# Solving the Cauchy problem in a cylinder by using regularizations of the factorized formulation

Jacques Henry, Andrès Fraguela Collar

<figure><img src="image_1.png" /></figure>

## To cite this version:

Jacques Henry, Andrès Fraguela Collar. Solving the Cauchy problem in a cylinder by using regularizations of the factorized formulation. 2025. hal-05209575

HAL Id: hal-05209575

https://inria.hal.science/hal-05209575v1

Preprint submitted on 14 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Solving the Cauchy problem in a cylinder
by using regularizations of the
factorized formulation<fnref n="1" />

J. Henry<fnref n="2" />

Centre INRIA de l'université de Bordeaux

A. Fraguela Collar

Benemérita Universidad Autónoma de Puebla

## Abstract

The Cauchy problem is a severely ill-posed problem that requires a regularization to be solved numerically. Usually the problem is regularized either by a quasi reversibility method or by minimizing the distance between solutions satisfying each of the boundary conditions with respect to the unknown boundary condition ([14]). Here we consider the factorized version of the direct problem ([8]) which yields an ill posed equation for the inverse problem. We propose four regularizations that we analyze using the Morozov principle ([16]).

## 1 Introduction

The inverse cardiac electrography technique is in full development. It aims at building a map of the electric potential on the epicardium by measuring of the electric potential on the thorax. This information and its evolution over time during a heartbeat are very useful for cardiologists because it is much richer than the classic electrocardiogram while also being non-invasive. This problem is modeled in the framework of electrostatics assuming a homogeneous thorax and devoid of charge. It is described by a Poisson equation. The idea of solving a boundary value problem by a spatial invariant embedding goes back to R. Bellman ([1]), ([2]). It uses the theory of optimal control of partial differential equations ([3]). This way leading to a factorization of the state equation was developed in ([4]), ([8]). From an electrical point of view it is necessary to find the distribution of potential on a body immersed in another knowing the distribution of potential on the outer surface of the surrounding body assuming zero normal current on this border. This is a Cauchy problem that is known to be ill-posed: the data must satisfy a condition of compatibility that we cannot

<footnote n="1">¹Partially supported by ECOS nord contract *Inverse problems in cardiology* (M19M02)</footnote>

<footnote n="2">²corresponding author</footnote>
1 formulate simply. Moreover the solution if it exists is unstable with respect to
2 the data. In practice, the solution to this problem involves regularization: the
3 construction of a close problem admitting a stable solution. Among the various
4 ways to build a regularization let cite quasireversibility ([17]). This method
5 involves the perturbation of the state equation using higher order operator
6 that satisfies all boundary value conditions. It was developed and analyzed
7 in ([18]) and ([19]). In ([12]) a Yosida regularization was used within a quasi-
8 reversibility approach. A large number of regularizations have been proposed.
9 Among them the Kohn Vogelius method ([14]) is very popular.

We propose several regularizations using the formulation of the direct problem set in a factorized form such as that in the book [8]. This new formulation was obtained using a spatial invariant embedding method. Using this factorized version is the main component for building new regularizations of the inverse problem.

In this factorized formulation, the Cauchy problem requires to solving a
parabolic equation in the backward direction, which reflects the ill-posed na-
ture of the initial problem. This equation can be regularized by adding a term
with higher order derivatives. However it is also proposed to introduce into
the regularizing term the difference with a solution of the parabolic problem
but set in the direction where there is stability. This new term can be consid-
ered as a disturbance that improves stability and decreases error. In addition
solving the parabolic equation in the unstable direction can be approached by
a regularized control problem where the control is the state to be estimated
and the cost is based on the distance to the measures. This is close to the
Kohn-Vogelius method ([14]) but is applied here to the factorized formulation.

The invariant embedding factorization method in ([8]) can be applied to
various geometries (e.g. the thorax). In this study which carries out different
regularizations we limit ourselves to a cylindrical geometry where invariant
embedding is very simple to implement and for which explicit calculations can
be made.

In the usual method of error analysis the estimate shows two terms

32 • a term in ε, the regularization parameter, that goes to 0 when ε goes to
33 0

34 • a term in $\delta C(\varepsilon)$ where $\delta$ is the order of magnitude of the measurement error and $C(\varepsilon)$ goes to $\infty$ as $\varepsilon$ goes to 0.

In the first method proposed here regulation concerns a term with higher order
derivatives with respect to the direction orthogonal to the embedding. This
analysis reveals a term in $C(\varepsilon)\delta$ but the coefficient $C(\varepsilon)$ grows very fast for $\varepsilon$
tending towards 0 which shows the nature of this inverse problem as a severely
ill posed problem, when the a priori information required for the solution has a
low level of smoothness. In the second method, a reference term is introduced into the regulation to consider the distance to the solution calculated in the backward direction.

Finally, another method is proposed where it is sought to jointly estimate the values of the solution on the part of the edge without measured data but also on that where there are measurements. By doing so we try to consider the fact that these measurements may be corrupted by errors. These results produce estimations on the edge without measurement less sensitive to measurement errors. We were inspired for this by [11] which shows that one can stabilize an unstable problem by plunging it into a control problem. On the other hand, the idea presented in [13] has also been taken up to jointly estimate the solution on the part of the edge without data but also admissible data close to the measurements on the measured part of the boundary.

In Section 2 we expose the problem statement in the case of a cylindrical domain and we recall the factorization method presented in [8]

In Section 3 we present and analyze four methods of regularization of the unstable Cauchy problem. For some of them the unstable part of the factorized problem is regularized by a parabolic problem and uses the reference to a stable problem set backward.

## 2 Problem statement and method

Consider Cauchy's problem for the Laplace equation in the "simplified" domain with a cylindrical shape. Indeed, for this geometry, invariant embedding is expressed in a simple manner. The case where the domain is between the skin and epicardium is studied in [8]. It is dealt with in a similar manner when considering the domains bounded by a family of surfaces sweeping over the domain by deforming continuously between the internal and external boundaries of the domain.

First we mathematically define the problem to be solved.

Laplace's problem is considered on the cylinder $\Omega$ of length $a$ and section $\mathcal{O}$ such that $\Omega = ]0; a[ \times \mathcal{O}$. Let $\Sigma = ]0; a[ \times \partial\mathcal{O}$ be the lateral border of the cylinder. The solution is considered null in this part of the domain. $\Gamma_0$ and $\Gamma_a$ denote the faces of the cylinder such that $\Gamma_0 = \{0\} \times \mathcal{O}$ et $\Gamma_a = \{a\} \times \mathcal{O}$ and the coordinate along the axis of the cylinder between these two surfaces is denoted by $x$. The Cauchy data of our problem are known on the surface in $x = 0$: $u = T$ et $\frac{\partial u}{\partial n} = \Phi$ on $\Gamma_0$ (where $n$ is the external normal to the
1 considered surface). The problem can then be written as follows:

$$ (\mathcal{P}_0) \left\{ \begin{array}{l} \Delta u = 0 \text{ in } \Omega \\ u = 0 \text{ on } \Sigma \\ u = T, \frac{\partial u}{\partial n} = \Phi \text{ on } \Gamma_0 \end{array} \right. \qquad (1) $$

2 In practice, for the inverse ECG problem $\Phi = 0$. We attempt to determine the

3 Dirichlet and Neumann conditions on the surface in $x = a$ which completes the

4 problem $(\mathcal{P}_0)$, that is to determine the data of potential $t$ and its derivative $\phi$

5 on the surface $\Gamma_a$ as follows:

$$ \left\{ \begin{array}{l} \Delta u = 0 \text{ in } \Omega \\ u = 0 \text{ on } \Sigma \\ u = T, \frac{\partial u}{\partial n} = \Phi \text{ on } \Gamma_0 \\ u = t, \frac{\partial u}{\partial n} = \phi \text{ on } \Gamma_a \end{array} \right. \qquad (2) $$

6 This Cauchy problem appears as an inverse problem of one of the following

7 boundary problems:

$$ (\mathcal{P}_1) \left\{ \begin{array}{l} \Delta u = 0 \text{ in } \Omega \\ u = 0 \text{ on } \Sigma \\ u = T, \text{ on } \Gamma_0 \\ \frac{\partial u}{\partial n} = \phi \text{ on } \Gamma_a \end{array} \right. \qquad (3) $$

8 or

$$ (\mathcal{P}_2) \left\{ \begin{array}{l} \Delta u = 0 \text{ in } \Omega \\ u = 0 \text{ on } \Sigma \\ \frac{\partial u}{\partial n} = \Phi \text{ on } \Gamma_0 \\ u = t, \text{ on } \Gamma_a \end{array} \right. \qquad (4) $$

9 **2.1 The boundary value problem in factorized form**

10 The method of invariant embedding presented in [8] provides an equivalent

11 form called factorized for problems $(\mathcal{P}_1)$ and $(\mathcal{P}_2)$. For this we consider the

12 family of problems similar to $(\mathcal{P}_2)$ but defined in the subcylinder $\Omega_x = ]0; x[ \times \mathcal{O}$

13 with the Dirichlet condition $u(x)$ given on the section of the cylinder in $x$,

14 $\Gamma_x = \{x\} \times \mathcal{O}$. This makes it possible to determine the operator $P(x)$ which

15 to the data $u(x)$ on $\Gamma_x$ for the problem on $\Omega_x$ provides the Neumann condition
1 $\frac{\partial u}{\partial x}$ on $\Gamma_x$ for the otherwise homogeneous problem. In [8], the properties of the

2 operator $P$ are studied. In particular it is affine and continuous from $H_{00}^{1/2}(\mathcal{O})$

3 which is defined in [6], to its dual $H_{00}^{1/2}(\mathcal{O})'$. It can be extended as an operator

4 from $H_0^1(\mathcal{O})$ to $L^2(\mathcal{O})$ and from $L^2(\mathcal{O})$ to $H^{-1}(\mathcal{O})$.

5 is as follows.

$$ \frac{\partial u}{\partial x} = P(x)u(x) + r(x), \qquad (5) $$

6 Equation (5) describes the relation between the Dirichlet and Neumann conditions on the $\Gamma_x$ boundary. It can also be considered as a differential equation

7 for $u$ with respect to $x$. The bilinear form associated with $P$ is coercive.

