HAL Id: hal-02174693
https://hal.archives-ouvertes.fr/hal-02174693v1
Preprint submitted on 5 Jul 2019 (v1), last revised 24 Jun 2021 (v2)
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Pollution sources reconstruction based on the
topological derivatve method
L Fernandez, A Novotny, R. Prakash, J. Sokolowski
To cite this version:
L Fernandez, A Novotny, R. Prakash, J. Sokolowski. Pollution sources reconstruction based on the topological derivatve method. 2019. �hal-02174693v1�
POLLUTION SOURCES RECONSTRUCTION BASED ON THE TOPOLOGICAL DERIVATIVE METHOD
L. FERNANDEZ, A.A. NOVOTNY∗, R. PRAKASH, AND J. SOKO LOWSKI
Abstract. The topological derivative method is used to solve a pollution sources recon-struction problem governed by a steady-state convection-diffusion equation. To be more precise, we are dealing with a shape optimization problem which consists of reconstruc-tion of a set of pollureconstruc-tion sources in a fluid medium by measuring the concentrareconstruc-tion of the pollutants within some subregion of the reference domain. The shape functional measur-ing the misfit between the known data and solution of the state equation is minimized with respect to a set of ball-shaped anomalies representing the pollution sources. The necessary conditions for optimality are derived with help of the topological derivative method which consists in expanding the shape functional asymptotically and then trun-cated it up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. Two different cases are considered. Firstly, when the velocity of the leakages is given and we reconstruct the support of the unknown sources, includ-ing their locations and sizes. In the second case, we consider the size of the pollution sources to be known and find out the mean velocity of the leakages and their locations. Numerical examples are presented showing the capability of the proposed algorithm in reconstructing multiple pollution sources in both cases.
1. Introduction
The topic adressed in this article belongs to a class of inverse problems where the main objective is to identify the location and size of pollution sources. In this area of research, the rate of impurity emission from the pollution sources using the measurements of the concentration of the pollutants in some monitoring region is also an interesting issue to investigate. In general, the associated forward problems are governed by a parabolic advection-dispersion-reaction equation in one [15] or more spatial dimensions [3, 24, 34]. Such inverse problems are motivated from many physical phenomena and real-life events such as leakages in nuclear plants, refinery and chemical laboratories; and fire outbreaks in buildings and forests, for example. However, most of the practical applications reported in the literature concern the monitoring the water quality of rivers and groundwater [3, 15, 24] and the air quality in large cities [34].
In this paper, we reconstruct a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within a subregion of the domain. To be more precise, we consider a two dimensional fluid domain where the impurities are getting injected with an unknown velocity through finitely many distributed unrecognized sources localized in a compactly supported region. The main idea is to measure the concentration of the pollutant away from their sources and use this information to completely characterize the sources and the intensity of the pollution, i.e. to determine the number of sources, their locations, their sizes and the velocity of impurity integration. It is not a difficult task to produce numerical results which show that if all these variables are unknown the uniqueness of the problem is lost and hence the well-posedness. This motivates us to consider two different inverse problems. In the first problem, we assume that the velocity field is given and reconstruct the topology of the pollution sources by recovering their
∗Corresponding Author
Key words and phrases. pollution sources reconstruction, shape optimization, topological derivative method, inverse problem.
locations and sizes. In the second one, the sizes of the pollution sources are assumed to be known and we determine their locations and the mean velocity of the respective leakages. In our simplified mathematical setting, we are dealing with an inverse reconstruction problem whose forward counterpart is governed by a steady-state convection-diffusion equation.
In this case, since the pollutant concentration is known in a small subregion of the fluid domain, the inverse problem can be written in the form of an over-determined boundary value problem. Following the classical well-known strategy, we rewrite our inverse problem as an optimization problem which consists in minimizing the misfit between the known measurement and the solution to the model problem with respect to the parameters related to the unknown sources of pollution. One can see [5, 6, 22] where the authors adopted this strategy for solving inverse problems of reconstruction of geometrical subdomains in different contexts. As we are interested in reconstructing the pollution sources, it becomes a topology optimization problem. Topology optimization is a broad field of investigation from engineering as well as mathematical point of view. Scientists used various approaches to solve different kind of inverse problems. The bibliography is quite exhaustive, but for a general idea on the subject, readers may refer to [1, 7, 10, 11, 12]. A quite general approach to deal with such inverse problems is based on the concept of topological derivatives. The reader may refer to the book by Novotny & Soko lowski [28] to be familiar with the notion of topological derivatives.
The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, inclusions, defects, source-terms and cracks. This relatively new concept has applications in many different fields such as shape and topology optimization, inverse problems, imaging processing, multi-scale material design and mechanical modeling including damage and fracture evolution phe-nomena. The topological derivative was rigorously introduced by Soko lowski & ˙Zochowski [31] and developed later by many authors [18, 27, 30, 33]. It can be seen as a particular case of the broader class of asymptotic methods fully developed in the book by Ammari & Kang [2], for instance. One can refer to the books by Maz’ya et al. [25] and Nazarov & Plamenevskij [26] for similar topics and discussions. In addition, one can also cite the paper by Soko lowski & ˙Zochowski [32] where the domain decomposition method was pro-posed in the domain of topological derivatives. More recently, the concept of second-order topological derivative was introduced in [14], which has been successfully used for solving a class of inverse reconstruction problems. A variety of applications of the topological derivative method in topology optimization, inverse imaging problem and inverse recon-struction problems can be found on the complementary book by Novotny, Soko lowski and ˙Zochowski [29]. Following the original ideas introduced in [16], in the current article, the second-order topological derivative is applied in the context of the proposed pollution sources reconstruction problem. The two main challenges in this article, in comparison to [16] are as follows: (i) the presence of the convective term in the governing forward equation; and (ii) the reconstruction of a vector quantity representing the mean velocity of a pollutant substance that escapes of a confined region characterizing a pollution source to be determined.
