Inverse source problem in a 2D linear evolution transport equation: detection of pollution source

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Inverse source problem in a 2D linearevolution transport equation: detectionof pollution sourceAdel Hamdi aa Laboratoire de Mathématiques LMI, Institut National desSciences Appliquées de Rouen, Avenue de l'Université, 76801Saint-Etienne-du-Rouvray, Cedex, France

Available online: 22 Nov 2011

To cite this article: Adel Hamdi (2012): Inverse source problem in a 2D linear evolution transportequation: detection of pollution source, Inverse Problems in Science and Engineering, 20:3, 401-421

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Inverse Problems in Science and EngineeringVol. 20, No. 3, April 2012, 401–421

Inverse source problem in a 2D linear evolution transport equation:

detection of pollution source

Adel Hamdi*

Laboratoire de Mathematiques LMI, Institut National des Sciences Appliquees de Rouen,Avenue de l’Universite, 76801 Saint-Etienne-du-Rouvray, Cedex, France

(Received 9 March 2011; final version received 30 October 2011)

This article deals with the identification of a time-dependent source spatiallysupported at an interior point of a 2D bounded domain. This source occurs in theright-hand side of an evolution linear advection-dispersion-reaction equation.We address the problem of localizing the source position and recovering thehistory of its time-dependent intensity function. We prove the identifiability of thesought source from recording the state on the outflow boundary of the controlleddomain. Then, assuming the source intensity function vanishes before reachingthe final control time, we establish a quasi-explicit identification method based onsome exact boundary controllability results that enable to determine the elementsdefining the sought source using the records of the state on the outflow boundaryand of its flux on the inflow boundary. Some numerical experiments on a variantof the surface water biological oxygen demand pollution model are presented.

Keywords: inverse source problem; exact boundary controllability; optimization;advection-dispersion-reaction equation; surface water pollution

1. Introduction

In the last few decades, we have started to see inverse problems be involved in several areasof science and engineering: in medicine, the inverse problem of electrocardiography, forexample, is employed to restore the heart activity from a given set of body surfacepotentials [1]. In seismology, inverse source problems are used to determine the hypocenterof an earthquake [2] as well as to study the dynamic problem of seismology, which is oneof the most topical problems of geophysical [3]. Here, one motivation for our studyconcerns an environmental application that consists of the identification of pollutionsources in surface water: in a river, for example, the presence of organic matter whichcould have origin such as city sewages, industrial wastes etc., usually drops to too low thelevel of the dissolved oxygen in the water. Since the lack of dissolved oxygen constitutes aserious threat to human health and to the diversity of the aquatic life, identifying pollutionsources could play a crucial role in preventing worse consequences regarding the perishingof many aquatic species as well as in alerting downstream drinking water stations aboutthe presence of an accidental pollution occurring in the upstream part of the river.

*Email: [email protected]

ISSN 1741–5977 print/ISSN 1741–5985 online

� 2012 Taylor & Francis

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In this article, we are interested in the inverse source problem that consists of localizinga sought pollution source spatially supported at an interior point of a 2D bounded domainand recovering the history of its time-dependent intensity function using some boundaryrecords related to the biological oxygen demand (BOD) concentration. This concentrationrepresents the amount of dissolved oxygen required by the micro-organism living in thewater to decompose the introduced organic substances [4,5]. Therefore, the more organicmaterial there is, the higher the BOD concentration. This article is organized as follows.Section 2 is devoted to stating the problem, assumptions and proving a technical lemmafor later use. In Section 3, we prove the identifiability of the sought source from recordingthe BOD concentration on the outflow boundary of the controlled domain. In Section 4,we establish an identification method that uses the records of the BOD concentration onthe outflow boundary and of its flux on the inflow boundary to determine the elementsdefining the sought source. Section 5 is reserved for deriving the identification results in thecase of a rectangular domain. Some numerical experiments on a variant of the surfacewater BOD pollution model are presented in Section 6.

2. Governing equations and problem statement

We consider a portion of a river represented by a bounded and connected open set � of IR2

with a Lipschitz boundary @�¼�N[�D. Here, �D denotes the inflow boundary of thedomain � whereas �N¼�L[�out with �L representing the lateral solid boundaries and�out the outflow boundary of �. The BOD concentration, denoted here by u, is governedby the following two-dimensional linear advection-dispersion-reaction equation [6,7]:

P½u� ¼ F in QT :¼ �� ð0,T Þ ð1Þ

where T is the final control time, F represents the pollution source and P is the linearparabolic partial differential operator defined as follows:

P½u� ¼ @tu� divðDruÞ þ Vruþ Ru ð2Þ

where V is the flow velocity, R is the reaction coefficient and D denotes the diffusiontensor. Here, V satisfies the so-called dry lateral solid boundary conditions which meansthat along �L, the normal component of the velocity field V vanishes:

V � � ¼ 0 on �L � ð0,T Þ ð3Þ

where � is the unit vector normal and exterior to the boundary @�. In the remainder,we assume some available mean flow velocity data: V¼ (V1,V2)

>. Then, according to [8,9],the diffusion tensor D is defined by the 2� 2 symmetric matrix D ¼ ½D1 D0

D0 D2� where the

entries D1, D0 and D2 are given by

D1 ¼DLV

21 þDTV

22

kVk22, D0 ¼

V1V2ðDL �DTÞ

kVk22and D2 ¼

DLV22 þDTV

21

kVk22ð4Þ

with kVk2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2

1 þ V22

qand DL, DT are the longitudinal and transverse diffusion coefficients.