8 The operator $P$ satisfies the Riccati equation:

$$ (P(x)u, u) \geq \alpha(x) \|u\|_{H_{00}^{1/2}(\mathcal{O})}^2, \quad \alpha \geq 0, \quad \forall u \in H_{00}^{1/2}(\mathcal{O}) \qquad (6) $$

9 The operator $P$ satisfies the Riccati equation:

$$ \frac{\partial P}{\partial x} + P^2 + \Delta_y = 0, \quad P(0) = 0. \qquad (7) $$

10 Here, $\Delta_y$ represents the Laplacian with $(n-1)$ variables acting on the functions

11 of $\mathcal{O}$. This equation is well posed for $x$ increasing from 0 to $a$. Residual $r$

12 satisfies the following equation:

$$ \frac{\partial r}{\partial x} + P(x)r(x) = 0, \quad r(0) = -\Phi \qquad (8) $$

13 This equation is of the abstract parabolic type owing to the positivity of $P$ and

14 is therefore well posed in the direction of increasing $x$. We can also consider

15 (5) as a well-posed parabolic equation in the direction of decreasing $x$ once $r$

16 is calculated including the final condition:

$$ u(a) = t. \qquad (9) $$

17 Thus, once Riccati equation (7) is solved, the solution of problem (4) is obtained using two uncoupled initial value problems (8) and (5) which are posed

18 one in the direction of increasing $x$ and the other in the opposite direction.

19 One can also use factorization with the help of the Neumann-Dirichlet operator $Q$ which is also presented in the book [8] (we invert here the direction of

20 immersion by the change $x \to a - x$). It verifies

21

22 immersion by the change $x \to a - x$). It verifies

$$ -\frac{\partial Q}{\partial x} + Q\Delta_y Q + I = 0, \quad Q(0) = 0, \qquad (10) $$

23 The factorized version of (3) is

$$ -\frac{\partial r'}{\partial x} + Q(x)\Delta_y r'(x) = 0, \quad r'(0) = T, \qquad (11) $$

$$ -Q(x)\frac{\partial u}{\partial x} + u = r', \quad u(a) = Q(a)\phi + r'(a). \qquad (12) $$
### 3 Regularization of the Cauchy problem thanks to the factorization using the Dirichlet-Neumann operator.

To solve approximately the inverse problem ($\mathcal{P}_0$) we propose four methods using the regularization of the equation using the Dirichlet-Neumann operator.

Then we make a comparison with the classical Kohn-Vogelius method.

Now, consider the problem of inverse ECG ($\mathcal{P}_0$) with zero flux on $\Gamma_0$ so $\Phi = 0$. Equation (8) implies $r = 0$. Therefore, in this case the Cauchy problem is summed up in the integration of Equation (5) in the sense of increasing $x$ with the initial condition $u(0) = T$.

This problem is of the backward parabolic type: therefore it is ill-posed. This is a way to highlight the ill-posed character of this Cauchy problem. We then propose regularization of this backward parabolic problem to solve the Cauchy problem. This problem is written

$$ \frac{\partial u}{\partial x} = P(x)u(x), \quad u(0) = T. \qquad (13) $$

Note that this formulation contains the Cauchy data of ($\mathcal{P}_0$) since $P(0) = 0$ which implies $\frac{\partial u}{\partial x}(0) = 0$.

#### 3.1 General properties of the inverse problem

**Definition 1** We shall say that $(T, \Phi)$ are compatible if there exists a solution to (1), that is if there exists $t \in L^2(\Gamma_a)$ such that the solution of

$$ \begin{align} \Delta u &= 0 \quad \text{in } \Omega \notag \\ u &= 0 \quad \text{on } \Sigma \notag \\ u &= T, \quad \text{on } \Gamma_0 \notag \\ u &= t, \quad \text{on } \Gamma_a \tag{14} \end{align} $$

satisfies furthermore

$$ \frac{\partial u}{\partial x}(0) = \Phi \quad \text{on } \Gamma_0. $$

In the following we consider only the case $\Phi = 0$. Let $X$ be the space of $T$ such that $(T, 0)$ is compatible.

**Definition 2** We say that the data $T$ and $0$ on $\Gamma_0$ are compatible with order
1. $k$ if there exists $t \in H^k(\Gamma_a)$ such that the solution of

$$ 
\begin{align} 
\Delta u &= 0 \quad \text{in } \Omega \\ 
u &= 0 \quad \text{on } \Sigma \\ 
u &= T, \quad \text{on } \Gamma_0 \\ 
u &= t, \quad \text{on } \Gamma_a 
\end{align} 
\tag{15} $$

satisfies furthermore

$$ \frac{\partial u}{\partial x}(0) = 0 \quad \text{on } \Gamma_0 $$

2.

3. We note $|\cdot|$ the norm in $L^2(\mathcal{O})$ and $\|\cdot\|$ the norm in $H_0^1(\mathcal{O})$. We explicitly used

4. the cylindrical geometry of $\Omega$. Let $\{\varphi_n\}_{n=1}^\infty$ be the basis of the eigenfunctions

5. of $-\Delta_y$ on $\mathcal{O}$ with eigenvalues $\lambda_n$. We get

$$ -\Delta_y \varphi_n = \lambda_n \varphi_n. \tag{16} $$

6. Let

$$ u(x) = \sum_{n=1}^{\infty} u^n(x) \varphi_n, \tag{17} $$

7. an expansion on this basis. It has been shown in [8] that operator $P$ is diagonalized in this basis

8.

$$ P(x)\varphi_n = \pi_n(x)\varphi_n, \quad x \in [0, a], \text{ with } \pi_n(x) = \alpha_n \tanh \alpha_n x, \quad \alpha_n = \sqrt{\lambda_n}. \tag{18} $$

9. The space of $T$ such that $(T, 0)$ is compatible is called $X$. Let $\|\cdot\|_X$ be

10. defined as

$$ \|T\|_X^2 = \sum_{n=1}^{\infty} |T_n|^2 \exp\left(2 \int_0^a \pi_n(s) ds\right). \tag{19} $$

11. **Proposition 3.1** *X is a Hilbert space with the norm $\|\cdot\|_X$.*

12. **Proof**

If $T \in X$, using the expansion on the basis $\{\varphi_n\}_{n=1}^\infty$ (17) we can solve equation (13) for each component.

$$ u^n(x) = \exp\left(\int_0^x \pi(\xi) \, d\xi\right) T_n. $$

13. Then it is clear that we can define $u$ in $L^2$ if $T$ belongs to $X$ and $X$ is a Hilbert

14. endowed with $\|\cdot\|_X$. $\square$
1 **Proposition 3.2** The space of compatible data $(X, 0)$ is dense in $(L^2(\mathcal{O}), 0)$.

*Proof* Assume that property is false. There exists $\eta \neq 0 \in L^2(\mathcal{O})$ such that

$$ (\eta, \bar{u}(0; u_1)) = 0, \quad \forall u_1 \in L^2(\mathcal{O}), \qquad (20) $$

$$ \text{with } \bar{u} \text{ verifying (62) and } \bar{u}(a) = u_1. \qquad (21) $$

2 Let be $\xi$ such that

$$ \frac{\partial \xi}{\partial x} + P\xi = 0, \quad \xi(0) = \eta. \qquad (22) $$

3 Then

$$ (\xi(a; \eta), u_1) = 0, \forall u_1 \text{ and so } \xi(a; \eta) = 0, \qquad (23) $$

4 but also

$$ \frac{\partial \xi}{\partial x}(a) = 0. \qquad (24) $$

5 By analyticity

$$ \xi(x) = 0 \quad \text{in a neighborhood of } a. \qquad (25) $$

6 Extending the reasoning to the whole interval

$$ \xi(x) = 0 \quad \text{in } [0, a] \text{ and so } \eta = 0. \quad \square \qquad (26) $$

## 3.2 Worst case error

8 Let $\delta \in L^2(\mathcal{O})$ be the measurement error of data $T$. By abuse of notation $\delta$

9 also denotes $|\delta|$. We call $S$ the operator

$$ u(a) \rightarrow S(u(a)) = u(0), \qquad (27) $$

10 where $u$ satisfies $(\mathcal{P}_2)$ and $\Phi = 0$. Following [15] we can define a worst case

11 error for error $\delta$ on the data $T$ and the a priori information $|u(a)| < E$.

$$ \mathcal{F}(\delta, E, |\cdot|) = \sup_{x \in X} \{|x| ||Sx| \le \delta, |u(a)| < E\} \qquad (28) $$

12 From theorem 1.21 of [15] we have

$$ \mathcal{F}(\delta, E, |\cdot|) \le \sqrt{\delta E} \qquad (29) $$

## 3.3 Parabolic regularization with integration in the unstable direction

### 3.3.1 Algorithm 1

16 The $P(x)$ operator is positive and of order one (it transforms functions of $H^1$

17 into functions of $L^2$). We will regularize (13) with the help of the Laplacian

18 $\Delta_y$ which is negative and of order two.

$$ \frac{\partial u}{\partial x} - P(x)u(x) - \varepsilon \Delta_y u(x) = 0, \quad u(0) = T, \quad T \in L^2(\mathcal{O}). \qquad (30) $$
1 **Proposition 3.3** The problem (30) admits a unique solution in $L^2(0, a; H_0^1(\mathcal{O})) \cap L^\infty(0, a; L^2(\mathcal{O}))$.

2 **Proof** It is sufficient to obtain classical estimates for a parabolic problem and then to apply the Galerkin method. Multiplying (30) by $u(x)$ and integrating from 0 to $x$ yields:

$$ \frac{1}{2}|u(x)|^2 - \int_0^x (Pu, u)(s) \, ds + \varepsilon \int_0^x |\nabla_y u|^2 \, ds = \frac{1}{2}|u(0)|^2. \quad (31) $$

6 owing to the continuity of $P$

$$ \begin{aligned} \int_0^x (Pu, u)(s) \, ds &\le C \int_0^x |\nabla_y u||u| \, ds \\ &\le \varepsilon \int_0^x |\nabla_y u|^2 \, ds + \frac{C^2}{4\varepsilon} \int_0^x |u|^2 \, ds. \end{aligned} \quad (32) $$

7 It can be proved (for example [8]) that $C \le 1$. So

$$ |u(x)|^2 \le |u(0)|^2 + \frac{1}{4\varepsilon} \int_0^x |u|^2 \, ds \quad (33) $$

8 By Gronwall inequality

$$ |u(x)|^2 \le |u(0)|^2 \exp \frac{x}{4\varepsilon}. \quad \square \quad (34) $$

9 We now study the behavior of the solution of Eq. (30) when $\varepsilon$ tends to 0. This solution is referred to as $u_\varepsilon$.

10 **Proposition 3.4** If the Cauchy data $(T, 0)$ on $\Gamma_0$ are compatible the solution $u_\varepsilon(a)$ of (30) converges in $H^{-2}(\mathcal{O})$ for $\varepsilon \to 0$ to $\tilde{u}(a)$ value on $\Gamma_a$ of the solution of (13). The error in norm $H^{-2}(\mathcal{O})$ is in $\varepsilon$. If the data are compatible with order $k$, then convergence occurs in $H^{k-2}(\mathcal{O})$.