The outline of this paper is as follows. The inverse problem is reformulated as a topol-ogy optimization problem in Section 2. Such reformulation of the problem allows to solve it through a topological derivatives-based approach. Thus, we introduce some tools from the theory of topological derivatives in the beginning of the Section 3, namely, the func-tionals associated to the unperturbed and perturbed domains, the ansatz for a scalar field of interest and some auxiliary boundary value problems. In Section 4, we state the main
result of this paper which consists of the asymptotic expansion of the shape functional. Estimates of the remainders, obtained in Section 4, are presented in Appendix A. Nu-merical results are presented in Section 5 where two reconstruction algorithms are devised from the asymptotic expansion of the shape functional. In addition, numerical experi-ments are conducted in order to test the capabilities of each of the proposed algorithms. Some concluding remarks are inferred in Section 6.
2. Problem formulation
Let Ω ⊂ R2 be an open and bounded domain with smooth boundary ∂Ω. Measurements
of the pollutant concentration field are collected in a subdomain Ωo ⊂ Ω. We assume that
there may be an unknown number (N∗ ∈ Z+) of isolated pollution sources ωi∗ within the domain Ω. See Figure 1(a). Therefore, there is a set ω∗ = ∪Ni=1∗ω∗i, whose components ω∗i satisfy ω∗i ∩ ω∗ j = ∅ for i 6= j and ω ∗ i ∩ ∂Ω = ∅ for each i, j ∈ {1, . . . , N ∗}. (a) (b)
Figure 1. Domain Ω with (a) and without (b) the set of isolated pollution sources ω∗.
We assume that the polluting substance is leaking in an incompressible flow with ve-locity V = (V1, V2) 6= (0, 0) at each ωi∗ ⊂ Ω, i = 1, . . . , N
∗. For a given Dirichlet data g
imposed on ∂Ω, the resulting pollutant concentration z in Ω is observed in the subdomain Ωo. In this set up, the inverse problem consists in finding χω∗ such that the pollutant
concentration z satisfies the following boundary value problem −∆z + χω∗V · ∇z = 0 in Ω,
z = g on ∂Ω, (2.1)
where the parameter χω∗ is defined as
χω∗ = 0 in Ω \ ω
∗,
1 in ωi∗, i = 1, . . . , N∗ (2.2) and the velocity V is considered as a null divergence vector field, namely, div V = 0 in ωi∗, for i = 1, . . . , N∗.
Now, for an initial guess χω of χω∗, we consider the pollutant concentration field u to
be the solution to the boundary value problem
−∆u + χωV · ∇u = 0 in Ω, u = g on ∂Ω, (2.3) where χω = 0 in Ω \ ω, 1 in ωi, i = 1, . . . , N. (2.4) The characteristic function χω∗ is unknown and hence z but we assume that z can be
measured in Ωo. Thus, if we look for an appropriate χω∗, then we wish u to agree with z
in Ωo, i.e., we want u = z|Ωo. In addition, in such a problem, we assume that the velocity
field V is known and then we reconstruct the support of ω∗ from partial measurements of the scalar field z taken on Ωo.
The inverse problem under investigation is written in the form of an ill-posed and over-determined boundary value problem. It is well known that such difficulty can be overcome by rewriting the inverse problem in the form of an optimization problem. In particular, if the velocity field V is a known data and we reconstruct the support of each ωi∗, i = 1, . . . , N∗, then the inverse problem (2.1) can be rewriting as a topology optimization problem, namely
Minimize ω⊂Ω Jω(u 1, . . . , uM) = M X m=1 Z Ωo (um− zm)2, (2.5)
where M ∈ Z+ is the number of observations, zm and um are the solutions of the
bound-ary value problems (2.1) and (2.3), respectively, corresponding to the Dirichlet data gm
with m = 1, . . . , M . Notice that, the minimizer of the topology optimization problem (2.5) produces the best approximation to ω∗, solution of the inverse problem (2.1), in an appropriate sense.
3. Topological derivatives-based approach
The inverse problem (2.1) has been written in the form of a topology optimization problem (2.5). A quite general approach for solution of such class of problems is based on the concept of topological derivative which requires the expansion of the shape functional Jω(u1, . . . , uM) with respect to the parameters depend upon a set of small perturbations.
Since the topological derivative does not depend on the initial guess of the unknown topology ω∗, we start with the unperturbed domain by setting ω = ∅. See Figure 1(b). More precisely, we consider
J0(u10, . . . , uM0 ) = M X m=1 Z Ωo (um0 − zm)2 , (3.1) where um
0 be the solution of the unperturbed boundary value problem
−∆um
0 = 0 in Ω,
um0 = gm on ∂Ω. (3.2) In this article, we are considering the topology optimization problem (2.5) for the ball-shaped pollution sources and hence we define the topologically perturbed counter-part of (3.2) by introducing N ∈ Z+ number of small circular pollution sources B
εi(xi) with
center at xi ∈ Ω and radius εi for i = 1, . . . , N . The respective geometric set can be
denoted as Bε(ξ) = N [ i=1 Bεi(xi), (3.3)
where ξ = (x1, . . . , xN) and ε = (ε1, . . . , εN). Moreover, we assume that Bε∩ ∂Ω = ∅,
Bε∩ Ωo = ∅ and Bεi(xi) ∩ Bεj(xj) = ∅ for each i 6= j and i, j ∈ {1, . . . , N }. The shape
functional associated with the topologically perturbed domain is written as
Jε(u1ε, . . . , uMε ) = M X m=1 Z Ωo (umε − zm)2 (3.4) with um
ε be the solution of the perturbed boundary value problem
−∆um
ε + χεV · ∇umε = 0 in Ω,
where the parameter χε is defined as χε = 0 in Ω \ Bε(ξ), 1 in Bε(ξ). (3.6)
By construction, we consider that the velocity field V satisfies the additional condition div V = 0 in Bε(ξ).
As mentioned earlier, the topological derivatives measure the sensitivity of the shape functional with respect to the parameters (ε, ξ) depending upon a set of small circular pollution sources Bε(ξ). Therefore, our idea is to obtain the number, location and radius
of the components of the set Bε(ξ) in order to produce the best approximation to the true
set of pollution sources ω∗ by using the concept of topological derivatives.