As in the most cases of interest we have DLDT 6¼ 0 and since det(D)¼DLDT, we assume inthe remainder that the matrix D is invertible. To the evolution equation (1), one has toappend initial and boundary conditions. For the initial condition, we could use withoutloss of generality no pollution occurring at the initial control time and thus a null initialBOD concentration. As far as the boundary conditions are concerned, an homogeneous

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Dirichlet condition on the inflow boundary seems to be reasonable since the convective

transport generally dominates the diffusion process. However, other physical consider-

ations suggest the use of a rather Neumann homogeneous condition on the remaining part

of the border. Therefore, we employ the following homogeneous conditions:

uð:, 0Þ ¼ 0 in �u ¼ 0 on

PD :¼ �D � ð0,T Þ

Dru � � ¼ 0 onP

N :¼ �N � ð0,T Þð5Þ

Note that due to the linearity of the operator P and according to the superposition

principle, the use of a non-zero initial condition and/or inhomogeneous boundary

conditions do not affect the results established in this article.In this study, we are interested in the inverse source problem that consists of the

identification of the source F involved in (1) using some boundary records related to

the state u. The main difficulty in such kind of inverse problem is the no identifiability of

the source F in its general form. To illustrate this difficulty, we introduce the following

common counterexample [10]: let f2D(�) be an infinitely differentiable function with

compact support in � and g¼�Df. Then, v(x, y, t)¼ tf (x, y) satisfies

@tv� Dv ¼ f þ tg in �� ð0,T Þvð�, 0Þ ¼ 0 in �v ¼ rv � � ¼ 0 on @�� ð0,T Þ

ð6Þ

without having necessarily fþ tg null. Therefore, same boundary records may lead to

identification of different sources. In the literature, to overcome this difficulty, authors

generally assume some available a priori information on the source F: for example,

time-independent sources F(x, t)¼ f (x) are treated by Cannon [11] using spectral theory,

then by Engl et al. [12] using the approximated controllability of the heat equation. The

results of this last article are generalized by Yamamoto [13,14] to sources of the form

F(x, t)¼ �(t) f (x) where f2L2 and the time-dependent function �2C1[0,T ] is assumed to

be known and satisfying the condition �(0) 6¼ 0. Furthermore, Hettlich and Rundell [15]

addressed the 2D inverse source problem for the heat equation with sources of the form

F(x, t)¼�D(x) where D is a subset of a disk. They proved the identifiability of D from

recording the flux at two different points of the boundary. El Badia and Hamdi [16,17]

studied the case of a 1D time-dependent point source F(x, t)¼ �(t)�(x�S ) where the

source position S and the time-dependent intensity function � are both unknown. They

proved the identifiability of the sought source from recording the state and its flux at two

interior points framing the source region. Those results have been recently improved by

Hamdi [18,19] to requiring only the record of the state at the two observation points, then

extended to the 2D stationary problem in [20]. Finally, the case where the source depends

on the state F(x, t)¼G(u(x, t)) was considered by DuChateau and Rundell [21] and

Cannon and DuChateau [22].In this article, we consider a 2D time-dependent point source F defined as follows:

Fðx, y, tÞ ¼ �ðtÞ�ðx� Sx, y� SyÞ ð7Þ

where � denotes the Dirac mass, S¼ (Sx,Sy) is an interior point to the domain � that

represents the source position and � is its time-dependent intensity function which is

assumed to be in L2(0,T ). In [23], it was shown for a similar problem that using a source F

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as introduced in (7) implies that (1)–(5) admits a unique solution u which belongs to

L2ð0,T;L2ð�ÞÞ \ Cð0,T;H�1ð�ÞÞ ð8Þ

Since the source position S is assumed to be interior to the domain �, the state u is smooth

enough on the boundary @� to define the following boundary observation operator:

M½F � :¼�Dru � � on

PD, u on

Pout

�ð9Þ

whereP

out :¼�out� (0,T ). This is the so-called direct problem.The inverse problem with which we are concerned here is that: given � the record of

Dru � � onP

D and f the record of u onP

out, find the elements Sx, Sy and the

time-dependent intensity function � defining the sought source F as introduced in (7)

such that

M½F � ¼�� on

PD, f on

Pout

�ð10Þ

As far as the regularity of the state u at a particular instance t2 (0,T ) is concerned, the

authors in [10,24] proved for a similar problem that the assumption

9T0 2 ð0,T Þ such that �ðtÞ ¼ 0 for all T 0 5 t � T ð11Þ

implies that we have

uð�,T�Þ 2 L2ð�Þ for all T 0 5T� � T ð12Þ

Let us introduce the new variable w defined from the state u as follows:

wðx, y, tÞ ¼ e�xþ�yuðx, y, tÞ in QT ¼ �� ð0,T Þ ð13Þ

where � and � are two real numbers. Then, since det(D)¼DLDT 6¼ 0, we choose the

coefficients � and � such that

D��

� �¼ �

1

2V,

��

� �¼ �

1

2 detðDÞ

D2 �D0

�D0 D1

� �V ð14Þ

which, in view of (4), leads to

� ¼D0V2 �D2V1

2 detðDÞ¼ �

V1

2DLand � ¼

D0V1 �D1V2

2 detðDÞ¼ �

V2

2DLð15Þ

Therefore, by changing the variable u to w, the problem (1)–(5) with the source F

introduced in (7) is rewritten as

@tw� divðDrwÞ þ �w ¼ �ðtÞe�xþ�y�ðx� Sx, y� SyÞ in QT

wð�, 0Þ ¼ 0 in �w ¼ 0 on

PD

Drw � � ¼ 0 onP

L

Drwþ1

2wV

� �� ¼ 0 on

Pout

ð16Þ

whereP

L :¼�L� (0,T ) and � is the real number defined by

� ¼ Rþ��

� �>D

��

� �¼)� ¼ R�

1

2V>

��

� �ð17Þ

In addition, for later use we need to prove the following lemma.

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LEMMA 2.1 Assume (11) holds and let T � be such that T 0<T �<T. The elements defining

the source F introduced in (7) are subject to

�� ¼ 1e2ð�Sxþ�SyÞ ¼ 2=1

ð18Þ

where �� ¼R T 0

0 e�RðT��tÞ�ðtÞdt and the coefficients 1, 2 are such that

1 ¼

Z�

wð�,T�Þv1ð�,T�Þd��

ZP�

D

v1Drw � � d� dtþ V � �

ZP�

out

wv1 d� dt

2 ¼

Z�


ZP�

D

v2Drw � � d� dt

ð19Þ

where w is the solution to (16) and vi ¼ e�RðT��tÞþð�1Þið�xþ�yÞ for i¼ 1, 2. Here,

P�D :¼

�D � ð0,T�Þ and

P�out :¼ �out � ð0,T

�Þ.