11 **Proof** Let $\tilde{u}$ be the solution of the exact problem (13) and $u_\varepsilon$ the solution of the regularized problem (30). We define $v_\varepsilon$ by

$$ v_\varepsilon = u_\varepsilon - \tilde{u}. \quad (35) $$

17 It verifies thanks to the fact that $\tilde{u}$ is solution of (13)

$$ \frac{\partial v_\varepsilon}{\partial x} - P(x)v_\varepsilon(x) - \varepsilon \Delta_y v_\varepsilon(x) = \varepsilon \Delta_y \tilde{u}, \quad v_\varepsilon(0) = 0. \quad (36) $$

18 Expand $u$ and $v_\varepsilon$ on the basis $\{\varphi_n\}_{n=1}^\infty$

$$ v_\varepsilon = \sum_n v^n \varphi_n, \quad \tilde{u} = \sum_n u^n \varphi_n. \quad (37) $$
1. By diagonalization, each component is solution of

$$v^{n'} - (\pi_n - \varepsilon\lambda_n)v^n = -\varepsilon\lambda_n u^n, \quad v^n(0) = 0, \qquad (38)$$

2. with

$$u^n(x) = u^n(a) \exp\left(\int_a^x \pi_n(s) ds\right). \qquad (39)$$

The equation (38) is integrated in

$$\begin{aligned} v^n(x) &= -\varepsilon\lambda_n u^n(a) \int_0^x \exp\left(\int_a^\xi \pi_n(s) ds\right) \exp\left(\int_\xi^x (\pi_n(s) - \varepsilon\lambda_n) ds\right) d\xi \\ &= -u^n(a) \exp\left(\int_a^x \pi_n(s) ds\right) (1 - \exp(-\varepsilon\lambda_n x)) \end{aligned}$$

3. therefore

$$v^n(a) = -u^n(a)(1 - \exp(-\varepsilon\lambda_n a)). \qquad (40)$$

4. thus

$$\|v_{\varepsilon}(a)\|^2 = \sum_{n} |v^n(a)|^2 < \sum_{n} |u^n(a)|^2 = \|\tilde{u}(a)\|^2, \qquad (41)$$

5. and assuming $\tilde{u}(a) \in L^2(\mathcal{O})$, using the inequality

$$0 \le 1 - \exp(-x) \le x \qquad (42)$$

for $x \ge 0$

$$\begin{aligned} \|v_{\varepsilon}(a)\|_{H^{-2}(\mathcal{O})}^2 &= \sum_n \frac{|v^n(a)|^2}{\lambda_n^2} \\ &= \sum_n |u^n(a)|^2 \frac{(1 - \exp(-\varepsilon\lambda_n a))^2}{\lambda_n^2} \\ &\le a^2 \|\tilde{u}(a)\|_{L^2(\mathcal{O})}^2 \varepsilon^2. \end{aligned} \qquad (43)$$

In the case where the data are compatible with order k

$$\begin{aligned} \|v_{\varepsilon}(a)\|_{H^{k-2}(\mathcal{O})}^2 &= \sum_n \frac{|v^n(a)|^2}{\lambda_n^{2-k}} \\ &= \sum_n \lambda_n^k |u^n(a)|^2 \frac{(1 - \exp(-\varepsilon\lambda_n a))^2}{\lambda_n^2} \\ &\le a^2 \|\tilde{u}(a)\|_{H^k(\mathcal{O})}^2 \varepsilon^2. \quad \square \end{aligned} \qquad (44)$$

Let us now study stability in relation to the error in the T data. It is assumed that only a perturbed version of the Cauchy condition for $\Gamma_0$ is known.

$$u(0) = T + \delta$$
1 **Proposition 3.5** The solution of the regularized problem (30) depends lin-

2 early and continuously on an error in the Cauchy data (T, 0).

3 **Proof** We take up the method for the demonstration of Proposition 3.4. Let $u_{\varepsilon}^{\delta}$

4 be the solution of the regularized problem with a perturbed initial condition.

$$
\frac{\partial u_{\varepsilon}^{\delta}}{\partial x} - P(x)u_{\varepsilon}^{\delta}(x) - \varepsilon \Delta_{y} u_{\varepsilon}^{\delta}(x) = 0, \quad u_{\varepsilon}^{\delta}(0) = T + \delta, \qquad (45)
$$

5 and let $v_{\varepsilon}^{\delta} = u_{\varepsilon}^{\delta} - u_{\varepsilon}$. This function verifies

$$
\frac{\partial v_{\varepsilon}^{\delta}}{\partial x} - P(x)v_{\varepsilon}^{\delta}(x) - \varepsilon \Delta_{y} v_{\varepsilon}^{\delta}(x) = 0, \quad v_{\varepsilon}^{\delta}(0) = \delta, \qquad (46)
$$

6 and its components $v_n^{0,\delta}$ on the basis of eigenfunctions

$$
\frac{dv_n^{0,\delta}}{dx} - \pi_n(x)v_n^{0,\delta}(x) + \varepsilon\lambda_n v_n^{0,\delta}(x) = 0, \quad v_n^{0,\delta}(0) = \delta_n, \qquad (47)
$$

$$
7 \qquad v_n^{0,\delta}(x) = \exp\left(\int_0^x (\pi_n(s) - \varepsilon\lambda_n) ds\right) \delta_n. \qquad (48)
$$

8 We have

$$
\frac{1}{2}|v_{\varepsilon}^{\delta}(x)|^2 - \int_{0}^{x} (P v_{\varepsilon}^{\delta}, v_{\varepsilon}^{\delta})(s) ds + \varepsilon \int_{0}^{x} |\nabla_y v_{\varepsilon}^{\delta}|^2 ds = \frac{1}{2}|\delta|^2. \quad (49)
$$

9 By the continuity of P and the inequality of Gronwall we have

$$
|v_{\varepsilon}^{\delta}(x)|^2 \leq |\delta|^2 \exp \frac{x}{4\varepsilon}. \quad \Box \qquad (50)
$$

10 Using the preceding results we can define an admissible regularization strategy

11 according to [15] (Definition 2.3).

12 **Proposition 3.6** *The choice*

$$
\epsilon = \epsilon_1(\delta), \tag{51}
$$

13 defines an admissible regularization strategy for $\epsilon_1$ that satisfies (55). Then we

14 have the following bound for the total error using this strategy in Algorithm 1

$$
|v^{1,tot}| \leq a \|\tilde{u}(a)\|_{H^2(\mathcal{O})} \epsilon_1(\delta) (1 + 4 \frac{\epsilon_1(\delta)}{a}). \quad (52)
$$

**Proof** Assuming $\tilde{u}(a)$ belongs to $H^2(\mathcal{O})$. The total error $v^{1,tot}$ on the estimate
of $\tilde{u}(a)$ is bounded by

$$
|v^{1,tot}| \leq |v_{\epsilon}^{\delta}(a)| + |v_{\epsilon}(a)| \qquad (53)
$$

$$
\leq |\delta| \exp \frac{a}{4\epsilon} + a \|\tilde{u}(a)\|_{H^2(\mathcal{O})} \epsilon. \quad (54)
$$
1 Let $\varepsilon_1(\delta)$ be defined by

$$ \frac{|\delta|}{4\varepsilon_1(\delta)^2} \exp \frac{a}{4\varepsilon_1(\delta)} = \|\tilde{u}(a)\|_{H^2(\mathcal{O})}. \qquad (55) $$

2 This represents the minimum of the right-handside of (54). Then we have

$$ |v^{1,tot}| \le a \|\tilde{u}(a)\|_{H^2(\mathcal{O})} \varepsilon_1(\delta) (1 + 4 \frac{\varepsilon_1(\delta)}{a}). \quad \square \qquad (56) $$

3 **3.4 Modulation of the regularizing term**

4 Regularization of Algorithm 1 verifies the required stability and convergence properties. However, it turns out that the dependence on $\varepsilon$ in (50) is very strong and numerically this parameter, which determines the level of regularization, is difficult to tune. In particular, for an excessively high value of $\varepsilon$ due to diffusion the solution of (30) goes to 0 for $x$ going to $\infty$ and all information on $u$ is lost. We can set parabolic regularization (30) by introducing in the regularization term the deviation from a reference value. If we knew the exact value $\tilde{u}$, taking it as reference we would have

$$ \frac{\partial u_{\varepsilon}}{\partial x} - P(x)u_{\varepsilon}(x) - \varepsilon \Delta_y (u_{\varepsilon} - \tilde{u})(x) = 0, \quad u_{\varepsilon}(0) = T = \tilde{u}(0). \qquad (57) $$

12 **Proposition 3.7** Assume $(T, 0)$ is a compatible Cauchy condition, and that

$$ u_{\varepsilon}(0) = T = \tilde{u}(0) \qquad (58) $$

then

$$ u_{\varepsilon} \equiv \tilde{u}, $$

13 **Proof** Let $w_{\varepsilon}$ be $u_{\varepsilon} - \tilde{u}$. It verifies an homogeneous equation

$$ \frac{\partial w_{\varepsilon}}{\partial x} - P(x)w_{\varepsilon}(x) - \varepsilon \Delta_y w_{\varepsilon}(x) = 0, \qquad (59) $$

14 with the initial condition $w_{\varepsilon}(0) = 0$. So

$$ w_{\varepsilon} \equiv 0. \quad \square \qquad (60) $$

15 Let us now consider the following more general situation: recall term $\bar{u}$ verifies the direct problem with an initial condition $t$ in $x = a$

$$ \frac{\partial u_{\varepsilon}}{\partial x} - P(x)u_{\varepsilon}(x) - \varepsilon \Delta_y (u_{\varepsilon} - \bar{u})(x) = 0, \quad u_{\varepsilon}(0) = T, \qquad (61) $$

17

$$ \frac{\partial \bar{u}}{\partial x} - P\bar{u} = 0, \quad \bar{u}(a) = t. \qquad (62) $$
1 We choose $t$ as

$$ \bar{u}(a) = t = u_{\varepsilon}(a). \qquad (63) $$

2 For data $T \in X$ this formulation, although it uses the regularized version

3 (61) of (14) provides the exact solution of the inverse problem. However, it is

4 difficult to apply the previous method to ensure the compatibility of Cauchy

5 conditions. In this formulation, the equations for $u_{\varepsilon}$ and $\bar{u}$ are coupled. We

6 demonstrate that this coupled system admits a unique solution.

7 **Proposition 3.8** It is assumed that the Cauchy data on $\Gamma_0$ are compatible.