In this article, we are interested in expanding the shape functional Jε defined in (3.4)
in power of ε. Therefore, we start by simplifying the difference between the perturbed shape functional Jε(u1ε, . . . , uMε ) and its unperturbed counter-part J0(u10, . . . , uM0 ) defined
in (3.4) and (3.1), respectively, as follows
Jε(uε) − J0(u0) = M X m=1 Z Ωo 2(um ε − u m 0 )(u m 0 − z m ) + (umε − um0 )2 , (3.7) where uε= (u1ε, . . . , uMε ) and u0 = (u10, . . . , uM0 ).
For m = 1, . . . , M , let us consider the following ansatz for the asymptotic expansion of um
ε with respect to the parameters corresponding to the small circular pollution sources
as described in Section 2 umε (x) = um0 (x) + N X i=1 |Bεi(xi)|h ε,m i (x) + N X i=1 N X j=1 |Bεi(xi)||Bεj(xj)|h ε,m ij (x) + N X i=1 N X j=1 |Bεi(xi)||Bεj(xj)| 2 ϕε,mij (x) + ˜umε (x), (3.8)
where |Bεi(xi)| is the Lebesgue measure (volume) of the two-dimensional ball Bεi(xi), i.e.,
|Bεi(xi)| = πε
2
i. Furthermore, for each i, j = 1, . . . , N and m = 1, . . . , M , the functions
hε,mi , hε,mij and ϕε,mij are the solutions of the following boundary value problems ∆hε,mi = 1 |Bεi(xi)| V · ∇um0 χBεi(xi) in Ω, hε,mi = 0 on ∂Ω, (3.9) ∆hε,mij = 1 |Bεi(xi)| V · ∇hε,mj χBεi(xi) in Ω, hε,mij = 0 on ∂Ω, (3.10) and ∆ϕε,mij = 1 |Bεi(xi)| V · ∇hε,mjj χBεi(xi) in Ω, ϕε,mij = 0 on ∂Ω, (3.11)
respectively. The last term of the expansion (3.8), namely ˜um
ε , must satisfy the problem
−∆˜umε + χεV · ∇˜umε = Φmε in Ω,
˜ um
ε = 0 on ∂Ω,
(3.12)
for each i, j = 1, . . . , N and m = 1, . . . , M ; with
such that Φmε,1 = − N X i=1 N X j=1 N X l=1 l6=j |Bεj(xj)||Bεl(xl)|V · ∇h ε,m jl χBεi(xi) (3.14) and Φmε,2 = − N X i=1 N X j=1 N X l=1 |Bεj(xj)||Bεl(xl)| 2 V · ∇ϕε,mjl χBεi(xi). (3.15)
In order to simplify further analysis, we write hε,mi as a sum of three functions pε i, qi
and ˜hε,mi in the form
hε,mi = (V (xi) · ∇um0 (xi))(pεi + qi) + ˜hε,mi . (3.16)
For a positive real number R ε, we consider the ball BR(xi) such that Bεi(xi) ⊂ Ω ⊂
BR(xi) with xi ∈ Ω for i = 1, . . . , N . Then, the function pεi is defined to be solution of
the problem ∆pεi = 1 |Bεi(xi)| χBεi(xi) in BR(xi), pε i = 1 2πln R on ∂BR(xi) (3.17)
Problem (3.17) can be solved analytically and its solution is given by
pεi = 1 4πε2 i kx − xik2− 1 2π 1 2− ln εi in Bεi(xi), 1 2πln kx − xik in BR(xi) \ Bεi(xi). (3.18)
From (3.18), one can observe that the solution pε
i does not depend on εi outside the ball
Bεi(xi). Therefore, we use the notation pi(x) := p
ε
i(x), ∀x ∈ Ω \ Bεi(xi). Additionally, qi
is the solution to the homogeneous boundary value problem ∆qi = 0 in Ω,
qi = −pi on ∂Ω
(3.19)
and ˜hε,mi solves the boundary value problem
∆˜hε,mi = 1 |Bεi(xi)|
[V · ∇um0 − V (xi) · ∇um0 (xi)] χBεi(xi) in Ω,
˜
hε,mi = 0 on ∂Ω.
(3.20)
Taking into account the decomposition (3.16), we can introduce the notations
hε,mi := (V (xi) · ∇u m 0 (xi))hεi + ˜h ε,m i in Bεi(xi), (V (xi) · ∇um0 (xi))hi+ ˜hε,mi in Ω \ Bεi(xi), (3.21) with hεi := pεi + qi and hi := pi + qi, (3.22) where pε
i and pi are the solutions to the problem (3.17) in Bεi(xi) and Ω \ Bεi(xi),
respec-tively.
Now, let us decompose ˜hε,mi solution of (3.20) into three parts, namely ˜
where ˜pε,mi is solution to a similar problem ∆˜pε,mi = 1 |Bεi(xi)| Wim· (x − xi) χBεi(xi) in BR(xi), ˜ pε,mi = 0 on ∂BR(xi), (3.24) with Wm
i coming out from the Taylor expansion of the source term of the boundary value
problem (3.20), namely
Wim := ((∇2um0 )V + (∇V )∇um0 )(xi). (3.25)
Problem (3.24) can be solved analytically and its solution is given by
˜ pε,mi = 1 8π kx − xik2− ε2i ε2 i − R 2− ε2 i R2 Wim· (x − xi) in Bεi(xi), ε2 i 8πR2 kx − xik2− R2 kx − xik2 Wim· (x − xi) in BR(xi) \ Bεi(xi). (3.26)
The next term in decomposition (3.23) given by ˜qm
i is solution of the following boundary
value problem ∆˜qm i = 0 in Ω, ˜ qm i = −˜pmi on ∂Ω, (3.27) where ˜pm i := ε −2 i p˜ ε,m
i in BR(xi) \ Bεi(xi) and hence the independence of ˜q
m
i with respect
to εi is related to the explicit solution ˜p ε,m
i of (3.26). Finally, the last term in (3.23) given
byh˜˜ε,mi has to be solution of ∆h˜˜ε,mi = 1 |Bεi(xi)| Tim(x) χBεi(xi) in Ω, ˜ ˜ hε,mi = 0 on ∂Ω, (3.28) where Tm
i (x) = O(kx − xik2) in appropriated norm.