Proof Assuming (11) holds, let T � be such that T 0<T �<T and v1, v2 be the two

functions defined in QT� ¼ �� ð0,T�Þ as follows:

viðx, y, tÞ ¼ e�RðT��tÞþð�1Þið�xþ�yÞ for i ¼ 1, 2 ð20Þ

where � and � are the two real numbers derived in (15). Then, using (14) we find

Drvi ¼ �ð�1Þi

2viV and divðDrviÞ ¼ �

1

2viV> ��

� �ð21Þ

Therefore, in view of (3) and (17), vi introduced in (20) satisfies for i¼ 1, 2 the following

adjoint system:

�@tvi � divðDrviÞ þ �vi ¼ 0 in QT�

við�,T�Þ ¼ eð�1Þ

ið�xþ�yÞ in �

vi onP�

D

Drvi � � ¼ 0 onP�

L

Drvi � � ¼ �ð�1Þi

2viV � � on

P�out

ð22Þ

Multiplying the first equation in (16) by vi and integrating by parts over QT� using

Green’s formula gives the following for i¼ 1, 2:

��eð1þð�1ÞiÞð�Sxþ�SyÞ ¼

Z�

wð�,T�Þvið�,T�Þd��

ZP�

D

viDrw � � d� dt

þ1

2

1� ð�1Þi

V � �

ZP�

out

viw d� dt ð23Þ


0 �ðtÞe�RðT��tÞ dt. By substituting the function vi in (23) with its value given in

(20), we obtain the following system:

�� ¼

Z�


ZP�

D

v1Drw � � d� dtþ V � �

ZP�

out

wv1 d� dt

��e2ð�Sxþ�SyÞ ¼

Z�


ZP�

D

v2Drw � � d� dt

ð24Þ

Then, using the two equations in (24), we find the results announced in (18)–(19). g


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3. Identifiability

Here, we prove the identifiability of the time-dependent point source F introduced in (7)from recording the state u on the outflow boundary

Pout. This theorem is inspired by the

results derived in [10].

THEOREM 3.1 The time-dependent point source F introduced in (7) is uniquely determinedby the record f of the state u on the boundary

Pout¼�out� (0,T ).

Proof Let ui be the solution to the problem (1)–(5) with the time-dependent point sourceFiðx, y, tÞ ¼ �iðtÞ�ðx� Si

x, y� SiyÞ and fi be the record of ui on

Pout for i¼ 1, 2. If f1¼ f2,

then the variable �u¼ u2� u1 satisfies a system similar to the problem (1)–(5) where theright-hand side of Equation (1) is F2�F1 and the boundary condition

Dr �u � � ¼ �u ¼ 0 onP

out ð25Þ

Let D be an open disk of IR2 such that (D\ @�)��out and Si ¼ ðSix,S

iyÞ =2D for i¼ 1, 2.

Therefore, the open O1¼ (D\�) is a subset of �n{Si}. Besides, the condition (25) enablesto extend �u by zero in O2� (0,T ) where O2 ¼ D \ ðIR

2 n�Þ. Then, �u satisfies

P½ �u� ¼ 0 in D� ð0,T Þ and �u ¼ 0 in O2 � ð0,T Þ ð26Þ

which implies, according to the Mizohata unique continuation theorem [25,26], that �u¼ 0in O1� (0,T ). Furthermore, with this the last result �u satisfies the same system than (26) in(�n{Si})� (0,T ) and {O}1� (0,T ). Therefore, a second application of the Mizohataunique continuation theorem leads to the conclusion that �u¼ 0 in (�n{Si})� (0,T ). Then,since �u2L2(�� (0,T )), it follows that �u¼ 0 in �� (0,T ). That implies F2¼F1, which isequivalent to having �2¼ �1 and S2

¼S1. g

4. Identification

In this section, we establish an identification method that determines the elements definingthe sought time-dependent point source F introduced in (7). This method not only requiresthe record f of the state u on the outflow boundary but also the record of its flux � on theinflow boundary. We proceed in two steps: the first step consists of proving a result thatexpresses Sx in terms of Sy. Then, the second step describes the recovery of thetime-dependent intensity function � once the source position S is fully known. We end thissection by giving an algorithm that details the identification procedure.

4.1. Step 1: link between the source parameters

A link between the source parameters is given by the following theorem.

THEOREM 4.1 Assume that (11) holds and let T � be such that T 0<T �<T. Then, given therecords {�, f } introduced in (10), the elements defining the source F are subject to

Sx ¼1

2�ln

P2

P1

� �� 2�Sy

� �and �� ¼ P1 ð27Þ

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0 �ðtÞe�RðT��tÞdt and P1, P2 are the following coefficients:

P1 ¼

Z�

uð�,T�Þd�þ V � �

ZP�

out

e�RðT��tÞf d� dt�

ZP�

D

e�RðT��tÞ� d� dt

P2 ¼

Z�

e2ð�xþ�yÞuð�,T�Þd��

ZP�

D

e�RðT��tÞþ2ð�xþ�yÞ� d� dt

ð28Þ

Proof According to (14), the variable w introduced in (13) satisfies

Drw � � ¼ e�xþ�yD ruþ u�

�

� ��

¼ e�xþ�y Dru�1

2uV

� �� ð29Þ

Since u¼ 0 onP

D, it then follows from (29) that Drw � �¼ e�xþ �yDru � � onP

D.

Therefore, using (13) and the records {�, f } introduced in (10), the proof of Theorem 4.1 is

an immediate consequence of Lemma 2.1. g

Remark 4.2 Note that for �¼ 0, which is according to (15) the case when V2¼ 0

(unidirectional velocity field), the computation of Sx using (27)–(28) does not require the

knowledge of Sy. In practice, this remark can be very useful especially in the case of an

accidental pollution source where the knowledge of Sx could lead to deduce Sy without any

need of an additional computation.However, as the state u is subject to only the knowledge of its value on

Pout and of its

flux Dru � � onP

D, the determination of the parameters Sx, Sy and �� using (27)–(28) is notso far possible since the coefficients P1 and P2 involve the unknown data u(�,T �).