8 The coupled system (61), (62) and (63) admits a unique solution. We also

9 have

$$ \bar{u}(0) = u_{\varepsilon}(0) = T. \qquad (64) $$

10 Reciprocally this condition is sufficient for the Cauchy problem to have a solution.

11

12 **Proof** We resume the expansion on the basis $\{\varphi_n\}_{n=1}^{\infty}$ and

$$ u_{\varepsilon}(x) = \sum_{n} u_{n}(x)\varphi_{n}. \qquad (65) $$

Here, we forget the reference to $\varepsilon$ to simplify notation. Let $T_n$ be a component of $T$ on this basis. For each component we have

$$ \bar{u}_n(x) = \exp\left(\int_a^x \pi_n(s) ds\right) u_n(a) \qquad (66) $$

$$ u_n(x) = \varepsilon \lambda_n \int_0^x \bar{u}^n(\xi) \exp\left(\int_\xi^x (\pi_n(s) - \varepsilon \lambda_n) ds\right) d\xi \qquad (67) $$

$$ + T_n \exp\left(\int_0^x (\pi_n(s) - \varepsilon \lambda_n) ds\right). \qquad (68) $$

We deduce from this

$$ u_n(x) = (1 - \exp(-\varepsilon \lambda_n x)) u_n(a) \exp\left(\int_a^x \pi_n(s) ds\right) + T_n \exp\left(\int_0^x (\pi_n(s) - \varepsilon \lambda_n) ds\right) \qquad (69) $$

$$ = \bar{u}_n(x) + \exp(-\varepsilon \lambda_n x) (T_n \exp\left(\int_0^x \pi_n(s) ds\right) - u_n(a) \exp\left(\int_a^x \pi_n(s) ds\right)), \qquad (70) $$

13 and

$$ u_n(a) = T_n \exp \int_0^a \pi_n(s) ds, \qquad (71) $$

14 for any $n$. Thus

$$ \sum_{n=1}^{\infty} T_n \varphi_n \exp\left(\int_0^x \pi_n(s) ds\right) = \bar{u}(x) \qquad (72) $$
is well defined in $L^2(Q)$ because of the assumption of compatibility with $T$ and
verifies (62). Therefore $u_\epsilon(0) = \bar{u}(0)$ is a necessary condition for compatibility.
This is also a sufficient condition. Indeed after (71)

$$
\bar{u}_n(x) = T_n \exp\left(\int_0^x \pi_n(s) ds\right) \qquad (73)
$$

$$
=u_n(a) \exp\left(\int_{a}^{x} \pi_{n}(s) d s\right) .
\qquad(74)
$$

Thus $\bar{u}$ is well defined in $L^2(Q)$ and verifies the Cauchy conditions in 0. The uniqueness follows from the previous calculation where one makes $T = 0$. $\square$

Once again, it should be stressed that, starting from compatible data $(T, 0)$ the integration of (61), (62) and (63) provides the exact value $\tilde{u}(a)$ on $\Gamma_a$. The relation $u_\epsilon(0) = \bar{u}(0)$ is not an assumption but the result of the computation. Therefore when solving the problem numerically, checking it is a way to prove that the solution is accurate.

We can also present the previous result in another way: starting from the compatible data $(T, 0)$ and integrating (62) for increasing $x$ (which is numerically unstable) or solving (61), (62) and (63) which depends on $\epsilon$ provides the same value of $\bar{u}(a)$.

3.4.1 Correction of the parabolic regularization : Algorithm 2

We propose here a method decoupling the equations for $u$ and $\bar{u}$ by successive corrections. We call $u_{\epsilon}^{0}$ the solution of (30). For the correction the following procedure is used: the reference of the regularization is calculated from the previous value of the state:

$$
\bar{u}^{1}(a) = u_{\epsilon}^{0}(a; T). \tag{75}
$$

Let $u_{\epsilon}^{1}$ be the solution of (61) with the recall term $\bar{u}^{1}$ solution of (62) and the initial condition (75) on $\Gamma_{a}$. Let us make the method explicit and forget the reference to $\epsilon$ to simplify the writing. The first iteration of relaxation yields
for $\bar{u}^{1}$

$$
\frac{\partial \bar{u}^1}{\partial x} - P \bar{u}^1 = 0, \quad \bar{u}^1(a) = u^0(a; T). \qquad (76)
$$

and for $u^1$

$$
\frac{\partial u^1}{\partial x} - P(x)u^1(x) - \epsilon \Delta_y (u^1 - \bar{u}^1)(x) = 0, \quad u^1(0) = T, \qquad (77)
$$

The implementation of this method does not assume that Cauchy data are compatible. However for the error estimate below we assume that. According to the compatibility hypothesis there exists a function $\tilde{u}$ such that

$$
\frac{\partial \tilde{u}}{\partial x} - P \tilde{u} = 0, \quad \tilde{u}(0) = T. \qquad (78)
$$
1 We consider the differences $v^0 = u^0 - \tilde{u}$, $\bar{v}^1 = \bar{u}^1 - \tilde{u}$ and $v^1 = u^1 - \tilde{u}$. These functions verify

$$ \frac{\partial v^0}{\partial x} - P(x)v^0(x) - \varepsilon \Delta_y u^0(x) = 0, \quad v^0(0) = 0, \qquad (79) $$

3 $\frac{\partial v^1}{\partial x} - P(x)v^1(x) - \varepsilon \Delta_y (v^1 - \bar{v}^1)(x) = 0, \quad v^1(0) = 0, \qquad (80)$

4 and for $\bar{v}^1$

$$ \frac{\partial \bar{v}^1}{\partial x} - P \bar{v}^1 = 0, \quad \bar{v}^1(a) = v^0(a). \qquad (81) $$

5 The estimate we are looking for is $u^1(a)$ with this method.

6 **Proposition 3.9** For a compatible $(T, 0)$ data of order 2 we have the following error estimate for the solution of the regularized problem with the value of reference: (77), (81)

$$ \|u^1(a) - \tilde{u}(a)\|_{H^{-2}(\mathcal{O})} \leq a \|\tilde{u}(a)\|_{H_0^1(\mathcal{O}) \cap H^2(\mathcal{O})} \varepsilon^2. \qquad (82) $$

9 If $(T, 0)$ is compatible of order $k+2$

$$ \|u^1(a) - \tilde{u}(a)\|_{H^{k-2}(\mathcal{O})} \leq a \|\tilde{u}(a)\|_{H^{k+2}(\mathcal{O})} \varepsilon^2. \qquad (83) $$

10 **Proof** Let $v_n^0$ be the components on the basis of $\{\varphi_n\}_{n=1}^\infty$ of $v^0$. We have

$$ \begin{aligned} v_n^0(x) &= - \int_0^x \varepsilon \lambda_n \tilde{u}_n(\xi) \exp \left( \int_\xi^x (\pi_n(s')) ds' \right) d\xi \\ &= - \varepsilon \lambda_n \tilde{u}_n(a) x \exp \int_a^x \pi_n(s) ds. \end{aligned} \qquad (84) $$

11 Let $\bar{v}_n^1$ be the components on the basis of $\{\varphi_n\}_{n=1}^\infty$ of $\bar{v}^1$ and the same for $v^1$. We have

$$ \bar{v}_n^1(x) = \exp \left( \int_a^x \pi_n(s) ds \right) v_n^0(a). \qquad (85) $$

$$ \begin{aligned} v_n^1(x) &= \int_0^x \varepsilon \lambda_n \bar{v}_n^1(\xi) \exp \left( \int_\xi^x (\pi_n(s') - \varepsilon \lambda_n) ds' \right) d\xi \\ &= \varepsilon \lambda_n v_n^0(a) \exp \int_a^x \pi_n(s) ds \int_0^x \exp(-\varepsilon \lambda_n(x-\xi)) d\xi \qquad (86) \\ &= v_n^0(a) \exp \int_a^x \pi_n(s) ds (1 - \exp(-\varepsilon \lambda_n x)) \end{aligned} $$

and therefore

$$ v_n^1(a) = (1 - \exp(-\varepsilon \lambda_n a)) v_n^0(a), \qquad (87) $$

$$ = -(1 - \exp(-\varepsilon \lambda_n a)) \varepsilon \lambda_n \tilde{u}_n(a), \qquad (88) $$
and using inequality (42)

$$
\begin{align*}
\|v^1(a)\|_{H^{-2}(\mathcal{O})}^2 &= \sum_n \frac{|v_n^1(a)|^2}{\lambda_n^2} \\
&= \varepsilon^2 \sum_n |\tilde{u}_n(a)|^2 (1 - \exp(-\varepsilon \lambda_n a))^2 \\
&\le a^2 \|\tilde{u}(a)\|_{H_0^1(\mathcal{O}) \cap H^2(\mathcal{O})}^2 \varepsilon^4. \tag{89}
\end{align*}
$$

In the case where the data are compatible of order k

$$
\begin{align*}
\|v^1(a)\|_{H^{k-2}(\mathcal{O})}^2 &= \sum_n \frac{|v_n^1(a)|^2}{\lambda_n^{2-k}} \\
&= \varepsilon^2 \sum_n \lambda_n^k |\tilde{u}_n(a)|^2 (1 - \exp(-\varepsilon \lambda_n a))^2 \\
&\le a^2 \|\tilde{u}(a)\|_{H^{k+2}(\mathcal{O})}^2 \varepsilon^4. \quad \Box \tag{90}
\end{align*}
$$

$$
\frac{\partial \bar{u}^2}{\partial x} - P \bar{u}^2 = 0, \quad \bar{u}^2(a) = u^1(a), \qquad (91)
$$

$$
\frac{\partial u^2}{\partial x} - P(x)u^2(x) - \varepsilon \Delta_y (u^2 - \bar{u}^2)(x) = 0, \quad u^2(0) = T. \qquad (92)
$$

$$
v_n^2(a) = (1 - \exp(-\varepsilon \lambda_n a))^3 u_n^0(a), \qquad (93)
$$

from where

$$
\begin{align}
\|v^2(a)\|_{H^{-2}(\mathcal{O})}^2 &= \sum_n \frac{|v_n^2(a)|^2}{\lambda_n^2} \notag \\
&\le a^6 \|\tilde{u}(a)\|_{H^4(\mathcal{O})}^2 \varepsilon^6. \tag{94}
\end{align}
$$

<footnote n="1">1 **Remark 1** In this calculation we see that there is compensation for the insta-</footnote>

<footnote>bility of the integration of v in the direct sense because of the positivity of P</footnote>

<footnote n="3">3 owing to the coupling with the backward integration of $\bar{v}$ from a to x.</footnote>

<footnote n="4">4 Thus the use of the reference $\bar{u}$ leads to an improvement in the precision of</footnote>

<footnote n="5">5 the estimate of the final state $u(a)$, but this improvement decreases for higher</footnote>

<footnote n="6">6 frequencies.</footnote>

<footnote n="7">7 This calculation can be repeated assuming a greater regularity for $\tilde{u}(a)$.</footnote>