Similarly to the decomposition proposed to the function hε,mi in (3.16), we write the function hε,mii as a sum of the functions pε
ii, qii and ˜hε,mii in the form
hε,mii = (V (xi) · ∇um0 (xi))(pεii+ qii) + ˜hε,mii . (3.29)
The function pε
ii is solution to the problem
∆pεii = 1 |Bεi(xi)| V · ∇pεi χBεi(xi) in BR(xi), pε ii = 0 on ∂BR(xi). (3.30)
Problem (3.30) can be solved analytically and its solution is given by
pεii= 1 (4πεi)2 kx − xik2− ε2i ε2 i −R 2− ε2 i R2 V · (x − xi) in Bεi(xi), 1 (4πR)2 kx − xik2− R2 kx − xik2 V · (x − xi) in BR(xi) \ Bεi(xi). (3.31)
From (3.31), one can observe that the solution pεii does not depend on εi outside the ball
Bεi(xi). Therefore, we use the notation pii(x) := p
ε
ii(x), ∀x ∈ Ω \ Bεi(xi). The functions
qii and ˜hε,mii are solutions to the following boundary value problems
∆qii = 0 in Ω,
qii = −pii on ∂Ω
and ∆˜hε,mii = 1 |Bεi(xi)| [(V (xi) · ∇um0 (xi))(V · ∇qi) + V · ∇˜hε,mi ]χBεi(xi) in Ω, ˜ hε,mii = 0 on ∂Ω, (3.33) respectively. Taking into account the decomposition (3.29) and the solution pεii given by (3.31), we introduce the notation
hε,mii := (V (xi) · ∇u m 0 (xi))hεii+ ˜h ε,m ii in Bεi(xi), (V (xi) · ∇um0 (xi))hii+ ˜hε,mii in Ω \ Bεi(xi), (3.34) with hεii:= pεii+ qii and hii:= pii+ qii, (3.35) where pε
ii and pii are the solutions to the problem (3.30) in Bεi(xi) and Ω \ Bεi(xi),
respectively.
Finally, in order to simplify further calculations related to the asymptotic development of the topologically perturbed cost functional (3.4), we introduce an adjoint state vm to
be the solution to the problem
−∆vm = (um
0 − zm)χΩo in Ω,
vm = 0 on ∂Ω. (3.36)
4. Asymptotic expansion of the shape functional Firstly, let us introduce the vector α ∈ RN, where each entry α
i denotes the Lebesgue
measure (volume) of the two-dimensional ball Bεi(xi), i.e,
α = (α1, . . . , αN) with αi = |Bεi(xi)|, for i = 1, . . . , N. (4.1)
By using the above notation, the expansion (3.8) can be rewritten in the following form
umε (x) = um0 (x) + N X i=1 αih ε,m i (x) + N X i=1 N X j=1 αiαjh ε,m ij (x) + N X i=1 N X j=1 αiα2jϕ ε,m ij (x) + ˜u m ε (x). (4.2) We proceed by replacing (4.2) in (3.7) to obtain
Jε(uε) − J0(u0) = 2 M X m=1 N X i=1 αi Z Ωo hε,mi (um0 − zm) + 2 M X m=1 N X i=1 N X j=1 αiαj Z Ωo hε,mij (um0 − zm) + 2 M X m=1 N X i=1 N X j=1 αiα2j Z Ωo ϕε,mij (um0 − zm) + M X m=1 N X i=1 N X j=1 αiαj Z Ωo hε,mi hε,mj + M X m=1 10 X `=1 E`m(ε), (4.3) where E1m(ε) = 2 Z Ωo ˜ umε (um0 − zm), (4.4) Em 2 (ε) = 2 N X i=1 N X j=1 N X l=1 αiαjαl Z Ωo hε,mi hε,mjl , (4.5) E3m(ε) = 2 N X i=1 N X j=1 N X l=1 αiαjα2l Z Ωo hε,mi ϕε,mjl , (4.6)
Em 4 (ε) = 2 N X i=1 αi Z Ωo hε,mi u˜mε , (4.7) Em 5 (ε) = N X i=1 N X j=1 N X l=1 N X p=1 αiαjαlαp Z Ωo hε,mij hε,mlp , (4.8) Em 6 (ε) = 2 N X i=1 N X j=1 N X l=1 N X p=1 αiαjαlα2p Z Ωo hε,mij ϕε,mlp , (4.9) E7m(ε) = 2 N X i=1 N X j=1 αiαj Z Ωo hε,mij u˜mε , (4.10) Em 8 (ε) = N X i=1 N X j=1 N X l=1 N X p=1 αiα2jαlα2p Z Ωo ϕε,mij ϕε,mlp , (4.11) E9m(ε) = 2 N X i=1 N X j=1 αiα2j Z Ωo ϕε,mij u˜mε (4.12) and Em 10(ε) = Z Ωo (˜umε)2. (4.13) By choosing vm and hε,m
i as test functions in the weak formulations of problems (3.9)
and (3.36), respectively, we obtain the following equation Z Ωo hε,mi (um0 − zm) = − 1 αi Z Bεi(xi) (V · ∇um0 )vm. (4.14)
Similarly, we choose vm and hε,mij as test functions in the weak forms of problems (3.10) and (3.36), respectively, to get
Z Ωo hε,mij (um0 − zm) = − 1 αi Z Bεi(xi) (V · ∇hε,mj )vm. (4.15) Finally, we have Z Ωo ϕε,mij (um0 − zm) = − 1 αi Z Bεi(xi) (V · ∇hε,mjj )vm, (4.16)
after considering vm and ϕε,mij as test functions in the weak formulations of problems (3.11) and (3.36), respectively. By using (4.14)-(4.16) in (4.3), we get
Jε(uε) − J0(u0) = −2 M X m=1 N X i=1 Z Bεi(xi) (V · ∇um0 )vm − 2 M X m=1 N X i=1 N X j=1 αj Z Bεi(xi) (V · ∇hε,mj )vm− 2 M X m=1 N X i=1 N X j=1 α2j Z Bεi(xi) (V · ∇hε,mjj )vm + M X m=1 N X i=1 N X j=1 αiαj Z Ωo hε,mi hε,mj + M X m=1 10 X `=1 E`m(ε). (4.17)
