle\begin{split}E_{g}:=\left\|\mathbf{u^{DA}}-\mathbf{\tilde{u}^{DD-
DA}}\right\|=\left\|\mathbf{u^{DA}}-\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}{\mathbf{\widetilde{u}_{i,k}}}\right\|\end{split}$

by applying the restriction and extension operator (Definitions 3 and 4) to
$\mathbf{u^{DA}}$, we get

$\displaystyle\begin{split}E_{g}=\left\|\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\left[(\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{k}))^{EO}-{\mathbf{\widetilde{u}_{i,k}}}\right]\right\|\end{split}$

from triangle inequality, it is

$\displaystyle\begin{split}E_{g}=\left\|\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\left[(\mathbf{u^{DA}}/(\Omega_{i}\times\Delta_{k}))^{EO}-{\mathbf{\widetilde{u}_{i,k}}}\right]\right\|\\\
\leq\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}\left\|(\mathbf{u_{i,k}^{DA}})^{EO}-{\mathbf{\widetilde{u}_{i,k}}}\right\|=\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}E_{i,k}\end{split}$
(82)

where $E_{i,k}$ is defined in (62). From Lemma 0.3.3 we have

$\displaystyle\begin{split}\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}E_{i,k}&\leq
c\cdot\sum_{i=1}^{N_{sub}}\sum_{k=1}^{N_{t}}((\Delta x)_{i}^{p}+\cdot(\Delta
t)_{k}^{q})\\\ &=c\cdot\left[N_{t}\cdot\sum_{i=1}^{N_{sub}}(\Delta
x)_{i}^{p}+N_{sub}\sum_{k=1}^{N_{t}}(\Delta t)_{k}^{q}\right]\end{split}$ (83)

consequently

$\displaystyle\begin{split}E_{g}\leq
c\cdot\left[N_{t}\cdot\sum_{i=1}^{N_{sub}}(\Delta
x)_{i}^{p}+N_{sub}\sum_{k=1}^{N_{t}}(\Delta t)_{k}^{q}\right].\end{split}$

By defining

$\displaystyle\begin{split}\Delta x&:=max_{i=1,\ldots,N_{sub}}(\Delta
x)_{i}\\\ \Delta t&:=max_{k=1,\ldots,N_{t}}(\Delta t)_{k}\end{split}$

we get

$\displaystyle\begin{split}E_{g}&\leq
c\cdot\left[N_{t}\cdot\sum_{i=1}^{N_{sub}}(\Delta
x)_{i}^{p}+N_{sub}\sum_{k=1}^{N_{t}}(\Delta t)_{k}^{q}\right]\\\ &\leq
c\cdot\left[N_{t}\cdot\sum_{i=1}^{N_{sub}}(\Delta
x)^{p}+N_{sub}\sum_{k=1}^{N_{t}}(\Delta t)^{q}\right]\\\ &\leq
c\cdot(N_{sub}\cdot N_{t})\cdot[(\Delta x)^{p}+(\Delta t)^{q}].\end{split}$

Hence, the (81) is proved.

In Figure 4, it is schematic description of results obtained from DD–4DVAR
local truncation analysis.

Now, we prove the stability of DD–4DVAR method by studying the propagation
error from each time interval to the next, assuming that the predictive model
is stable.

###### Assumption 0.3.2

(Stability of model discrete scheme) The discrete scheme applied to
$\mathcal{M}$ in (1) is stable scheme i.e. $\exists D>0$ such that

$\|u^{M}-v^{M}\|\leq D\cdot e_{0},$ (84)

where

  * •

$u^{M}$: is solution of $M$ in (9);

  * •

$v^{M}$: is solution of $\bar{M}$, where $\bar{M}$ is obtained by considering
initial error $e_{0}$ on initial condition;

  * •

$e_{0}$: is initial error on initial condition of $M$ in (9).