<footnote n="9">9 By the same calculation, integrating the equation for $v_n^2$ taking into account
10 the initial condition $v_n^2(0) = 0$</footnote>

<footnote n="11">11 Now consider the errors on the estimate of the value on $\Gamma_a$ due to the un-</footnote>

<footnote n="12">12 certainty $\delta \in L^2(\mathcal{O})$ in the measurement on $\Gamma_0$. By abusing notation $\delta$ also</footnote>
1 represents the $L^2$ norm of this uncertainty. We refer to $u^{0,\delta}$ and $u^{1,\delta}$ as the

2 solutions of (30) and (77) respectively for a perturbed initial condition $\delta$. Let

3 $v^{0,\delta} = u^{0,\delta} - u^0$ and $v^{1,\delta} = u^{1,\delta} - u^1$ be these errors. They verify

$$ \frac{\partial v^{0,\delta}}{\partial x} - P(x)v^{0,\delta}(x) - \varepsilon \Delta_y v^{0,\delta}(x) = 0, \quad v^{0,\delta}(0) = \delta, \qquad (95) $$

$$ \frac{\partial v^{1,\delta}}{\partial x} - P(x)v^{1,\delta}(x) - \varepsilon \Delta_y (v^{1,\delta}(x) - \bar{v}^{1,\delta}(x)) = 0, \quad v^{1,\delta}(0) = \delta, \qquad (96) $$

5 with

$$ \frac{\partial \bar{v}^{1,\delta}}{\partial x} - P(x) \bar{v}^{1,\delta}(x) = 0, \quad \bar{v}^{1,\delta}(a) = v^{0,\delta}(a), \qquad (97) $$

setting

$$ \bar{v}^{1,\delta} = \bar{u}^{1,\delta} - \bar{u}^1. $$

**Proposition 3.10** *Assuming that the data $(T, 0)$ are compatible we have the following bound on the estimate of $u(a)$ using the methods (77) and (76) for a noise $\delta \in L^2(\mathcal{O})$*

$$ \|u^{1,\delta}(a) - u^1(a)\|_{L^2(\mathcal{O})} \leq 2 \exp\left(\frac{a}{2\varepsilon}\right) \|\delta\|_{L^2(\mathcal{O})} $$

6 **Proof** As for the proof of Proposition 3.5 we expand on the basis of eigenfunctions

7 $\{\varphi_n\}_{n=1}^\infty$ of $-\Delta_y$. Let $v_n^{0,\delta}$, $v_n^{1,\delta}$ and $\bar{v}_n^{1,\delta}$ be their corresponding components.

8 They verify

$$ \frac{dv_n^{0,\delta}}{dx} - \pi_n(x)v_n^{0,\delta}(x) + \varepsilon\lambda_n v_n^{0,\delta}(x) = 0, \quad v_n^{0,\delta}(0) = \delta_n, \qquad (98) $$

$$ \frac{dv_n^{1,\delta}}{dx} - \pi_n(x)v_n^{1,\delta}(x) + \varepsilon\lambda_n(v_n^{1,\delta}(x) - \bar{v}_n^{1,\delta}(x)) = 0, \quad v_n^{1,\delta}(0) = \delta_n, \qquad (99) $$

$$ \frac{d\bar{v}_n^{1,\delta}}{dx} - \pi_n(x)\bar{v}_n^{1,\delta}(x) = 0, \quad \bar{v}_n^{1,\delta}(a) = v_n^{0,\delta}(a). \qquad (100) $$

They give the solution

$$ v_n^{0,\delta}(x) = \delta_n \exp\left(\int_0^x (\pi_n(s) - \varepsilon\lambda_n)ds\right), \qquad (101) $$

$$ v_n^{1,\delta}(x) = v_n^{0,\delta}(a) \exp\left(\int_a^x \pi_n(s)ds\right)(1 - \exp(-\varepsilon\lambda_n x)) \qquad (102) $$

$$ + \delta_n \exp\left(\int_0^x (\pi_n(s) - \varepsilon\lambda_n)ds\right), \qquad (103) $$
and thus in $x = a$

$$v_n^{1,\delta}(a) = v_n^{0,\delta}(a)(1 - \exp(-\varepsilon\lambda_n a)) \quad (104)$$

$$+ \delta_n \exp\left(\int_0^a (\pi_n(s) - \varepsilon\lambda_n)ds\right) \quad (105)$$

$$=(2 - \exp(-\varepsilon\lambda_n a))\delta_n \exp\left(\int_0^a (\pi_n(s) - \varepsilon\lambda_n)ds\right) \quad (106)$$

1. The last term of the right-hand member can be bounded as follows:

$$\exp\left(\int_0^a \pi_n(s) ds\right) = \cosh(a\alpha_n). \quad (107)$$

So we have

$$\exp\left(\int_0^a (\pi_n(s) - \varepsilon\lambda_n) ds\right) \leq \exp\left(a(\sqrt{\lambda_n} - \varepsilon\lambda_n)\right) \quad (108)$$

$$\leq \exp\left(\frac{a}{2\varepsilon}\right). \quad (109)$$

2. The last inequality is independent of $n$. Thus

$$\|v^{1,\delta}(a)\|_{L^2}^2 = \sum_n \|v_n^{1,\delta}(a)\|^2 \leq 4 \exp\left(\frac{a}{\varepsilon}\right)\delta^2. \quad \square \quad (110)$$

3. Then choosing $\varepsilon_1(\delta)$ such that

$$2\varepsilon_1(\delta)^3 \|\tilde{u}(a)\|_{H^2(\mathcal{O})} = \exp\left(\frac{a}{2\varepsilon_1}\right)\delta, \quad (111)$$

4. uniquely defines an admissible regularization strategy for the method defined

5. in (76) and (77) assuming $\tilde{u}(a) \in H^2(\mathcal{O})$. Then from (110) and (89) we obtain

6. a bound on the total error $v^{1,tot}$

$$\|v^{1,tot}\| \leq 2 \exp\left(\frac{a}{2\varepsilon}\right)\delta + a \|\tilde{u}(a)\|_{H^2(\mathcal{O})} \varepsilon^2 \quad (112)$$

7. **Proposition 3.11** If $\delta$ is sufficiently small, there exists an interval $\varepsilon < \bar{\varepsilon}$ such that the upper bound of the total error for Algorithm 2 (112) is less

8. than that for Algorithm 1.

9.

10. **Proof** Consider the function of $\varepsilon$ defined by the right hand side of (112)

$$F_1(\varepsilon) = 2 \exp\left(\frac{a}{2\varepsilon}\right)\delta + a \|\tilde{u}(a)\|_{H^2(\mathcal{O})} \varepsilon^2 \quad (113)$$

11. By comparison with the same function for algorithm 1 (55)

$$F_0(\varepsilon) = \exp\left(\frac{a}{4\varepsilon}\right)\delta + a \|\tilde{u}(a)\|_{H^2(\mathcal{O})} \varepsilon. \quad (114)$$
1. For $\varepsilon$ small enough or large enough we have

$$F_1(\varepsilon) \geq F_0(\varepsilon). \qquad (115)$$

2. For $\delta$ small enough there exist $\underline{\varepsilon} < \bar{\varepsilon}$ such that

$$F_1(\varepsilon) \leq F_0(\varepsilon) \quad \text{on} \quad \underline{\varepsilon} < \varepsilon < \bar{\varepsilon}. \quad \square \qquad (116)$$

3. Therefore we see that after one step of correction for a good choice of the regularization parameter we obtain a better estimate which is less sensitive to measurement errors. This benefit decreases exponentially with $a$ which is the distance between the surface of estimation of the state and that of the measurement. However we cannot hope that by iterating this procedure we can asymptotically reach the exact value $\tilde{u}(a)$ unless the measurement $T$ is compatible.

## 3.5 Regularization via a control problem with state equation to be integrated backwards in a stable manner

We propose now another regularization procedure (Algorithm 3) based on a control problem. Consider a control problem with state $\bar{u}$ defined by

$$\frac{\partial \bar{u}}{\partial x} - P\bar{u} = 0, \quad \bar{u}(a) = u_1, \qquad (117)$$

12. with the control variable $u_1$ and with cost function to minimize

$$\tilde{J}_{\eta}(u_1) = \frac{1}{2} \|\bar{u}(u_1, 0) - T\|^2 + \frac{\eta}{2} \|u_1\|^2. \qquad (118)$$

13. This problem is well posed and the argument of the minimum $\tilde{u}_{1,\eta}$ is the sought estimate of $u(1)$. It is exactly $u(1)$ if $(T, 0)$ is compatible and $\eta = 0$.

15. Setting

$$C_n = \exp\left(\int_a^0 \pi_n(x) dx\right) = \frac{1}{\cosh \sqrt{\lambda_n} a}, \qquad (119)$$

16. $C_n$ decreases with respect to $n$ and goes to 0 as $n \to \infty$. We now apply Morozov's discrepancy principle.

18. **Proposition 3.12** Assume the observation $T$ satisfies

1.

$$D'^2 = \frac{1}{4} \sum_n \frac{T_n^2}{C_n^4} < \infty \qquad (120)$$

and assume that there is an error of amplitude $\delta$ in $L^2(\mathcal{O})$ norm on the observation $T$ then the total error is bounded by

$$O(\delta^{1/2})$$

19. or if the observation $T$ satisfies
2.

$$D^2 = \sum_n \frac{T_n^2}{C_n^6} < \infty \qquad (121)$$

then the bound is in

$$O(\delta^{2/3})$$

1. **Proof** Let $C$ be the operator transforming the value of the solution of (76) at $x=1$ into its value at $x=0$. Then the optimal value of the control problem (76), (118) satisfies

2.

3.

$$\eta u_1 + C(Cu_1 - T) = 0. \qquad (122)$$

4. For each component we have by integration of (77) and (76)

$$u_{1,\eta}^{(n)} = T_n \frac{C_n}{\eta + C_n^2} \qquad (123)$$

5. We can then estimate the deviation from the exact value due to the regularization.

6.