Taking into account the decompositions of the functions hε,mi , ˜hε,mi and hε,mii given by (3.16), (3.23) and (3.29), respectively, we rewrite (4.17) as follows
Jε(uε) − J0(u0) = −2 M X m=1 N X i=1 Z Bεi(xi) (V · ∇um0 )vm− 2 M X m=1 N X i=1 αi Z Bεi(xi) (V · ∇˜pε,mi )vm − 2 M X m=1 N X i=1 αi(V (xi) · ∇um0 (xi)) Z Bεi(xi) (V · ∇pεi)vm − 2 M X m=1 N X i=1 αi(V (xi) · ∇um0 (xi)) Z Bεi(xi) (V · ∇qi)vm − 2 M X m=1 N X i=1 N X j=1 j6=i αj(V (xj) · ∇um0 (xj)) Z Bεi(xi) (V · ∇hj)vm − 2 M X m=1 N X i=1 α2i(V (xi) · ∇um0 (xi)) Z Bεi(xi) (V · ∇pεii)vm + M X m=1 N X i=1 N X j=1 αiαj(V (xi) · ∇um0 (xi))(V (xj) · ∇um0 (xj)) Z Ωo hihj + M X m=1 17 X `=1 Em ` (ε). (4.18)
Here, the seven new remainders are defined as
Em 11(ε) = −2 N X i=1 αi Z Bεi(xi) V · (ε2i∇qm i + ∇ ˜ ˜ hε,mi )vm, (4.19) Em 12(ε) = −2 N X i=1 N X j=1 j6=i αj Z Bεi(xi) (V · ∇˜hε,mj )vm, (4.20) E13m(ε) = −2 N X i=1 N X j=1 α2j(V (xj) · ∇um0 (xj)) Z Bεi(xi) (V · ∇qjj)vm, (4.21) Em 14(ε) = −2 N X i=1 N X j=1 α2j Z Bεi(xi) (V · ∇˜hε,mjj )vm, (4.22) Em 15(ε) = −2 N X i=1 N X j=1 j6=i α2j(V (xj) · ∇um0 (xj)) Z Bεi(xi) (V · ∇pε,mjj )vm, (4.23) Em 16(ε) = N X i=1 N X j=1 αiαj Z Ωo [(V (xi) · ∇um0 (xi))hi˜hε,mj + (V (xj) · ∇um0 (xj))hj˜hε,mi ] (4.24) and E17m(ε) = k2 N X i=1 N X j=1 αiαj Z Ωo ˜ hε,mi ˜hε,mj . (4.25)
Now, by considering the Taylor’s expansions of the functions ∇um
0 , ∇qi, ∇hj and vm
around the point xi up to the appropriated order as done in [16], we get
Jε(uε) − J0(u0) = −2 M X m=1 N X i=1 αi(V (xi) · ∇um0 (xi))vm(xi) − 1 2π M X m=1 N X i=1 α2i V (xi) · ∇2um0 (xi)∇vm(xi) + 1 4π M X m=1 N X i=1 α2i (V (xi) · (∇2u0m(xi)V (xi)))vm(xi) + 1 4π M X m=1 N X i=1 α2i (V (xi) · (∇V (xi)∇um0 (xi)))vm(xi) − 1 4π M X m=1 N X i=1 α2i(V (xi) · ∇um0 (xi))(V (xi) · ∇vm(xi)) − 2 M X m=1 N X i=1 α2i(V (xi) · ∇um0 (xi))(V (xi) · ∇qi(xi))vm(xi) + 1 8π M X m=1 N X i=1 α2i(V (xi) · ∇um0 (xi))kV (xi)k2vm(xi) − 2 M X m=1 N X i=1 N X j=1 j6=i αiαj(V (xj) · ∇um0 (xj))(V (xi) · ∇hj(xi))vm(xi) + M X m=1 N X i=1 N X j=1 αiαj(V (xi) · ∇um0(xi))(V (xj) · ∇um0 (xj)) Z Ωo hihj + Em(ε), (4.26) with Em(ε) = Em 0 (ε) + M X m=1 17 X `=1 Em ` (ε). (4.27)
The new remainder Em
0 (ε) contains the residual terms arising from different Taylor’s
ex-pansions arguing in a similar fashion to get the remainders Em
10(ε)–E19m(ε) in [16].
We are now in position to state the main result of this paper. Theorem 1. For i = 1, . . . , N and m = 1, . . . , M , let um
0 , qi, hi and vm be the functions
defined in (3.2), (3.19), (3.22) and (3.36), respectively. Additionally, let V be the null divergence velocity field introduced in problem (2.1). Then, the asymptotic expansion for the topologically perturbed shape functional Jε(uε) defined in (3.4), with respect to
αi = |Bεi(xi)|, i = 1, . . . , N , is given by (4.26), where the remaining terms, given by
(4.27), are such that |Em(ε)| = o(|ε|4), for m = 1, . . . , M , whose justification can be found in Appendix A.
5. Numerical experiments
This section consists of two reconstruction algorithms and their respective numerical examples. Firstly, we propose the following contextualization of the inverse problem in order to recall some of its characteristic elements. Let us consider a pipeline network within a reference domain Ω which contains a N∗ number of leakage points through which
a polluting substance escapes with a velocity V . Each leakage point ωi∗ (i = 1, . . . , N∗) can be imagined as a pollution source which has to be reconstructed by a ball-shaped geometrical subdomain Bεi(xi). Hence, it can be completely carachterized by the center
xi and radius εi of Bεi(xi), for i = 1, . . . , N
∗. Moreover, the vector field V at the point x i
provides us an additional information about each leakage.
Since the asymptotic expansion in (4.26) depends on (a) the number, (b) the locations, (c) the sizes of the pollution sources and (d) the velocity of the pollution, we solve two different problems keeping in mind the physical aspects of the current scenario.