###### Definition 8

(Propagation error from $\Delta_{k-1}$ to $\Delta_{k}$) Let us consider

  * •

$(\Delta_{1},\Delta_{2},\ldots,\Delta_{N_{t}})$: be the decomposition of
$\Delta$ (DD module);

  * •

$\mathbf{\widetilde{u}^{DD-DA}}$: be DD–4DVAR solution in $\Omega\times\Delta$
defined in (37);

  * •

${\mathbf{\tilde{v}^{DD-DA}}}$: be DD–4DVAR solution in $\Omega\times\Delta$
computed by adding perturbation $e_{k}$ to initial condition of
$P_{i,k}^{M_{i,k},n}$ defined in (22).

We define

$\bar{E}_{k}:=\|\mathbf{\tilde{u}^{DD-DA}}/\Delta_{k}-{\mathbf{\tilde{v}^{DD-
DA}}}/\Delta_{k}\|$ (85)

the propagation error from $\Delta_{k-1}$ to $\Delta_{k}$.

In the following, result of stability of DD–4DVAR method.

###### Theorem 0.3.4

(Stability of DD–4DVAR method) If $e_{0}$, the error on initial condition of
${M}$ in (9), is equal to zero i.e.

$e_{0}=0$ (86)

then, $\forall k=1,\ldots,N_{t}$ $\exists\ C_{k}>0$ such that

$\bar{E}_{k}\leq C_{k}\cdot\bar{e}_{k}$

where

  * •

$\bar{E}_{k}$: is propagation error from $\Delta_{k-1}$ to $\Delta_{k}$
defined in (85);

  * •

$C_{k}$: is a constant dependind on the model reduction and on ASM modules of
DD–4DVAR method;

  * •

$\bar{e}_{k}$: perturbation on initial condition of $P_{i,k}^{M_{i,k}}$
defined in (22).

Proof. To simplify the notations in the proof, we consider
$\bar{e}_{k}=\bar{e},\ \forall k=1,\ldots,N_{t}$. From (35), (59) and triangle
inequality, we get

$\displaystyle\begin{split}\bar{E}_{k}:=\|\mathbf{\tilde{u}^{DD-
DA}}/\Delta_{k}-{\mathbf{\tilde{v}^{DD-
DA}}}/\Delta_{k}\|&\leq\|(u_{i,k}^{M_{i,k},\bar{n}})^{EO}/\Delta_{k}-(v_{i,k}^{M_{i,k},\bar{n}})^{EO}/\Delta_{k}\|\\\
&+\|(u_{i,k}^{M_{i,k},\bar{n}-1})^{EO}/\Delta_{k}-(v_{i,k}^{M_{i,k},\bar{n}-1})^{EO}/\Delta_{k}\|\\\
&+\|(u_{i,k}^{ASM,\bar{n}})^{EO}/\Delta_{k}-(v_{i,k}^{ASM,\bar{n}})^{EO}/\Delta_{k}\|.\end{split}$
(87)

From (74) and (61), we can neglect the dependency on outer loop iteration
$\bar{n}$ i.e.

$\displaystyle\begin{split}\bar{E}_{k}\leq&2\cdot\|(u_{i,k}^{M_{i,k}})^{EO}/\Delta_{k}-(v_{i,k}^{M_{i,k}})^{EO}/\Delta_{k}\|+\|(u_{i,k}^{ASM})^{EO}/\Delta_{k}-(v_{i,k}^{ASM})^{EO}/\Delta_{k}\|.\end{split}$
(88)

From Assumption 0.3.2, we get that $\exists\bar{D}>0$ such that

$\displaystyle\begin{split}\|(u_{i,k}^{M_{i,k}})^{EO}/\Delta_{k}-(v_{i,k}^{M_{i,k}})^{EO}/\Delta_{k}\|\leq\bar{D}\cdot
e_{0}.\end{split}$ (89)

where $e_{0}$ is the error on initial condition of ${M}$ in (9). By adding and
subtracting $\mathbf{u^{DA}}/\Delta_{k}$ to
$\left[(u_{i,k}^{ASM})^{EO}/\Delta_{k}-(v_{i,k}^{ASM})^{EO}/\Delta_{k}\right]$
and from triangle inequality, it is