Under assumption 1

$$|v_{\eta}^{(n)}| = |u_{1,\eta}^{(n)} - \tilde{u}_n(a)| \qquad (124)$$

$$= \frac{\eta}{C_n(\eta + C_n^2)} |T_n| \qquad (125)$$

$$\leq |T_n| \frac{\sqrt{\eta}}{2C_n^2} \qquad (126)$$

(127)

and

$$\sum |v_{\eta}^{(n)}|^2 \leq \eta \sum \frac{T_n^2}{4C_n^4}, \qquad (128)$$

$$\|v^{\eta}\| \leq D' \sqrt{\eta} \qquad (129)$$

7. with

$$D' = \frac{1}{2} \sqrt{\sum_n \frac{T_n^2}{C_n^4}} \qquad (130)$$

With respect to an error $\delta$ on the measurement $T$

$$|v_{\eta,\delta}^{(n)}| = |u_{1,\eta}^{(n)} - u_{1,\eta,\delta}^{(n)}| \qquad (131)$$

$$= |\delta_n \frac{C_n}{C_n^2 + \eta}| \qquad (132)$$

8. Then

$$\|v_{\eta,\delta}\| \leq \frac{\|\delta\|}{2\sqrt{\eta}} \qquad (133)$$
1. The total error is then bounded by

$$ \|v_{\eta, \delta}^{\text{tot}}\| \le \frac{\delta}{2\sqrt{\eta}} + D'\sqrt{\eta}, \qquad (134) $$

2. so the optimal value for $\eta$ is

$$ \eta_{\text{opt}} = \frac{\delta}{2D'}, \qquad (135) $$

3. and the bound for the total error

$$ \|v_{\delta}^{\text{tot}}\| \le \sqrt{2D'\delta}, \qquad (136) $$

Under assumption 2

$$ |v_{\eta}^{(n)}| = |u_{1,\eta}^{(n)} - \tilde{u}_n| \qquad (137) $$

$$ = |T_n| \left| \frac{\eta}{C_n(\eta + C_n^2)} \right| \qquad (138) $$

$$ \le \frac{T_n}{C_n^3} \eta \qquad (139) $$

So

$$ \sum |v_{\eta}^{(n)}|^2 \le \eta^2 \sum \frac{T_n^2}{C_n^6}, \qquad (140) $$

$$ \|v^{\eta}\| \le D\eta, \qquad (141) $$

4. where

$$ D = \sqrt{\sum_n \frac{T_n^2}{C_n^6}}. \qquad (142) $$

5. With respect to an error $\delta$ on the measurement $T$

$$ v_{\eta, \delta}^{(n)} = \delta_n \frac{C_n}{\eta + C_n^2} \le \frac{\delta_n}{2\sqrt{\eta}}. \qquad (143) $$

6. So

$$ \|v_{\eta, \delta}\| \le \frac{\delta}{2\sqrt{\eta}}. \qquad (144) $$

7. The total error on $u_1$ is bounded by

$$ \|v_{\eta, \delta}^{\text{tot}}\| \le \frac{\delta}{2\sqrt{\eta}} + D\eta. \qquad (145) $$

8. The choice that ensures the best warantee on the total error is

$$ \eta_{\text{opt}} = \left( \frac{\delta}{4D} \right)^{(2/3)} \qquad (146) $$

9. giving a bound on the total error

$$ \|v_{\delta}^{\text{tot}}\| \le \frac{3}{\sqrt[3]{2}} D^{1/3} \delta^{2/3} \qquad (147) $$

10. **Remark 2** Condition (120) is the same as (b) in Theorem 2.17 p.50 in [15].
## 3.6 Regularization via a control problem with integrations of the state equations both in the stable and unstable directions: Algorithm 4

Now let us consider the more general situation where the measure $T$ on $\Gamma_0$ is a general function in $L^2(\mathcal{O})$. It is proposed to regularize not only the evolution of $u(x)$ but also the data of its initial condition $u_0$. We are looking for an initial data close to measure $T$ but also close to admissible data. We use it as a state equation

$$ \frac{\partial u}{\partial x} - P(x)u(x) - \varepsilon \Delta_y (u - \bar{u})(x) = 0, \quad u(0) = u_0, \qquad (148) $$

$$ \frac{\partial \bar{u}}{\partial x} - P\bar{u} = 0, \quad \bar{u}(a) = u(a). \qquad (149) $$

The difficulty in this approach is that we are not sure of the existence of a solution for this system for any value of $u_0$ outside of $X$. Therefore, it is proposed to relax the condition in $a$ and to use as an additional control

$$ u_1 = \bar{u}(a), \qquad (150) $$

such that the equations on $u$ and $\bar{u}$ are independent. We now consider minimization of the functional.

$$ \hat{J}_{\eta}(u_0, u_1) = \frac{1}{2} \|\bar{u}(u_1, 0) - T\|^2 + \frac{\eta}{2} \|u_1\|^2 \qquad (151) $$

$$ + \gamma \left( \frac{1}{2} \|u_0 - T\|^2 + \frac{1}{2} \|(u(u_0; a) - u_1)\|^2 \right), \qquad (152) $$

where $\bar{u}$ verifies (117) with $\bar{u}(a) = u_1$. Thus the functional $\hat{J}_{\eta}(u_0, u_1)$ is well defined for any value of $u_0, u_1$. Here $\gamma > 0$ is a parameter to be tuned. In this approach, $\bar{u}$ represents the best approximation of the state $\tilde{u}$ and $\bar{u}(a) = u_1$ is the desired estimate of the state on $\Gamma_a$. By construction $\bar{u}(0)$ is a compatible data because

$$ \frac{\partial \bar{u}}{\partial x}(0) = P(0)\bar{u}(0) = 0. $$

**Remark 3** The coefficients $\frac{1}{2}$ in formula (152) allow us to simplify the calculations but are not necessary.

**Remark 4** Let us note again that if the data $(T, 0)$ are compatible regularization in $\varepsilon$ does not introduce any modification in the calculation of $\bar{u}(a)$. In this case the minimum of the functional $\hat{J}_0(u_0, u_1)$ (for $\varepsilon = 0$) is obtained for $u_0 = T, u_1 = u(u_0; a)$.

**Remark 5** We remark that the value $\hat{J}_0 = 0$ can only be reached if $\bar{u}(u_1, 0) = u_0, u_0 = T$ and $u(u_0; a) = u_1$ if the data $(T, 0)$ are compatible.
### 1.3.6.1 Existence of an optimal control

**Proposition 3.13** The functional $\hat{J}_{\eta}$ reaches its minimum in $(u_{0,\eta}, u_{1,\eta})$ which we still denote by $(u_0, u_1)$. Furthermore it defines a continuous mapping $G_{\eta} \in \mathcal{L}(L^2(\mathcal{O}), L^2(\mathcal{O}))$ yielding the estimate $u_1$ of $u(a)$ from the knowledge of the data $T$.

$$u_1 = G_{\eta}(T).$$

*Proof* By projection on the basis $\{\varphi_n\}_{n=1}^{\infty}$ we can write the necessary conditions of optimality that verify the components $u_n^0, u_n^1$ of $u_0, u_1$ without using adjoint states for each component. Still we have to show that the series built so far are converging.

$$u^n(x) = u_n^0 \exp(-\int_0^x (\varepsilon\lambda_n - \pi_n(s)) ds) + u_n^1 \exp(\int_a^x \pi_n(s) ds)(1 - \exp -\varepsilon\lambda_n x) \quad (153)$$

$$\bar{u}^n(0) = u_n^1 \exp\left(\int_a^0 \pi_n(x) dx\right) \quad (154)$$

### 2. Setting

$$A_n = \exp(-\int_0^a (\varepsilon\lambda_n - \pi_n(x)) dx, \quad B_n = -\exp -\varepsilon\lambda_n a, \quad (155)$$

we have

$$B_n \to 0, \quad C_n \to 0 \quad \text{for } n \to \infty. \quad (156)$$

$$A_n = -\frac{B_n}{C_n} \to 0 \quad \text{for } n \to \infty \quad (157)$$

Using (155) and (119) we get

$$\begin{aligned} \hat{J}_{\eta}(u_0, u_1) &= \frac{1}{2} \left[ \sum_{n=0}^{\infty} (\gamma |A_n u_0^{(n)} + B_n u_1^{(n)}|^2 + \gamma |u_0^{(n)} - T_n|^2 + |C_n u_1^{(n)} - T_n|^2) + \eta |u_1^{(n)}|^2 \right] \\ &= \frac{1}{2} \left[ \sum_{n=0}^{\infty} \left( u_0^{(n)}, u_1^{(n)} \right) \begin{pmatrix} \gamma(A_n^2 + 1) & \gamma A_n B_n \\ \gamma A_n B_n & \gamma B_n^2 + C_n^2 + \eta \end{pmatrix} \begin{pmatrix} u_0^{(n)} \\ u_1^{(n)} \end{pmatrix} \right] \\ &\quad + \left[ \sum_{n=0}^{\infty} (-\gamma T_n, -T_n C_n) \begin{pmatrix} u_0^{(n)} \\ u_1^{(n)} \end{pmatrix} + T_n^2 \right] \end{aligned} \quad (158)$$

For the gradient components

$$\nabla_{u_0}^{(n)} \hat{J}_{\eta} = \gamma(A_n^2 + 1)u_0^{(n)} + \gamma A_n B_n u_1^{(n)} - \gamma T_n \quad (159)$$

$$\nabla_{u_1}^{(n)} \hat{J}_{\eta} = \gamma A_n B_n u_0^{(n)} + (\gamma B_n^2 + C_n^2 + \eta) u_1^{(n)} - T_n C_n \quad (160)$$
The optimum of $\hat{J}_{\eta}$ is characterised by

$$ \left(1 + \frac{B_n^2}{C_n^2}\right) u_0^{(n)} - \frac{B_n^2}{C_n} u_1^{(n)} = T_n \quad (161) $$

$$ -\frac{\gamma B_n^2}{C_n} u_0^{(n)} + (\gamma B_n^2 + C_n^2 + \eta) u_1^{(n)} = T_n C_n. \quad (162) $$

1. Eliminating $u_0^{(n)}$ we get

$$ u_1^{(n)}(\gamma C_n^2 B_n^2 + (\eta + C_n^2)(B_n^2 + C_n^2)) = ((\gamma + 1)B_n^2 + C_n^2)C_n T_n. \quad (163) $$

2. setting

$$ H_n = \frac{C_n^2 B_n^2}{C_n^2 + B_n^2}, \quad (164) $$

3. therefore

$$ u_1^{(n)}(\gamma H_n + \eta + C_n^2) = \left( \frac{\gamma B_n^2}{B_n^2 + C_n^2} + 1 \right) C_n T_n \quad (165) $$

$$ |u_1^{(n)}| < \left(1 + \frac{\gamma B_n^2}{B_n^2 + C_n^2}\right) \frac{C_n T_n}{2\sqrt{\eta} \sqrt{C_n^2 + \gamma H_n}} \quad (166) $$

So

$$ \|u_1\|_{L^2(\mathcal{O})} = \sqrt{\sum_{n=1}^{\infty} |u_1^{(n)}|^2} \quad (167) $$

$$ \leq \frac{\gamma + 1}{2\sqrt{\eta}} \|T\|_{L^2(\mathcal{O})} \quad (168) $$

5. This computation shows that for $T \in L^2(\mathcal{O})$ the estimate of $\bar{u}(a)$ is defined in

6. $L^2(\mathcal{O})$. $\square$

### 3.6.2 Bounding the error due to regularization

8. We need to make additional regularity assumptions on the measurement $T$ to

9. estimate the error due to regularization with $\eta$. This assumption is set by