In the first case, we assume that the velocity field V is given and we reconstruct the topology of the pollution sources by recovering their locations ξ and sizes α. We deal with this problem in Section 5.1 under the nomenclature support velocity reconstruction. In Section 5.2, we consider the second case with the name mean velocity reconstruction. Here, we assume that the sizes α of the pollution sources are known and determine the mean velocity V (ξ) of the respective leakages.
Numerical experiments are conducted in the following scenario. The reference geomet-rical domain is given by a square Ω = (−0.5, 0.5) × (−0.5, 0.5) which is discretized with three-node finite elements. The mesh is generated from a grid of 160 × 160 squares. Each square is divided into four triangles which leads us to a resulting mesh comprising 102400 elements and 51521 nodes. Measurements of scalar fields of interest are taken in the subdomain Ωo which is a union of four small regions in the neighborhood of the corners
of the square domain Ω. See Figure 2(a). The boundary ∂Ω of the reference domain is excited by imposing different Dirichlet data, namely, g1 = x, g2 = y and g3 = xy.
The auxiliary boundary value problems are solved over the resulting mesh. From these solutions the topological derivatives can be numerically evaluated at any point of the mesh. However, due the high complexity of the algorithm [23], a sub-grid X consists of uniformly distributed points is extracted from the original mesh and then used as a set of adimissible locations where a combinatorial search is performed which leads to the optimal solution defined in X. In the case of the examples presented below, we consider a fixed sub-grid X comprising 165 points, as illustrated in Figure 2(b).
(a) (b)
Figure 2. Numerical scenario: (a) the set Ωo (in gray color) as part of the
reference domain Ω and (b) the set of admissible locations X.
In order to obtain noisy synthetic data, the true target function zm, corresponding to
the solution of the problem (2.1) for a given Dirichlet data gm, is corrupted with white
Gaussian noise, where the resulting level of noise in the measurement of zm is given by
µ∗ = kz m µ − zmkL2(Ω o) kzmk L2(Ω o) × 100%, (5.1)
with zµm denotes the corrupted target function used as synthetic data.
In all our numerical results, we represent the (true and reconstructed) pollution sources by black, the subdomain Ωo by gray and the remaining domain by white colors.
5.1. Support velocity reconstruction. In this section, we present our first algorithm followed by two numerical examples concerning the reconstruction of the topology of the pollution sources. The solution algorithm is devised from the topological asymptotic expansion of the shape functional Jε(uε) given by (4.26). After disregarding all terms of
order o(|α|2) in (4.26), the resulting truncated expansion, in its matrix form, is written
as
δJ (α, ξ, N ) := α · d(ξ) + 1
2H(ξ)α · α, (5.2) where the entries of the vector d ∈ RN and the matrix H ∈ RN ×N are given by
di := −2 M X m=1 (V (xi) · ∇um0 (xi))vm(xi), (5.3) and Hii := − 1 π M X m=1 V (xi) · ∇2um0 (xi)∇vm(xi) + 1 2π M X m=1 (V (xi) · (∇2um0 (xi) V (xi)))vm(xi) + 1 2π M X m=1 (V (xi) · (∇V (xi) ∇um0 (xi)))vm(xi) − 1 2π M X m=1 (V (xi) · ∇um0 (xi))(V (xi) · ∇vm(xi)) − 4 M X m=1 (V (xi) · ∇um0 (xi))(V (xi) · ∇qi(xi))vm(xi) + 1 4π M X m=1 (V (xi) · ∇um0 (xi))kV (xi)k2vm(xi) + 2 M X m=1 (V (xi) · ∇um0 (xi))2 Z Ωo h2i, (5.4) Hij := −4 M X m=1 (V (xj) · ∇um0 (xj))(V (xi) · ∇hj(xi))vm(xi) + 2 M X m=1 (V (xi) · ∇um0 (xi))(V (xj) · ∇um0 (xj)) Z Ωo hihj, (5.5) if i 6= j, respectively, for i, j = 1, . . . , N .
The truncated asymptotic expansion δJ (α, ξ, N ), given by (5.2), is a quadratic form in α. Hence, in order to minimize (5.2), we proceed with the derivative of δJ (α, ξ, N ) with respect to the variable α to write the first-order optimality condition
hDαδJ, βi = [d(ξ) + Hs(ξ)α] · β = 0, ∀β ∈ RN, (5.6)
with Hs = (H + H>)/2, where H> denotes the transpose of the matrix H. From (5.6),
we obtain the linear system
α = −(Hs(ξ))−1d(ξ). (5.7) Note that, the quantity α, solution of (5.7), becomes a function of the locations ξ, i.e., α = α(ξ). Let us replace α = α(ξ) in (5.2) to obtain
δJ (α(ξ), ξ, N ) = 1
2d(ξ) · α(ξ). (5.8) Therefore, the optimal locations and sizes of the pollution sources, given by the pair of vectors (α?,ξ?), can be obtained by
ξ? := argmin
ξ∈X
where X denotes the set of admissible locations for the pollution sources. The above procedure written in pseudo-code format can be found in [23].
5.1.1. Numerical examples. Two numerical examples are presented below. The first one tests the robustness of the reconstruction in the presence of noisy data while the second one analyses the sensitivity of the reconstruction with respect to the number of measurements.
Example 1: reconstruction of a single leakage from the noisy data. We consider a target domain containing a single leakage ω∗ located at x∗ = (0.2, 0.1) whose radius is ε∗ = 0.03, as shown in Figure 3(a). In addition, it is assumed that the pollution flows with constant velocity V = (2, 2). The reconstruction of the topology of ω∗ is performed from only one measurement by choosing g1 as Dirichlet data. The pollution source is accurately
reconstructed in the absence of noise, as can be seen in Figure 3(b). We observe that, for levels of noise up to 0.1%, the proposed algorithm is able to find the exact center x∗ and size of ω∗. The reconstruction starts to be degraded for levels of noise greater than 0.3%. These results are demonstrated in Figures 3(c)-3(d).
(a) (b) µ∗= 0% (c) µ∗= 0.1% (d) µ∗= 0.3%
Figure 3. Example 1: Target (a) and results (b)-(d).