To determine the two integrals in (28) involving the unknown data u(�,T �),

we introduce for a given fixed T � in (T 0,T ) the following two exact boundary

controllability problems: for i¼ 1, 2 find a boundary control i such that the solution to

@t’i � divðDr’iÞ þ �’i ¼ 0 in QT�T :¼ �� ðT�,T Þ’ið�,T

�Þ ¼ 0 in �’i ¼ i on

PT�TD :¼ �D � ðT

�,T Þ

Dr’i � � ¼ 0 onPT�T

N :¼ �N � ðT�,T Þ

ð30Þ

satisfies ’ið�,T Þ ¼ eð�1Þið�xþ�yÞ in � ð31Þ

Moreover, using s¼TþT �� t, ’ið�, sÞ ¼ ’ið�, tÞ and ið�, sÞ ¼ ið�, tÞ we obtain the two

equivalent problems: for i¼ 1, 2, find i such that the solution ’i to

�@s’i � divðDr’iÞ þ �’i ¼ 0 in QT�T

’ið�,T Þ ¼ 0 in �’i ¼ i on

PT�TD

Dr’i:� ¼ 0 onPT�T

N

ð32Þ

satisfies ’ið�,T�Þ ¼ eð�1Þ

ið�xþ�yÞ in � ð33Þ

Then, we prove the following proposition.


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PROPOSITION 4.3 Assume that (11) holds, T � 2 (T 0,T ) and for i¼ 1, 2 there exists i inL2ð

PT�TD Þ solution to the exact boundary controllability problem (30)–(31). Then, given the

records {�, f } introduced in (10), we haveZ�

uð�,T�Þd� ¼1

2V � �

ZPT�T

out

e�xþ�y’1f d� ds�

ZPT�T

D

e�xþ�y 1� d� ds

Z�

uð�,T�Þe2ð�xþ�yÞ d� ¼1

2V � �

ZPT�T

out

e�xþ�y’2f d� ds�

ZPT�T

D

e�xþ�y 2� d� ds

ð34Þ

where ’i is the solution to (32) with the boundary control i andPT�T

out :¼ �out � ðT�,T Þ.

Proof In view of (11) and (12), the variable w introduced in (13) satisfies in QT�T a systemsimilar to (16) where the right-hand side of the first equation is zero and the initialcondition is w(�,T �)2L2(�). Therefore, multiplying the first equation of this system by ’i,the solution to (32) and integrating by parts over QT�T using Green’s formula givesZ

�

wð�,T�Þ’ið�,T�Þd� ¼

1

2V � �

ZPT�T

out

’iw d� ds�

ZPT�T

D

iDrw � � d� ds, i ¼ 1, 2 ð35Þ

Then, using the initial condition ’ið�,T�Þ ¼ eð�1Þ

ið�xþ�yÞ in (35) and as established in (29)

that Drw � �¼ e�xþ �yDru � � onPT�T

D , we obtain in view of (13) and by employing therecords {�, f } the results announced in (34). g

4.1.1. Determination of the HUM boundary control i

As far as solving the exact boundary controllability problem (30)–(31) is concerned, we usethe Hilbert uniqueness method (HUM) introduced by Lions [27,28] to determine theso-called HUM boundary control i. To this end, we introduce the adjoint problem

�@tz� divðDrzÞ þ �z ¼ 0 in QT�T

zð�,T Þ ¼ z0 in �z ¼ 0 on

PT�TD

Drz � � ¼ 0 onPT�T

N

ð36Þ

where T � 2 (T 0,T ) and z02H1(�). We start by establishing a necessary and sufficientcondition on i to be a solution to the exact boundary controllability problem (30)–(31).

THEOREM 4.4 The solution ’i2L2(T �,T;H2(�)) to the problem (30) with a boundary

control i 2 L2ðPT�T

D Þ satisfies (31) if and only if

h i,Drz � �iL2ðPT�T

DÞþ eð�1Þ

ið�xþ�yÞ, z0

D EL2ð�Þ¼ 0 ð37Þ

for all z2L2(T �,T;H2(�)) solution to the adjoint problem (36) with z02H1(�).

Proof In view of (30), (36) and using Green’s formula, we find

�h’i, ziL2ð�Þ

�TT�¼

Z T

T�h@t’i, ziL2ð�Þ þ h’i, @tziL2ð�Þ

dt

¼

Z T

T�

hdivðDr’iÞ � �’i, ziL2ð�Þ þ h’i,�divðDrzÞ þ �ziL2ð�Þ

dt

¼ �h i,Drz � �iL2ðPT�T

DÞ

ð38Þ

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Therefore, since ’i(�,T�)¼ 0 in �, the necessary condition is an immediate consequence

of (38). Furthermore, by subtracting (37) from (38), we obtain

h’ið�,T Þ, z0iL2ð�Þ ¼ he

ð�1Þið�xþ�yÞ, z0iL2ð�Þ, for all z0 2 H1ð�Þ ð39Þ

which implies that ’ið�,T Þ ¼ eð�1Þið�xþ�yÞ in �. g

To determine the HUM boundary control i for i¼ 1, 2, we introduce the function Jidefined for all z solution to the adjoint problem (36) with z(�,T )¼ z02H1(�) as follows:

Jiðz0Þ ¼

1

2kDrz � �k2

L2ðPT�T

DÞ� h’ið�,T Þ, z

0iL2ð�Þ ð40Þ

Note that, as established in [29], one proves that the function Ji introduced in (40) is

strictly convex and coercive. That implies the existence and the unicity of its minimizer. In

the remainder, for i¼ 1, 2 we denote by z0i the minimizer of the function Ji. Then,

according to the first-order optimality condition, we have the following for all z0 in H1(�):

rJiðz0i Þ � z

0 ¼ hDrzi � �,Drz � �iL2ðPT�T


0iL2ð�Þ

¼ 0 ð41Þ

where zi and z are the solutions to the adjoint problem (36), respectively, with zið�,T Þ ¼ z0iand z(�,T )¼ z0. We define the HUM boundary control i as follows [29,30]:

i¼�Drzi � �. Therefore, in view of (41), it is clear that i fulfils (37), and thus it is a

solution to the exact boundary controllability problem (30)–(31).Consequently, to determine the HUM boundary control i, we need to compute the

minimizer z0i of the function Ji. To this end, we introduce the following two operators