10. Kirsch in ([15] Theorem 2.9 p36)

11. **Proposition 3.14** *Assuming the measurement $T$ of $u$ on $\Gamma_0$ satisfies*

$$ D^{\prime\prime 2} = \sum_n T_n^2 (\cosh a \sqrt{\lambda_n})^4 < \infty \quad (169) $$

12. then we have the bound

$$ \|\tilde{u}(a) - u_1\| < C\sqrt{\eta} \quad (170) $$
1 **Proof** Let $v_{\eta,\gamma} = u_1 - \tilde{u}(a)$.

$$v_{\eta,\gamma} = - \frac{\eta(B_n^2 + C_n^2)T_n}{C_n(\gamma C_n^2 B_n^2 + C_n^2(C_n^2 + B_n^2) + \eta(B_n^2 + C_n^2))}, \quad (171)$$

2 we have

$$v_{\eta,\gamma} = - \frac{\eta T_n}{C_n(\gamma H_n + C_n^2 + \eta)}. \quad (172)$$

Then

$$\sum_n |v_{\eta,\gamma}^{(n)}|^2 \le \eta^2 \sum_n \left( \frac{T_n^2}{C_n^2(\gamma H_n + C_n^2 + \eta)^2} \right) \quad (173)$$

$$\le \frac{D''^2}{\gamma + 1} \eta \quad (174)$$

3 By comparison with formula (129) we see that the error due to this regular-

4 ization produces a smaller error for all the components than for algorithm 3

5 because $H_n > 0$. The compatibility of the data (169) is sufficient to ensure

6 the convergence of the series. Then

$$\|v_{\eta}\|_{L^2(\mathcal{O})} < \frac{D''}{\sqrt{\gamma + 1}\sqrt{\eta}} \quad (175)$$

7 **Remark 6** This result is independent of $\varepsilon$. However, in practice, the speed of

8 convergence may depend on this.

9 **Remark 7** The regularity (169) that we have assumed on the exact measure-

10 $T$ also implies a regularity on $u_1$.

11 **3.6.3 Influence of an error on the data**

Let us consider the effect of a perturbation $\delta$ on the measure $T$ on $\Gamma_0$. Let $v_{\eta,\delta} = u_{1,\delta} - u_1$ where $u_{1,\delta}$ is the value of the optimal $u_1$ relative to observation $T + \delta$. For the $n^{th}$ component we have

$$v_{\eta,\delta}^{(n)} = \frac{((\gamma + 1)B_n^2 + C_n^2)C_n\delta_n}{\gamma C_n^2 B_n^2 + (\eta + C_n^2)(B_n^2 + C_n^2)} \quad (176)$$

$$= \frac{C_n \delta_n}{C_n^2 + \eta \frac{B_n^2 + C_n^2}{(\gamma+1)B_n^2 + C_n^2}} \quad (177)$$

$$\le \frac{\delta_n}{2\sqrt{\eta}} \sqrt{\frac{(\gamma + 1)B_n^2 + C_n^2}{B_n^2 + C_n^2}} \quad (178)$$

$$\le \frac{\delta_n}{2\sqrt{\eta}} \sqrt{1 + \gamma}. \quad (179)$$
1 Comparing this last formula to (143) we see that the measurement error may
2 be amplified if $\gamma$ is large. The total error on $u_1$ is bounded by

$$
\|u_{1,\delta} - \tilde{u}(a)\|_{L^2(\mathcal{O})} \le D'' \frac{\sqrt{\eta}}{\sqrt{\gamma+1}} + \frac{\delta}{2\sqrt{\eta}} \sqrt{1+\gamma}. \quad (180)
$$

3 From (180) we define a regularization strategy by taking

$$
\eta(\delta) = \frac{\delta(1 + \gamma)}{2D''}. \tag{181}
$$

4 From definition 2.3 in [15] it is clear that the regularization strategy (181) is
5 admissible because of (180). Following this choice we can bound the total error
6 by the level of noise in the data as

$$
\|u_{1,\delta} - \tilde{u}(a)\|_{L^2(\mathcal{O})} \le \sqrt{2D''} \sqrt{\delta} \quad (182)
$$

**Remark 8** The bound (182) on the error is coarse because it does not consider the convergence of cosh $\sqrt{\lambda_n}a$ towards $\infty$. This means that this method is less sensitive to high frequency errors on the data than the estimate given by (145). In practice one can hope for better results. Once again the parameter $\varepsilon$ does not appear. The correponding regularization helps to ensure the stability of the numerical integration of the equation for $u$. This was not considered in this analysis.

14 **3.6.4 Comparison of the bounds of error for Algorithm 3 and 4**

15 To summarize Algorithm 3 it includes two regularizations:

16 • The first one aims to stabilize the integration in $x$ of (13) using a reference
17 to a backward stable problem as in Algorithm 2. The tuning parameter
18 is $\varepsilon$.

• In the second regularization we relax the initial conditions for $u$ and $\bar{u}$. The tuning parameter was $\eta$. The results for analyzing this regularization strategy involve only $\eta$ because the bounds are uniform in $\varepsilon$.

22 **3.7 Comparison with the Kohn-Vogelius method**

23 The Kohn-Vogelius method is largely used for solving inverse problems of the
24 class considered here. In this method two states are considered: $u_D$ that
25 satisfies the Dirichlet boundary condition on the measured part of the domain
26 boundary and $u_N$ that satisfies the Neumann boundary condition on the same
27 part. Both satisfy a Dirichlet boundary condition $u_1$ on the unmeasured part
1 of the boundary which is sought to minimize the difference $u_D - u_N$.

2 The Kohn-Vogelius method with regularization reads

$$ \min_{u_1} J(u_1) \qquad (183) $$

3 where

$$ J(u_1) = \frac{1}{2} \left[ \int_0^a \| \frac{d}{dx}(u_N - u_D) \|^2 dx + \int_0^a \| \nabla_y (u_N - u_D) \|^2 dx \right] + \frac{\eta}{2} \|u_1\|^2, \qquad (184) $$

4 where $u_N, u_D$ satisfy

$$ \Delta u = 0, \quad u|_{\Sigma} = 0 \qquad (185) $$

5 with the boundary conditions

$$ u_D|_{\Gamma_0} = T, \quad \frac{\partial u_N}{\partial n}|_{\Gamma_0} = 0, \quad u_D|_{\Gamma_a} = u_N|_{\Gamma_a} = u_1 \qquad (186) $$

Expanding as previously on the basis $\{\varphi_n\}_{n=1}^\infty$ we have

$$ u_D = \sum_n u_D^n \varphi_n, \quad u_D^n = u_1^n \frac{\sinh \sqrt{\lambda_n} x}{\sinh a\sqrt{\lambda_n}} - T^n \frac{\sinh \sqrt{\lambda_n} (x-a)}{\sinh a\sqrt{\lambda_n}}, \qquad (187) $$

$$ u_N = \sum_n u_N^n \varphi_n, \quad u_N^n = u_1^n \frac{\cosh \sqrt{\lambda_n} x}{\cosh a\sqrt{\lambda_n}}. \qquad (188) $$

Considering the $n^{th}$ component

$$ u_N^n - u_D^n = 2(u_1^n \frac{\sinh \sqrt{\lambda_n} (x-a)}{\sinh 2a\sqrt{\lambda_n}}) + T_n \frac{\sinh \sqrt{\lambda_n} (x-a)}{\sinh a\sqrt{\lambda_n}} \qquad (189) $$

$$ = 2 \frac{(u_1^n - \cosh(a\sqrt{\lambda_n})T_n) \sinh \sqrt{\lambda_n} (x-a)}{\sinh 2a\sqrt{\lambda_n}}, \qquad (190) $$

6 and

$$ \frac{d}{dx}(u_N^n - u_D^n) = 2\sqrt{\lambda_n} \frac{(u_1^n - \cosh(a\sqrt{\lambda_n})T_n) \cosh \sqrt{\lambda_n} (x-a)}{\sinh 2a\sqrt{\lambda_n}}. \qquad (191) $$

Hence

$$ J(u_1) = \frac{1}{2} \sum_n \frac{(u_1^n - T_n \cosh a\sqrt{\lambda_n})^2}{\sinh 2a\sqrt{\lambda_n}} + \frac{\eta}{2} |u_1|^2 \qquad (192) $$

$$ = \frac{1}{2} u_1^T A u_1 + u_1^T F + G \qquad (193) $$
1 where A a diagonal matrix, F a vector are defined by

$$A_{n,n} = \frac{1}{2 \sinh 2a\sqrt{\lambda_n}} + \frac{\eta}{2}, \quad F_n = \frac{-T_n \cosh a\sqrt{\lambda_n}}{\sinh 2a\sqrt{\lambda_n}}, \qquad (194)$$

2 (G is a constant). The data $(T, 0)$ are compatible if

$$T = u_N(0), \qquad (195)$$

3 that is if

$$\sum_n T_n^2 (\cosh(a\sqrt{\lambda_n}))^2 < \infty. \qquad (196)$$

4 We can compute the $n^{th}$ component of the optimal value $\hat{u}_1^\eta$ by

$$\hat{u}_1^{n,\eta} = \frac{T_n \cosh a\sqrt{\lambda_n}}{1 + \eta \sinh 2a\sqrt{\lambda_n}} = \frac{T_n \cosh a\sqrt{\lambda_n}}{1 + 2\eta \sinh a\sqrt{\lambda_n} \cosh a\sqrt{\lambda_n}}. \qquad (197)$$

Then we have

$$\|\hat{u}_1^\eta\|^2 = \sum_n |\hat{u}_1^{n,\eta}|^2 \qquad (198)$$

$$< \sum_n \frac{T_n^2}{4\eta^2(\sinh a\sqrt{\lambda_n})^2} < \infty, \qquad (199)$$

5 this proves that $\hat{u}_1^\eta$ is well defined in $L^2(\mathcal{O})$.

6 **Remark 9** Formulas (163) and (197) provide another way to estimate the

7 state on $\Gamma_a$.

8 **3.7.1 Error due to regularization**

9 Let $\tilde{u}$ be the exact value of $u$. It has the components $\tilde{u}_n(a) = T_n \cosh(a\sqrt{\lambda_n})$ on $\Gamma_a$.