$\displaystyle\begin{split}\|(u_{i,k}^{ASM})^{EO}/\Delta_{k}-(v_{i,k}^{ASM})^{EO}/\Delta_{k}\|\leq\|\mathbf{u^{DA}}/\Delta_{k}-(u_{i,k}^{ASM})^{EO}/\Delta_{k}\|\\\
+\|\mathbf{u^{DA}}/\Delta_{k}-(v_{i,k}^{ASM})^{EO}/\Delta_{k}\|.\end{split}$
(90)

From (90) and (64), we get

$\displaystyle\begin{split}\|(u_{i,k}^{ASM})^{EO}/\Delta_{k}-(v_{i,k}^{ASM})^{EO}/\Delta_{k}\|\leq\mu_{k}^{DD-
DA}\cdot E_{1}^{ASM}+\bar{\mu}_{k}^{DD-DA}\cdot\bar{E}_{1}^{ASM}\end{split}$
(91)

where

$\displaystyle\begin{split}E_{1}^{ASM}&=\|\mathbf{u^{DA}}/\Delta_{1}-(u_{i,1}^{ASM})^{EO}/\Delta_{1}\|\\\
\bar{E}_{1}^{ASM}&=\|\mathbf{u^{DA}}/\Delta_{1}-(v_{i,1}^{ASM})^{EO}/\Delta_{1}\|.\end{split}$
(92)

and

$\displaystyle\begin{split}\mu_{k}^{DD-
DA}&:=\left[1+\frac{1}{\sigma_{0}^{2}}\mu^{2}(\mathbf{V})\mu^{2}(G/\Delta_{k})\right]\mu({M}/\Delta_{k})\\\
\bar{\mu}_{k}^{DD-
DA}&:=\left[1+\frac{1}{\sigma_{0}^{2}}\mu^{2}(\mathbf{V})\mu^{2}(G/\Delta_{k})\right]\mu(\bar{M}/\Delta_{k})\\\
\end{split}$ (93)

with $\sigma_{0}$ observational error variance,
$\mathbf{B}=\mathbf{V}\mathbf{V^{T}}$ covariance matrix of the error on
background to $\Omega$, $G$ matrix defined in (• ‣ 2), $M$ defines in (9) and
$\bar{M}$ is discrete model obtained by considering initial error $e_{0}$ on
initial condition of $M$. By applying (64) to $E_{1}^{ASM}$ and
$\bar{E}_{1}^{ASM}$ in (92), we get

$\displaystyle\begin{split}\|(u_{i,k}^{ASM})^{EO}/\Delta_{k}-(v_{i,k}^{ASM})^{EO}/\Delta_{k}\|\leq\mu_{k}^{DD-
DA}\cdot\mu_{1}^{DD-DA}{e}_{0}+\bar{\mu}_{k}^{DD-DA}\cdot\bar{\mu}_{k}^{DD-
DA}\bar{e}.\end{split}$ (94)

From (88), (89) and (91), it is

$\displaystyle\begin{split}\bar{E}_{k}\leq 2\cdot\bar{D}\cdot
e_{0}+\mu_{k}^{DD-DA}\cdot\mu_{1}^{DD-DA}{e}_{0}+\bar{\mu}_{k}^{DD-
DA}\cdot\bar{\mu}_{1}^{DD-DA}\bar{e}\end{split}$

and from hypothesis in (86), we get

$\displaystyle\begin{split}\bar{E}_{k}\leq\bar{\mu}_{k}^{DD-
DA}\cdot\bar{\mu}_{1}^{DD-DA}\bar{e}\end{split}$ (95)

Consequently, for $k=1,\ldots,N_{t}$ we get that $\exists\ C_{k}>0$ such that

$\displaystyle\begin{split}\bar{E}_{k}\leq C_{k}\cdot\bar{e}\end{split}$ (96)

where

$C_{k}:=\bar{\mu}_{k}^{DD-DA}\cdot\bar{\mu}_{1}^{DD-DA}.$ (97)

The thesis is proved.