[29,30]: GT: H1ð�Þ �!L2ðPT�T

D Þ such that, to a given z0, associates GT(z0)¼�Drz � �

where z is the solution to the adjoint problem (36) with the initial data z(�,T )¼ z0. The

second operator is G�T : L2ðPT�T

D Þ �!H1ð�Þ such that, to a given boundary control ,G�Tð Þ ¼ ’ð�,T Þ, the solution to the problem (30) taken at the final time T. Hence,

according to (38), we find

hG�Tð Þ, z0iL2ð�Þ ¼ �h ,Drz � �iL2ð

PT�T

DÞ¼ h ,GTðz

0ÞiL2ðPT�T

DÞ

ð42Þ

which implies that GT and G�T are two adjoint operators. Then, in view of (41), we obtain

rJiðz0i Þ � z

0 ¼ hGTðz0i Þ,GTðz

0ÞiL2ðPT�T


0iL2ð�Þ

¼ hG�TGTðz0i Þ � ’ið�,T Þ, z

0iL2ð�Þ

¼ 0 ð43Þ

for all z0 in H1(�). Therefore, the minimizer z0i of Ji can be determined from solving

G�TGTðz0i Þ ¼ ’ið�,T Þ where G�TGT : H1ð�Þ �!H1ð�Þ ð44Þ

Inner product method: To determine the minimizer z0i , the inner product method [29] uses

a matrix A as a representation to the controllability operator G�TGT introduced in (44).

The entries of this matrix A are defined by following proposition.


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PROPOSITION 4.5 Let (ek)k0 be a complete orthonormal family of L2(�). If ek2H1(�)

for all k 0, then the entries of the matrix A are given for all l 0 and k 0 as follows:

Alk ¼

ZPT�T

D

Drzk � �

Drzl � �

d� dt ð45Þ

where zk and zl are the solutions to (36) with the initial data zk(�,T )¼ ek and zl(�,T )¼ el.

Proof Let (ek)k0 be a complete orthonormal family of L2(�). Then, for all two elementsX, Y of L2(�), we have X ¼

Pk0 xkek and Y ¼

Pk0 ykek. Therefore, AX¼Y leads toX

k0

xkAek ¼Xk0

ykek¼)Xk0

xkhAek, eliL2ð�Þ ¼ yl, l 0

That implies Alk ¼ hAek, eliL2ð�Þ. Since for all k 0 the function ek is in H1(�), using thecontrollability operator introduced in (44) and in view of (42) we find

Alk ¼ hAek, eliL2ð�Þ ¼ hG�TGTðekÞ, eliL2ð�Þ ¼ hGTðekÞ,GTðel Þi

L2ðPT�T

DÞ

for all l 0 and k 0. This is the result announced in (45). g

Therefore, using the inner product method, we can determine for i¼ 1, 2 theminimizer z0i of the function Ji introduced in (40) from solving the following linear system:

Ai ¼ bi ð46Þ

where A is the matrix defined in Proposition 4.5, the kth component of the vector i isik ¼ hz

0i , ekiL2ð�Þ and of the vector bi is bik ¼ h’ið�,T Þ, ekiL2ð�Þ. As far as the complete

orthonormal family (ek)k0 is concerned, one could use the normalized eigenfunctions thatsolve the following eigenvalue problem:

�divðDrekÞ þ �ek ¼ kek in �ek ¼ 0 on �D

Drek � � ¼ 0 on �N

ð47Þ

That implies the solution zk to the adjoint problem (36) with the initial data zk(�,T )¼ ek isgiven by zkð�, tÞ ¼ e�kðT�tÞek, which in view of (45) leads to find

Alk ¼

ZPT�T

D

e�ðlþkÞðT�tÞDrek � �

Drel � �

d� dt

¼1� e�ðlþkÞðT�T

�Þ

l þ k

Z�D

Drek � �

Drel � �

d� ð48Þ

for all l 0 and k 0.

4.2. Step 2: recovery of the time-dependent intensity function k

Here, we assume the source position S¼ (Sx,Sy) to be fully known and focus on recoveringthe history of the time-dependent intensity function �. Let Dt> 0 be a given time step andassume there exists an integer N0> 0 such that N0Dt¼T 0. We denote by tm, form¼ 1,k,N0, the regularly distributed discrete times tm¼mDt and by ’mþ1 the solution tothe problem (30) in Qmþ1 :¼�� (0, tmþ1) with the initial data ’mþ1(�, 0)¼ 0 and the

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time-independent boundary control ¼ e��x��y onPmþ1

D :¼ �D � ð0, tmþ1Þ. Then, using

s¼ tmþ1� t, the function ’mþ1ð�, sÞ ¼ ’mþ1ð�, tÞ satisfies the following system:

�@s’mþ1 � divðDr’mþ1Þ þ �’mþ1 ¼ 0 in Qmþ1

’mþ1ð�, 0Þ ¼ ’mþ1ð�, tmþ1Þ and ’mþ1ð�, tmþ1Þ ¼ 0 in �

’mþ1 ¼ e��x��y onXmþ1D

Dr’mþ1 � � ¼ 0 onXmþ1N

ð49Þ

wherePmþ1

N :¼ �N � ð0, tmþ1Þ. Multiplying the first equation in (16) by ’mþ1 and

integrating by parts over Qmþ1 using Green’s formula gives

e�Sxþ�Sy

Z tmþ1

0

�ðtÞ’mþ1ðS, tÞdt ¼1

2V � �

ZPmþ1

out

’mþ1w d� dt�

ZPmþ1

D

e��x��yDrw � � d� dt

ð50Þ

Therefore, to recover the time-dependent intensity function �, we proceed as follows: using

the trapezoidal rule, we find the following for 1�m�N0:

Z tmþ1

0

�ðtÞ’mþ1ðS, tÞdt ¼Xmk¼0

Z tkþ1

tk

�ðtÞ’mþ1ðS, tÞdt

Dt2

Xmk¼0

�k’mþ1ðS, tkÞ þ �kþ1’mþ1ðS, tkþ1Þ

¼ DtXmk¼1

�k’mþ1ðS, tkÞ ð51Þ

where �m¼ �(tm). In (51), we used �(t0)¼ 0. Besides, using the records {�, f } given in (10)

and in view of (50), we introduce the following notations: for m¼ 1, . . . ,N0:

dmþ1 ¼ e�ð�Sxþ�SyÞ1

2V � �

ZPmþ1

out

’mþ1w d� dt�

ZPmþ1

D

e��x��yDrw � � d� dt

!