Example 2: reconstruction of multiple leakages. In this example, the pipeline network within the reference domain contains four leakages which are located at the points x∗1 = (0, 0.3), x∗2 = (0.3, 0), x∗3 = (0, −0.3) and x∗4 = (−0.3, 0). The sizes of the leakages are equal, namely, ε∗i = 0.03, for i = 1, . . . , 4. We illustrate the target domain in Figure 4(a). The pollution substance escapes from the pipeline network with a velocity given by V = (1, 10x2). We start by considering only one measurement associated to the Dirichlet data g1. After comparing Figures 4(a) and 4(b), we observe that the algorithm fails in reconstructing the target domain. For a large number of pollution sources, the information obtained from only one measurement is not sufficient to accurately reconstruct them. Therefore, we improve the number of measurements by exciting the boundary of the domain with the Dirichlet data g2 and g3. As we can observe in Figures 4(c) and
4(d), the pollution sources were successfully reconstructed when we have considered three measurements simultaneously. In this case, we have obtained the exact location of the leakages while their sizes are approximately equal to the true values, namely, ε?
i = 0.0299,
for i = 1, 2, 3, 4.
(a) (b) M = 1 (c) M = 2 (d) M = 3
5.2. Mean velocity reconstruction. Here, we present our second algorithm concerning the mean velocity reconstruction for leakages whose sizes are known. We demonstrate the effectiveness of the algorithm with an numerical example at the end of this section. Analogous to the mathematical steps taken in Section 5.1, we start by considering the topological asymptotic expansion of the shape functional Jε(uε) given by (4.26). Firstly,
all the terms of order o(|α|2) together with the fourth and the seventh terms on the
right-hand side of (4.26) are disregarded. This procedure allows us to obtain a linear system with respect to the variable of interest similar to (5.7). Next, we take the resulting truncated expansion and rewrite the velocity field in the form
V (xi) =
αiV (xi)
αi
= αiV (xi) (5.10)
in order to work with the two-dimensional flow rate of the leakages at the points xi,
i = 1, . . . , N . Taking into account these two steps, we obtain the truncated expansion
δJ (V , ξ, N ) := V · d(ξ) + 1
2H(ξ)V · V , (5.11) where d, V ∈ R2N and H ∈ R2N ×2N can be seen as the first-order topological derivative, the mean flow of the leakages and the higher-order topological derivative, respectively. More precisely, for a fixed number N of leakages to be reconstructed, the unknown flow vector V ∈ R2N is given by V := (V1, . . . , VN)> where each entry Vi is a vector in
R2 written as V i := (V 1 i, V 2 i)
>, for i = 1, . . . , N . Thus, the components of V
i must be
interpreted as follows: V1i and V2i are the flow rate of the polluting substance at the point xi in the x- and y-directions, respectively. Similarly, the vector d ∈ R2N is written as
d := (d1, . . . , dN)> where each entry di is a vector in R2 given by
di := −2α2i M X m=1 ∇um 0 (xi)vm(xi) − 1 2πα 3 i M X m=1 ∇2um 0 (xi)∇vm(xi), (5.12)
for i = 1, . . . , N . Each entry Hij ∈ R2×2 of the matrix H ∈ R2N ×2N
H := H11 · · · H1N .. . . .. ... HN 1 · · · HN N (5.13) is given by Hii := 1 2πα 4 i M X m=1 ∇2um 0 (xi)vm(xi) − 1 2πα 4 i M X m=1 ∇um 0 (xi) ⊗ ∇vm(xi) − 4α4i M X m=1 (∇um0 (xi) ⊗ ∇qi(xi))vm(xi) + 2α4i M X m=1 ∇um0 (xi) ⊗ ∇um0(xi) Z Ωo h2i (5.14) and Hij := −4α2iα 2 j M X m=1 (∇um0 (xi) ⊗ ∇hi(xj))vm(xj) + 2α2iα2j M X m=1 ∇um 0 (xi) ⊗ ∇um0 (xj) Z Ωo hihj, (5.15) for i 6= j with i, j = 1, . . . , N .
The truncated expansion δJ (V , ξ, N ), given by (5.11), is a quadratic form in V . Hence, in order to minimize (5.11), we differentiate δJ (V , ξ, N ) with respect to V to get the first-order optimality condition
hDVδJ, βi = [d(ξ) + Hs(ξ)V ] · β = 0, ∀β ∈ R2N, (5.16) with Hs = (H + H>)/2. From (5.16), we obtain the linear system
V = −(Hs(ξ))−1d(ξ). (5.17) Note that, the quantity V , solution of (5.17), becomes a function of the locations ξ, i.e., V = V (ξ). Let us replace V = V (ξ) in (5.11) to obtain
δJ (V (ξ), ξ, N ) = 1
2d(ξ) · V (ξ). (5.18) Therefore, the optimal locations and the velocity field of each leakage, given by the pair (V?,ξ?), can be obtained by
ξ? := argmin
ξ∈X
δJ (V (ξ), ξ, N ) and V? := V (ξ?), (5.19)
where X denotes the set of admissible locations for the pollution sources. Since the quantity V? is obtained, the velocity field V? at the point ξ? can be calculated through
the formula introduced in (5.10), for each leakage ω∗i, i = 1, . . . , N .
By construction, the resulting algorithm is analogous to the one developed in Section 5.1. The main difference between them is the reconstruction variable. In the first algo-rithm, we have reconstructed the size of each leakage which is a scalar quantity. In the second one, the unknown variable is a vector quantity, namely, the velocity at the leakage points.
5.2.1. Numerical example. A numerical example is presented below. In this case, we demonstrate the capability of the algorithm in reconstructing the mean velocity field when multiple leakages are considered in the target domain.
Example 3: reconstruction of the velocity field for multiple leakages. In this example, the target domain contains two leakages of same size ε∗1 = ε∗2 = 0.03 whose locations are given by the points x∗1 = (−0.1, 0.3) and x∗2 = (0.1, −0.2). See Figure 5(a). The velocity of the leakage is given by the vector field V = (−5y2+ 5y, 1). Consequently, the velocity of each leakage, ω∗1 and ω2∗, is equal to V (x∗1) = (1.05, 1) and V (x∗2) = (−1.2, 1), respectively. The reconstruction is obtained from three measurements with the help of all Dirichlet data g1, g2 and g3. In this case, we successfully reconstructed the center of each pollution source, as can be seen in Figure 5(b). The velocities of the leakages were accurately obtained, namely, the quantitative results are V (x?