From Theorem 0.3.4, we get the stability of DD–4DVAR method.

Remark (Condition number of DD–4DVAR method) We note that in Theorem 0.3.4, to
prove the stability of DD–4DVAR method, we study the propagation error
according to forward error analysis. In particular, we get that the constant
$C_{k},\ \forall k=1,\ldots,N_{t}$ in (97) depends on the quantity
$\bar{\mu}_{k}^{DD-DA}$ defined in (93), which is in turn depends on condition
numbers of $M$ in (9), $G$ in (• ‣ 2) and
$\mathbf{B}=\mathbf{V}\mathbf{V^{T}}$ covariance matrix of the error on
background in $\Omega$. As a consequence of the forward error analysis, we may
say that the quantity $\bar{\mu}_{k}^{DD-DA}$ can be regarded as the condition
number of DD–4DVAR method.

Figure 4: Schematic description of DD–4DVAR local truncation analysis.

##  0.4 Validation analysis

Validation is performed on the high performance hybrid computing architecture
of the SCoPE (Sistema Cooperativo Per Elaborazioni scientifiche
multidiscipliari) data center, located in the University of Naples Federico
II. Specifically, the HPC architecture is composed by 8 nodes which consist of
distributed memory DELL M600 blades. The blades are connected by a 10 Gigabit
Ethernet technology and each of them is composed of 2 Intel<EMAIL_ADDRESS>quadcore processors sharing the same local 16 GB RAM memory for a number of 8
cores per blade and of 64 total cores. Experimental results allows to deduce
that the experimental order of DD–4DVAR consistency corresponds to the
theoretical one obtained in Theorem 0.3.3 and DD–4DVAR is well-conditioned. We
consider the following experimental set up.

4DVAR DA set up.

  * •

$\Omega=(0,1)\subset\mathbb{R}$: spatial domain;

  * •

$\Delta=[0,1.5]\subset\mathbb{R}$: time interval;

  * •

$N_{p}$: numbers of inner nodes of $\Omega$ defined in (3);

  * •

$N$: numbers of instants of time in $\Delta$;

  * •

$n_{obs}=64$: number of observations considered at each step $l=0,1,\ldots,N$;

  * •

$y\in\mathbb{R}^{N\cdot n_{obs}}$: observations vector at each step
$l=0,1,\ldots,N$;

  * •

$H_{l}\in\mathbb{R}^{n_{obs}\times N_{p}}$: piecewise linear interpolation
operator whose coefficients are computed using the nodes of $\Omega$ nearest
to the observation values;

  * •

$G\in\mathbb{R}^{N\cdot n_{obs}\times N_{p}}$: obtained as in (• ‣ 2) from the
matrix $H_{l}$, $l=0,1,\ldots,N$;

  * •

$\sigma_{m}^{2}=0.5$, $\sigma_{0}^{2}=0.5$: model and observational error
variances;

  * •

$\textbf{B}\equiv B_{l}=\sigma_{m}^{2}\cdot C$: covariance matrix of the error
of the model at each step $l=0,1,\ldots,N$, where $C\in\mathbb{R}^{N_{p}\times
N_{p}}$ denotes the Gaussian correlation structure of the model errors in
(98);

  * •

${R}_{l}=\sigma_{0}^{2}\cdot I_{n_{obs},n_{obs}}\in\mathbb{R}^{n_{obs}\times
n_{obs}}$: covariance matrix of the errors of the observations at each step
$l=0,1,\ldots,N-1$.