¼ e�ð�Sxþ�SyÞ1

2V � �

ZPmþ1

out

e�xþ�y’mþ1 f d� dt�

ZPmþ1

D

�d� dt

!ð52Þ

Here,Pmþ1

out :¼ �out � ð0, tmþ1Þ and according to (29), we used Drw � �¼ e�xþ�yDru � � onPmþ1D . Then, with reference to (50) and by employing (51)–(52) we obtain the following

recursive formula:

�m 1

’mþ1ðS, tmÞ

dmþ1Dt�Xm�1k¼1

�k’mþ1ðS, tkÞ

!for all m ¼ 1, . . . ,N0 ð53Þ


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Furthermore, to compute ’mþ1ðS, tkÞ for k¼ 1, . . . ,m we multiply the first equation in

@t�k � divðDr�kÞ þ ��k ¼ �ðt� tkÞ�ðx� Sx, y� SyÞ in Qmþ1

�kð�, 0Þ ¼ 0 in �

�k ¼ 0 onPmþ1D

Dr�k � � ¼ 0 onPmþ1N

ð54Þ

by ’mþ1 and integrate by parts over Qmþ1 using Green’s formula. That leads to

’mþ1ðS, tkÞ ¼ �

ZPmþ1

D

e��x��yDr�k � �d� dt ð55Þ

Moreover, in view of (11)–(12) and using the orthonormal basis (ej)j made by thenormalized eigenfunctions of the system introduced in (47), we express �k the solutionto (54) as follows:

�kð�, tÞ ¼ Hðt� tkÞXj0

ej ðSÞe�j ðt�tkÞej, for k ¼ 1, . . . ,N0 ð56Þ

where H is the Heaviside function and j is the eigenvalue associated to ej.

4.3. Identification procedure

According to the two previous steps, we see that given Sy, one can determine the sought Sx

using (27) of step 1 and the time-dependent intensity function � by employing the recursiveformula derived in (53) of step 2. Therefore, starting from some initial guess S0

y, we use thefollowing minimization problem to identify Sy and thus Sx and �:

min05Sy5‘

1

2ku� f k2

L2ðP

outÞþ1

2kDru � ��k2

L2ðP

DÞ

ð57Þ

where ‘ is the width of �. We detail the identification procedure in the followingalgorithm:

BeginData: the records � and f

(1) For i¼ 1 to 2 do

. Compute the minimizer z0i of the function Ji introduced in (40).

. Set the HUM boundary control i¼�Drzi . � where zi is the solution tothe adjoint problem (36) with the initial data zið�,T Þ ¼ z0i .

. Compute ’i, the solution to (30) with the boundary control i.

(2) Initialization of Sny.

(3) Compute the associated Snx from (27) and �nm, for m¼ 1, . . . ,N0, using (53).

(4) Determine un the solution to (1)–(5) with the source Fn ¼ �n�ðx� Snx, y� Sn

yÞ.(5) Test: if kun � f k2

L2ðP

outÞþ kDrun � ��k2

L2ðP

DÞis small enough, go to End

Otherwise, correct Sny and go to 3.

End

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5. Identification of F in a rectangular domain

Here, we apply the identification method established in the previous section to the case

where the controlled portion of the river is a rectangular domain: �¼ (0,L)� (0, ‘) and the

mean velocity vector V is perpendicular to the inflow boundary, i.e. V2¼ 0. In such

situation, the longitudinal and transverse diffusion directions coincide with the x and y

cartesian axes. And thus, according to (4), the diffusion tensor D is represented by the

2� 2 diagonal matrix of entries D1¼DL and D2¼DT. Therefore, using the dimensional

analysis method [31], the solution to the system

@tu0 � divðDru0Þ þ Vru0 þ Ru0 ¼ �ðtÞ�ðx� Sx, y� SyÞ in IR2 � ð0,T Þu0ð�, 0Þ ¼ 0 in IR2 ð58Þ

can be expressed as follows:

u0 ¼ HðtÞ�ðtÞ ?ðtÞHðtÞ

4�tffiffiffiffiffiffiffiffiffiffiffiffiD1D2

p e�ðx�Sx�V1 tÞ

2

4D1 t�ð y�SyÞ

2

4D2 t�Rt

ð59Þ

where H is the Heaviside function and ?(t) denotes the convolution product with respect to

the variable t. Furthermore, using �u such that

@tu� divðDruÞ þ Vruþ Ru ¼ 0 in QT

uð�, 0Þ ¼ 0 in �u ¼ �u0 on

PD

Dru � � ¼ �Dru0 � � onP

N

ð60Þ

implies that the solution to the problem (1)–(5) with the source F introduced in (7) is

u¼ uþ u0. Besides, using the separation of variables method, we determine the normalized

eigenfunctions solutions to the eigenvalue problem introduced in (47) for �¼ (0,L)� (0, ‘)and V2¼ 0 as follows: for all n 0 and m 0

enmðx, yÞ ¼ cnm sin ð2nþ 1Þ�

2Lx

�cos m

�

‘y

�where cnm ¼

ffiffiffiffiffiffi2

L‘

rfor m ¼ 0

2ffiffiffiffiffiffiL‘p for m4 0

8>><>>: ð61Þ

and the associated eigenvalues nm are such that

nm ¼ �þD1ðð2nþ 1Þ�=2LÞ2 þD2ðm�=‘ Þ2

ð62Þ

Let M and N be two sufficiently large integers. For simplicity, we employ the following

notations: for n¼ 0, . . . ,N� 1; m¼ 0, . . . ,M� 1 and p¼ 0, . . . ,N� 1; q¼ 0, . . . ,M� 1

ek ¼ enm and k ¼ nm where k ¼ nMþmel ¼ epq and l ¼ pq where l ¼ pMþ q

ð63Þ

Then, since the unit normal vector exterior to the inflow boundary �D is �¼ (�1, 0)>, we

find according to (48) that for l¼ 0, . . . ,NM� 1 and k¼ 0, . . . ,NM� 1

Alk ¼ D21

1� e�ðlþkÞðT�T�Þ

l þ k

Z ‘

0

@xekð0, yÞ@xel ð0, yÞdy

¼ D21


l þ kcnmcpqð2nþ 1Þð2pþ 1Þ

�

2L

�2Z ‘

0

cosm�

‘y

�cos

q�

‘y

�dy ð64Þ


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As according to (61) we have cnm¼ cpm for all integers n, m and p, the square symmetric

matrix A of order NM is defined for l¼ 0, . . . ,NM� 1 and k¼ 0, . . . ,NM� 1 as follows:

Alk ¼

�D1

22L3


l þ kð2nþ 1Þð2pþ 1Þ if m ¼ q

0 if m 6¼ q

8<: ð65Þ

We use the expansions of ’i(�,T ) and of the minimizer z0i of the function Ji introduced

in (40) in the complete orthonormal family (ek)k0: z0i ¼PNM�1

k¼0 ikek and ’ið�,T Þ ¼PNM�1k¼0 bikek. Then, in view of the linear system (46) and for each 0�m�M� 1,

we determine the N components i,mn ¼ inMþm for n¼ 0, . . . ,N� 1 from solving

Ami,m ¼ bi,m where Ampn ¼

�D1

22L3

1� e�ðpmþnmÞðT�T�Þ

pm þ nmð2pþ 1Þð2nþ 1Þ ð66Þ

and the vector bi,m is such that bi,mp ¼ h’ið�,T Þ, epmiL2ð�Þ for p¼ 0, . . . ,N� 1. As far as the

linear system introduced in (66) is concerned, we prove the following result.

PROPOSITION 5.1 For all m¼ 0, . . . ,M� 1, the real N�N matrix Am involved in the linear

system introduced in (66) is symmetric and positive definite.

Proof According to (66), we have for all p¼ 0, . . . ,N� 1 and n¼ 0, . . . ,N� 1

Ampn ¼

�D1

22L3

ð2pþ 1Þð2nþ 1Þ

Z T

T�e�ðpmþnmÞðT�tÞdt ð67Þ

Therefore, for all vector x>¼ (x0, . . . , xN�1) of IRN we have

x>Amx ¼

�D1

22L3

Z T

T�

XN�1p¼0

e�pmðT�tÞð2pþ 1Þxp

!2

dt 0 ð68Þ

Furthermore, since D1 6¼ 0, it follows from (68) that

x>Amx ¼ 0,XN�1p¼0

e�pmðT�tÞð2pþ 1Þxp ¼ 0 for almost all t 2 ðT�,T Þ ð69Þ

As for all m¼ 0, . . . ,M� 1 the sequence (pm)p is strictly increasing, then the second

equation in (69) implies that xp¼ 0 for all p¼ 0, . . . ,N� 1. g

Once the NM components ik are determined from solving the linear system (66) for

m¼ 0, . . . ,M� 1, the solution zi to the adjoint problem (36) with the initial data

zið�,T Þ ¼ z0i where z0i ¼PNM�1

k¼0 ikek is then given by

ziðx, y, tÞ ¼XNM�1

k¼0

ike�kðT�tÞekðx, yÞ for ðx, y, tÞ 2 ð0,LÞ � ð0, ‘ Þ � ðT�,T Þ ð70Þ

Hence, for i¼ 1, 2 and in view of (70) the HUM boundary control i¼�Drzi � � solutionto the exact boundary controllability problem introduced in (30)–(31) for �¼ (0,L)� (0, ‘)

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and V2¼ 0 is defined onPT�T

D ¼ �D � ðT�,T Þ as follows:

ið y, tÞ ¼ D1@xzið0, y, tÞ

¼ D1

XN�1n¼0

XM�1m¼0

inMþme�nmðT�tÞ@xenmð0, yÞ

¼ ðtÞXN�1n¼0

ð2nþ 1Þe�D1

ð2nþ1Þ �2L

2ðT�tÞ

inMffiffiffi2p þ

XM�1m¼1

inMþme�D2

m�‘

2ðT�tÞ cos

m�

‘y

� !

where (t)¼ (D1�/l1/2L3/2)e��(T�t).

6. Numerical experiments

We carry out numerical experiments in the case of a rectangular domain: �¼ (0,L)� (0, ‘)where the inflow boundary �D coincides with the y-coordinate axis and the lower lateralboundary coincides with the x-coordinate axis. Furthermore, we assume the mean velocityvector V to be perpendicular to the inflow boundary �D, i.e. V2¼ 0. That implies, asmentioned in Section 4, �¼ 0 and the diffusion tensor is represented by the 2� 2 diagonalmatrix D with entries D1¼DL and D2¼DT.

For numerical computation, we reduce the domain QT¼�� (0,T ) to the unit cubeQ¼ (0, 1)3 using the following undimensioned variables:

x1 ¼x

L, x2 ¼

y

‘and x3 ¼

t

Tð71Þ

Then, to determine ’ the solution to the problem (30) with a boundary control ,we compute �(x1,x2, x3)¼ ’(x1L, x2‘, x3T ) solution to

@x3��TD1

L2@x1x1��

TD2

‘2@x2x2�þ T�� ¼ 0 in ð0, 1Þ2 � ðx�3, 1Þ

�ðx1, x2, x3 ¼ x�3Þ ¼ 0 in ð0, 1Þ2

�ðx1 ¼ 0,x2,x3Þ ¼ ðx2‘,x3TÞ on ð0, 1Þ � ðx�3, 1Þ@x2�ðx1,x2 ¼ 0, x3Þ ¼ @x2�ðx1, x2 ¼ 1,x3Þ ¼ 0 on ð0, 1Þ � ðx�3, 1Þ@x1�ðx1 ¼ 1,x2, x3Þ ¼ 0 on ð0, 1Þ � ðx�3, 1Þ

8>>>>><>>>>>:

ð72Þ

where x�3 ¼ T�=T. Given three positive integers Nx, Ny and NT, we discretize the problem(72) using the steps Dx1¼ 1/Nx, Dx2¼ 1/Ny, Dx3¼ 1/NT and the five-point finite differencemethod with the Crank–Nicolson scheme. For numerical experiments, we takeL¼ 1000m, ‘¼ 50m, V1¼ 0.5ms�1, DL¼ 30m2 s�1 and DT¼ 10m2 s�1. We supposecontrolling the rectangular domain �¼ (0,L)� (0, ‘) for T¼ 14400 s (4 h). To generate therecords {�, f }, we use (59)–(60) with a source located at S¼ (Sx,Sy) and loading thefollowing time-dependent intensity:

�ðtÞ ¼

X3n¼1

cne�vnðt��nÞ

2

if t � T 0

0 otherwise

8><>: ð73Þ

where c1¼ 1.2, c2¼ 0.4, c3¼ 0.6 and v1¼ 10�6, v2¼ 5� 10�5, v3¼ 10�6. The coefficients �iare such that �1¼ 4.5� 103, �2¼ 6.5� 103, �3¼ 9� 103. We use Nx¼ 10, Ny¼ 5and NT¼ 210. That means, we employ four sensors on each of the inflow and the


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outflow boundaries. Those sensors are recording the evolution of Dru � � and of u with atime step Dt¼T/NT. Here, we used N0

¼N�¼ 180 which implies that T 0¼T �¼N0Dt and

equivalent to say that T 0¼T �¼ (6/7)T. As far as the computation of the boundary

control i is concerned, we employed N¼M¼ 5.As mentioned in Remark 4.2, for �¼ 0, which is the case in our numerical experiments

since we are using V2¼ 0, the determination of the x-coordinate Sx from (27) does notrequire the knowledge of Sy. Then, we carry out numerical experiments with Sy assumed tobe known and focus on studying numerically how does the introduction of a Gaussiannoise on the used records {�, f } affect the identification results of Sx using (27) in step 1and of the time-dependent intensity function � from (53) in step 2. In the sequel, we startby presenting for some introduced Gaussian noise the identified x-coordinate of the sourceposition, denoted here by Sxident , and the two curves representing the used intensity functionintroduced in (73) and the computed � using the recursive formula derived in (53). We alsocompute for each introduced Gaussian noise the following relative errors:

ErrorLam ¼

��ident ��exact

��2��exact

��2

and ErrorPosition ¼

��Sx � Sxident

��Sx

ð74Þ

where the vectors �exact ¼ ð�ðt1Þ, . . . , �ðtN0 ÞÞ> with � is the function introduced in (73) and

�ident ¼ ð�1, . . . , �N0Þ> identified from (53). Then, we draw the relative errors computed

from (74) with respect to the intensity of the introduced Gaussian noise. We carry outnumerical tests for two different source locations: S1¼ (200, 1) and S2¼ (600, 49). We startby presenting the numerical experiments corresponding to the first location S1. Then, wegive those related to the second location S2.

The numerical experiments presented above seem to be saying that the establishedidentification method enables us to identify the elements defining the sought time-dependent point source F. However, those numerical results corresponding to the sourceS1¼ (200, 1) which is located rather in the upstream part of the river seem to be relativelysensitive with respect to the introduction of a high intensity of Gaussian noise on the usedmeasures. In fact, the accuracy in Figures 1 and 2 starts to deteriorate when the intensity ofthe introduced Gaussian noise becomes relatively high. This tendency is confirmed byFigure 3. The deterioration of the accuracy in this case may be explained by the following:since the convective transport usually dominates the diffusion process, then the maininformation on the source activity seems to be the data recorded on the outflow boundary.Therefore, the bigger the distance separating the source position and the outflowboundary, the more sensitive the data recorded on this boundary.

In Figures 4–6 we present the numerical experiments associated to the second sourcelocation S2¼ (600, 49).

The analysis of the numerical experiments corresponding to the second source locationto be S2 seems to be confirming our intuition for the numerical results associated to theupstream source location S1. Indeed, we remark that even with a higher intensity of theintroduced Gaussian noise on the used measures, the results in Figures 4 and 5 are moreaccurate than those presented in Figures 1 and 2. Moreover, comparing to those presentedin Figure 3, the behaviour of the relative errors on the source location and on the intensityfunction given in Figure 6 confirms our expectation that for the downstream sourcelocation S2 the identified results will be more accurate and also stable with respect tothe introduction of a Gaussian noise on the used measures. This expectation comes fromthe fact that in the case of a downstream location, the quality of the signal observed on the

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0 2000 4000 6000 8000 10,000 12,000 14,000

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2.5Identification of the source intensity function

Time steps

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Exact intensity function

Identified intensity function

Figure 2. Graph of location S1: Noise 5%, Sxident ¼ 328m, ErrorLam ¼69.3%.

0 2000 4000 6000 8000 10,000 12,000 14,000

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1.5


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Figure 1. Graph of location S1: Noise 3%, Sxident ¼ 281m, ErrorLam ¼ 35.01%.


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0 2000 4000 6000 8000 10,000 12,000 14,000

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0 1 2 3 4 5 6 7 8 9 100.0

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1.4Relative error with respect to the noise intensity

Noise intensity

Rel

ativ

e er

ror

Relative error on the source position

Relative error on the intensity function

Figure 3. Graph of location S1: relative errors with respect to the introduced Gaussian noise.

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0 5 10 15

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0.7Relative error with respect to the noise intensity

Noise intensity

Rel

ativ

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ror

Relative error on the source position

Relative error on the intensity function

Figure 6. Graph of location S2: relative errors with respect to the introduced Gaussian noise.

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outflow boundary could be much better than the one corresponding to a rather upstreamsource location.

7. Conclusion

In this article, we studied the inverse source problem that consists of localizing atime-dependent source spatially supported at an interior point of a 2D bounded domainand recovering the history with respect to the time of its intensity function. This sourceoccurs in the right-hand side of an evolution linear advection-dispersion-reactionequation. We proved the identifiability of the sought source from recording the state onthe outflow boundary. Then, we established an identification method based on some exactboundary controllability results. This method requires the records of the state on theoutflow boundary and of its flux on the inflow boundary of the controlled domain. Somenumerical experiments in the case where the controlled domain is rectangular and themean velocity vector is perpendicular to the inflow boundary are presented. The analysisof those numerical experiments shows that the proposed identification method is accurateand relatively stable with respect to the introduction of a Gaussian noise on the usedrecords.

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Inverse source problem in a 2D linear evolution transport equation: detection of pollution source

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