11 **Proposition 3.15** Assume $(T, 0)$ are compatible. Then

$$\hat{u}_1^\eta \to \tilde{u}(a) \quad \text{weakly in } L^2(\mathcal{O}) \text{ as } \eta \to 0 \qquad (200)$$

12 Furthermore if

$$\sum_n T_n^2 (\cosh a\sqrt{\lambda_n})^2 2 \sinh 2a\sqrt{\lambda_n} < \infty \qquad (201)$$

13 then

$$\|\hat{u}_1^\eta - \tilde{u}(a)\|_{L^2(\mathcal{O})} < C\sqrt{\eta}\|\tilde{u}(a)\|_{L^2(\mathcal{O})}. \qquad (202)$$
1 **Proof** From (197) we have

$$ \hat{u}_1^{n,\eta} - \tilde{u}_n(a) = -\eta T_n \frac{\sinh 2a\sqrt{\lambda_n} \cosh a\sqrt{\lambda_n}}{1 + \eta \sinh 2a\sqrt{\lambda_n}}. \quad (203) $$

2 Hence each component of $\hat{u}_1^\eta$ converges to $\tilde{u}_1(a)$ when $\eta$ goes to 0. We have

$$ \sum_n |\hat{u}_1^{n,\eta} - \tilde{u}_n(a)|^2 \le \sum_n T_n^2 (\cosh a\sqrt{\lambda_n})^2 < \infty. \quad (204) $$

3 Therefore $\hat{u}_1^\eta - \tilde{u}(a)$ has a weak limit for $\eta$ going to 0 which must be 0. Define

4 the bilinear form $a$ by

$$ a(u, v) = \sum_n A_{n,n}(u_n, v_n) \quad (205) $$

From the definitions of $\hat{u}_1$ and $\tilde{u}$ we have

$$ \sum_n (\hat{u}_1^{n,\eta} - \tilde{u}_n(a))^2 \left(\frac{\eta}{2} + A_{n,n}\right) = \eta \sum_n (\hat{u}_1^{n,\eta} - \tilde{u}_n(a), \tilde{u}_n(a)) \quad (206) $$

$$ \le \frac{1}{2} \left( \sum_n A_{n,n} (\hat{u}_1^{n,\eta} - \tilde{u}_n(a))^2 + \frac{\eta^2}{2} \sum_n \frac{\tilde{u}_n(a)^2}{A_{n,n}} \right) \quad (207) $$

5 consequently

$$ \sum_n (\hat{u}_1^{n,\eta} - \tilde{u}_n(a))^2 \le \eta \sum_n \frac{\tilde{u}_n(a)^2}{A_{n,n}}. \quad \Box \quad (208) $$

6 **Remark 10** The comparison with the method presented in Section 3.4.1 shows

7 that as the convergence for a vanishing regularization was in $L^2$ strong for the

8 former, the convergence here is weaker as we have only a weak convergence for

9 the Kohn-Vogelius method for general admissible data. Here we need additional

10 regularity on the data (201) to obtain the estimate in norm. This additional

11 regularity is called "smoothness assumption" in [15]. It should be noticed that

12 condition (201) is stronger than condition (169) for Algorithm 3.

13 **3.7.2 Influence of an error on the data and total error**

14 As in Section 3.6.3 we evaluate the influence of an error $\delta$ of the data $T$ on
15 the estimate $\hat{u}_1$ of the state on $\Gamma_a$.

**Proposition 3.16** Let $\delta \in L^2(\mathcal{O})$ be the error on the data $T$ on $\Gamma_0$. Then
the corresponding error on the estimate on $\Gamma_a$ is bounded by

$$ \frac{\|\delta\|}{2\eta(\sinh a\sqrt{\lambda_1})} $$

16 The total error is bounded by (215)
1. **Proof** By (197) we have

$$ \hat{u}_1^{n, \eta, \delta} - \hat{u}_1^{n, \eta} = \frac{\cosh a\sqrt{\lambda_n}}{1 + \eta \sinh 2a\sqrt{\lambda_n}} \delta_n, \quad (209) $$

which yields

$$ \sum_n |\hat{u}_1^{n, \eta, \delta} - \hat{u}_1^{n, \eta}|^2 \le \sum_n \frac{\delta_n^2}{4\eta^2(\sinh a\sqrt{\lambda_n})^2} \quad (210) $$

$$ \le \frac{\|\delta\|^2}{4\eta^2(\sinh a\sqrt{\lambda_1})^2}. \quad (211) $$

For the total error

$$ \|\hat{u}_1^{\eta, \delta} - \tilde{u}(a)\|_{L^2(\mathcal{O})} \le \|\hat{u}_1^{\eta, \delta} - \hat{u}_1^{\eta}\|_{L^2(\mathcal{O})} + \|\hat{u}_1^{\eta} - \tilde{u}(a)\|_{L^2(\mathcal{O})} \quad (212) $$

$$ \le \frac{\|\delta\|}{2\eta(\sinh a\sqrt{\lambda_1})} + C\sqrt{\eta}\|\tilde{u}(a)\|_{L^2(\mathcal{O})}. \quad (213) $$

2. This bound is similar to (145) but less sharp because the effect of the error on

3. the data is in $\frac{1}{\eta}$ instead of $\frac{1}{\sqrt{\eta}}$. Hence we define an admissible regularization

4. strategy by

$$ \eta(\delta) = \sqrt[3]{\frac{\delta^2}{2(\sinh a\sqrt{\lambda_1})^2 C^2 \|\tilde{u}(a)\|_{L^2(\mathcal{O})}^2}}. \quad (214) $$

5. Using this strategy we can bound the total error by

$$ \|\hat{u}_1^{\eta(\delta), \delta} - \tilde{u}(a)\|_{L^2(\mathcal{O})} \le C' \delta^{1/3} \quad (215) $$

6. **4 Conclusion and summary of results**

7. This paper presents a preliminary study for the reconstruction of the potential

8. map on the heart starting from the measures of potential on the chest. The error

9. bounds for the case of a cylindrical geometry are presented in the following

10. table.
1

* Algorithm 1: Regularization using a parabolic state equation (30). An admissible regularization strategy is given in (52) and (55).

* Algorithm 2: Regularization with modulation of the regularizing term with reference to a backward problem (57), (77), (76). An admissible regularization strategy is (111) which leads to a total error bound (112).

* Algorithm 3: A control problem where the control acts on the final state and the objective is to minimize the discrepancy between the measurement and the initial state. An admissible regularization strategy is (135) which leads to a bound on the total error (136).

* Algorithm 4: Regularization strategy through a control problem acting on both the initial and final state (77), (117), (150), (152). An admissible regularization strategy is (181) under assumption (169); it depends on the square root of the level of noise on the data. This strategy seems better than that obtained using the Kohn-Vogelius method (214).

* Kohn-Vogelius method (183) (184). The regularization strategy gives a slower convergence in term of the size of the error on the data (214) because it is in $O(\delta^{1/3})$ and it requires a stronger assumption (201) on the regularity of the state to be estimated.

Therefore, these theoretical estimates show the better behavior of the proposed methods compared to the Kohn-Vogelius method. Some numerical experiments will be undertaken to confirm these results.

Furthermore we would like to extend them to more general domains. Some extensions were described in [4] dealing with a general space invariant embedding method for the direct problem. Their use for the inverse problem will be investigated.

Finally the time dependent problem will be investigated. When using an ECG cardiologists pay much attention to the time relation between the signals of various electrodes. Combining this with a spatial information from an inverse problem would enhance the diagnosis.

13

14

**Data availability statement:**
1

No new data were created or analysed in this study.

2

3

**Conflict of Interest declaration:**

4

The authors declare that they have no affiliations with or involvement in any
organization or entity with any financial interest in the subject matter or
materials discussed in this manuscript.

5

6

7

**References**

8

9

[1] E. Angel and R. Bellman, *Dynamic Programming and Partial Differential Equations*, Academic Press, 1972.

10

[2] R. Bellman, *Dynamic Programming*, Princeton University Press, 1957.

11

12

[3] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, *Representation and Control of Infinite Dimensional systems*, Birkhäuser, 2007.

13

14

15

[4] J. Henry and A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, *Nonlinear Analysis. Theory, Methods & Applications, 59* (2004), 629 - 647.

16

17

[5] J.L. Lions, *Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles*, Dunod, 1968.

18

19

[6] J.L. Lions and E. Magenes, *Problèmes aux limites non homogènes et applications*, vol 1, Dunod, 1968.

20

21

22

[7] L. Tartar, Sur l'étude directe d'équations non linéaires intervenant en théorie du contrôle optimal, *Journal of Functional Analysis* **17**, 1,1-47,1974.

23

24

25

26

[8] J. Henry, A.M. Ramos, *La méthode de factorisation des problèmes aux limites par plongement invariant*, ISTE éditions 2016. English translation *Factorization of Boundary Value Problems Using the Invartant Embedding Method* ISTE press 2016

27

28

29

[9] A. Cimetière, F. Delvare, M. Jaoua, F. Pons, *Solution of the Cauchy problem using iterated Tikhonov regularization*, Inverse Problems, **17**(2001) 553-570

30

31

32

33

[10] M. Addouche, N. Bouaroudj, F. Jday, J. Henry, N. Zemzemi, Analysis of the ECGI inverse problem solution with respect to the measurement boundary size and the distribution of noise, *Mathematical Modelling of Natural Phenomena* **14**(2) January 2018
1

2

[11] J.-L. Lions, Contrôle des systèmes distribués singuliers Gauthier Villars
1983

3

4

5

[12] A. Benrabah, N.Boussetila, F; Rebbani, Regularization of an abstract ill-posed Cauchy problem via general quasi-reversibility method, *Applied sciences Vol 18, 2016*, pp. 1-17

6

7

8

9

[13] Eduardo Hernández-Montero, Andrés Fraguela-Collar, Jacques Henry An
optimal quasi solution for the Cauchy problem for Laplace equation in
the framework of inverse ECG. *Mathematical Modelling of Natural Phenomena, In press, 10.1051/mmnp/2018062*. hal-01933948

10

11

12

[14] R. Kohn, M. Vogelius, Relaxation of a variational method for impedance computed tomography, *Communications on pure and applied mathematics, 40, 6* 1987

13

14

[15] A. Kirsch, *An Introduction to the Mathematical Theory of Inverse Problems*, Applied Mathematical Sciences, 120, Springer.

15

16

[16] V.A. Morozov, *Methods for Solving Incorretly Posed Problems* Springer-Verlag 1984

17

18

[17] R.Lattès and J-L Lions, *Méthode de quasi-réversibilité et applications* Dunod Paris 1967.

19

20

21

[18] L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation, *Inverse Problems 22 (2006)* 413-430

22

23

24

[19] G. Mophou, M. Warma, Quasi-reversibility methods of optimal control for ill-posed final value diffusion equations, *Journal of Mathematical Analysis and Applications 517, 2023* 126618