  * •

$\textbf{R}\in\mathbb{R}^{N\cdot n_{obs}\times N\cdot n_{obs}}$: a diagonal
matrix obtained from the matrices $R_{l}$, $l=0,1,\ldots,N-1$.

DD-4DVAR set up: we consider the following set up:

  * •

$p$: number of cores;

  * •

$N_{sub}\equiv p$: number of spatial subdomains;

  * •

$N_{t}$: number of time intervals;

  * •

$n_{1}=n_{nsub}=1$ and $n_{2}=n_{3}=\ldots=n_{nsub-1}=2$: number of subdomains
adjacent to $\Omega_{1},\ \Omega_{nsub}$ and $\Omega_{2},\
\Omega_{3},\ldots,\Omega_{nsub-1}$, respectively;

  * •

$\delta$: number of inner nodes of overlap regions defined in (18);

  * •

$N_{loc}$: inner nodes of subdomains defined in (19);

  * •

$\Delta x$ and $\Delta t$: spatial and temporal step sizes of $M_{i,k}$
defined in (23);

  * •

$C:=\\{c_{i,j}\\}_{i,j=1,\ldots,N_{p}}\in\mathbb{R}^{N_{p}\times N_{p}}$: the
Gaussian correlation structure of the model error where

$c_{i,j}=\rho^{|i-j|^{2}},\quad\rho=exp\left(\frac{-\Delta
x^{2}}{2}\right),\quad|i-j|<N_{p}/2\quad\begin{array}[]{ll}\textrm{for}\
i,j=1,\ldots,{N_{p}}\end{array}.$ (98)

We introduce

$e^{p}=e^{p}(\delta,\Delta x,\Delta
t,N_{sub},N_{t}):=\|\mathbf{u^{DA}}-\mathbf{\tilde{u}^{DD-DA}}\|_{2},$ (99)

where $\mathbf{u^{DA}}$ denotes the minimum of the 4DVAR (global) functional
$\mathbf{J}$ in (7) while $\mathbf{\tilde{u}^{DD-DA}}$ is obtained by
gathering minimum of the local 4DVar functionals $\mathbf{J}_{i,k}$ in (27) by
considering different values of $\delta$ defined in (18) and $p\equiv
N_{sub}$. $\mathbf{u^{DA}}\in\mathbb{R}^{N_{p}\times N}$ is computed by
running DD-4DVar algorithm for $N_{sub}=1$, while $\mathbf{\tilde{u}^{DD-
DA}}\in\mathbb{R}^{N_{p}\times N}$ is computed by gathering local solutions
obtained by running DD-4DVAR algorithm for different values of $N_{sub}>1$
with $\delta\geq 0$, as shown in Figure (5).

In the following, we present experimental results of the consistency and
stability of DD–4DVAR method considering the initial boundary problem of the
SWEs 1D. The discrete model is obtained using Lax–Wendroff scheme [4] on
$\Omega\times\Delta$ where the orders of convergence in space and time are
$p=q=2$.

  * •

Consistency of DD–4DVAR method.

From Table 1.1, we get333By fixing the values of $N_{sub}=N_{t}=4$ and
$\delta=2$, we get $e^{p}(2,\Delta x,\Delta t,4,4)={e}^{p}(\Delta x,\Delta t)$
i.e. $e^{p}$ defined in (99) depends only on $\Delta x$ and $\Delta t$.

$e^{p}\left(\frac{\Delta x}{d},\frac{\Delta
t}{d}\right)\approx\frac{e^{p}(\Delta x,\Delta t)}{d^{2}}\quad
d=1,2,4,6,8,10.$ (100)

As shown in Table 1.1 and Figure 1.6, the experimental order of consistency
corresponds to the theoretical one obtained in Theorem 0.3.3.

  * •

Stability of DD–4DVAR method.

In Table 1.2 and Figure 1.7, we report values of $\bar{E}_{k}$ for different
values of perturbation $\bar{e}_{k}$ on initial condition of
$P_{i,k}^{M_{i,k}}$ defined in (22). From the values in Table 1.2, we
experimentally estimate $C_{k}$ in (97), in particular

$C_{k}\approx 2.00\times 10^{1}\quad\forall k=1,\ldots,N_{t}.$

Consequently, DD–4DVAR with the initial boundary problem of SWEs 1D is well-
conditioned.

$d$ | $\frac{\Delta x}{d}$ | $\frac{\Delta t}{d}$ | $e^{p}\left(\frac{\Delta x}{d},\frac{\Delta t}{d}\right)$ | $\frac{e^{p}(\Delta x,\Delta t)}{d^{2}}$
---|---|---|---|---
1 | $7.87\times\times 10^{-3}$ | $1.09\times 10^{-1}$ | $1.53\times 10^{-2}$ | $1.53\times 10^{-2}$
2 | $3.92\times\times 10^{-3}$ | $5.47\times 10^{-2}$ | $9.01\times 10^{-4}$ | $3.83\times 10^{-3}$
4 | $1.96\times\times 10^{-3}$ | $2.74\times 10^{-2}$ | $6.45\times 10^{-4}$ | $9.56\times 10^{-4}$
6 | $1.30\times\times 10^{-3}$ | $1.83\times 10^{-2}$ | $3.65\times 10^{-4}$ | $2.39\times 10^{-4}$
8 | $9.78\times\times 10^{-4}$ | $1.37\times 10^{-2}$ | $3.99\times 10^{-4}$ | $4.25\times 10^{-4}$
10 | $7.81\times\times 10^{-4}$ | $1.10\times 10^{-2}$ | $3.77\times 10^{-4}$ | $1.53\times 10^{-4}$
Table 1: Fixed $N_{p}=640$ number of inner nodes in $\Omega$, $N=9$ number of instants of time in $\Delta$, $N_{sub}=4$ number of spatial subdomain and $N_{t}=4$ time intervals. We report the values of $e^{p}$ defined in (99) for different values of $\Delta x$ and $\Delta t$ spatial and temporal step sizes of $M_{i,k}$ defined in (23). $\bar{e}_{k}$ | $\bar{E}_{k}$
---|---
$3.03\times 10^{-6}$ | $6.06\times 10^{-5}$
$3.03\times 10^{-5}$ | $6.06\times 10^{-4}$
$3.03\times 10^{-4}$ | $6.06\times 10^{-3}$
$3.03\times 10^{-3}$ | $6.06\times 10^{-2}$
Table 2: Fixed $N_{p}=640$ number of inner nodes in $\Omega$, $N=20$ number of
instants of time in $\Delta$, $N_{sub}=4$ number of spatial subdomains and
$N_{t}=4$ time intervals. For $k=1,2,3,4$, we report the values of
$\bar{E}_{k}$ defined in (85) for different values of perturbation
$\bar{e}_{k}$ to initial condition of $P_{i,k}^{M_{i,k}}$ defined in (22).
Figure 5: Decomposition of spatial domain $\Omega\subset\mathbb{R}$ in two
subdomains $\\{\Omega_{i}\\}_{i=1,2}$ by identifying overlap region
$\Omega_{12}$ defined in (13) and interfaces $\Gamma_{12}$ and $\Gamma_{21}$
defined in (14). On the left case $\delta=0$ i.e. no inner nodes in
$\Omega_{12}$, on the right case $\delta=2$ i.e. two inner nodes in overlap
region $\Omega_{12}$.
Figure 6: Plot of values of $e^{p}\left(\frac{\Delta x}{d},\frac{\Delta
t}{d}\right)$ (orange dashed line) and $\frac{e^{p}(\Delta x,\Delta
t)}{d^{2}}$ (blue full line) for $d=1,2,4,6,8,10$ reported in Table 1.1.
Figure 7: Plot of values $(\bar{e}_{k},\bar{E}_{k})$ reported in Table 1.2.

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