1) = (1.0418, 0.9972) and
V (x?
2) = (−1.1946, 0.9908).
(a) (b) M = 3.
6. Conclusions
As concluding remarks concerning the proposed reconstruction algorithms, we highlight some important points. Briefly, for a given velocity field V and number N of trial pollution sources, the reconstruction algorithm is able to find their locations ξ? and sizes α? = α(ξ?) in one step. Similarly, in the second algorithm, for a given number N of pollution sources whose sizes are assumed to be known, the reconstruction algorithm returns their locations ξ? and velocities V? = V (ξ?) in one step. In addition, such algorithms are independent of
initial guess and they are capable of reconstructing the pollution sources in the presence of noise data. We point out some interesting features/aspects of the type of algorithm proposed above: (a) when the number N∗ of target subdomains is unknown, the algorithm can be started based on an assumption that there exists N > N∗ subdomains and then we should find a (N − N∗) of trial balls with negligible sizes [17]; (b) if the center of the target subdomain does not belong to the set of nodes of the sub-mesh, namely x∗ ∈ X, the/ algorithm returns a location x?which is the closest to x∗ [17]; and (c) since a combinatorial search over all the n-points of the set X has to be performed, this exhaustive search becomes rapidly infeasible for n N as N increases [23]. Finally, the proposed method approximates the unknown set of hidden anomalies by several balls which can be seen as a limitation of our approach. However, it can be used to get a good initial guess for more sophisticated iterative approaches based on level-sets methods [4, 8, 9, 19, 20, 21], for instance.
Appendix A. Justification for the main result
The justification for the result stated through Theorem 1 is presented in two steps. Firstly, we prove a priori estimates related to the auxiliary states hiε,m, ˜hε,mi , h˜˜ε,mi , hε,mij , ˜
hε,mij , ϕε,mij and ˜um
ε for i, j = 1, . . . , N and m = 1, . . . , M . Then, in the final part of
this section, the previously obtained results are used to estimate the remainders left in the asymptotic expansion of the topologically perturbed cost functional. These estimates justify our topological asymptotic expansion (4.26).
A.1. A priori estimates. In order to simplify the presentation, we denote all the con-stants independent of ε, i and m as C for i = 1, . . . , N and m = 1, . . . , M , whose value changes according to the place it is used. Furthermore, we use the fact that αi ∼ ε2i, when
necessary.
Lemma 2. Let Ω be an open and bounded domain in R2 and let B
ε be a ball of radius ε,
such that Bε ⊂ Ω. Then, for a function ϕ ∈ H1(Ω), the following estimate holds true
kϕkL2(Bε) ≤ CεδkϕkH1(Ω) , (A.1)
with 0 < δ < 1 and the constant C independent of the small parameter ε. Proof. From the H¨older inequality, we have
kϕkL2(B
ε) ≤ Cε
1/qkϕk
L2p(Ω) , (A.2)
for all p, q ∈ [1, +∞] satisfying 1/p + 1/q = 1. By choosing q > 1 and p accordingly, the Sobolev embedding theorem [13, Ch. IV, §8, Sec. 1.2, pp 140] implies H1(Ω) ⊂ L2p(Ω)
with a continuous embedding. Therefore, we have kϕkL2(B
ε)≤ Cε
1/qkϕk
H1(Ω) , (A.3)
Lemma 3. For i = 1, . . . , N and m = 1, . . . , M , let hε,mi and ˜hε,mi be the weak solu-tions of problems (3.9) and (3.20), respectively. Then, there exists a positive constant C independent of ε such that
k˜hε,mi kH1(Ω) ≤ Cεδii, (A.4) khε,mi kL2(Ωo) ≤ C(1 + εδii), (A.5) khε,mi kL2(B εi) ≤ C(εi| log εi| + ε 2δi i ), (A.6) khε,mi kL2(B εj) ≤ C(εj + ε δi i ε δj j ), if i 6= j, (A.7) k∇hε,mi kL2(B εi) ≤ C(1 + εi+ ε δi i ), (A.8) k∇hε,mi kL2(B εj) ≤ C(εj + ε δi i ), if i 6= j, (A.9) khε,mi kH1(Ω) ≤ C(| log εi|1/2+ εδii−1/2), (A.10) k˜˜hε,mi kH1(Ω) ≤ Cεδii+1, (A.11) k∇˜hε,mj kL2(B εi) ≤ C(εiε 2 j + ε δj+1 j ), if i 6= j, (A.12)
for any 0 < δi, δj < 1, with i, j = 1, . . . , N and m = 1, . . . , M .
Proof. By taking ˜hε,mi as a test function in the weak formulation of (3.20), we get Z Ω |∇˜hε,mi |2 = −1 αi Z Bεi(xi)
[V · ∇um0 − V (xi) · ∇um0 (xi)] ˜hε,mi . (A.13)
From the Cauchy-Schwarz inequality and the interior elliptic regularity of the function um
0 , there exists a positive constant C independent of ε, i and m such that
Z Ω |∇˜hε,mi |2 ≤ Cε−2 i kV · ∇u m 0 − V (xi) · ∇um0 (xi)kL2(B εi)k˜h ε,m i kL2(B εi) ≤ Cε−2i kx − xikL2(B εi)k˜h ε,m i kL2(B εi) ≤ Cε δi i k˜h ε,m i kH1(Ω), (A.14)
where we have used Lemma 2, with ϕ = ˜hε,mi , to obtain the last inequality. From the Poincar´e inequality, we have
Ck˜hε,mi k2H1(Ω) ≤
Z
Ω
|∇˜hε,mi |2. (A.15)
Combining the results obtained in (A.15) and (A.14), we get (A.4).
By considering the decomposition (3.16) as well as the fact that pε
i and qi are independent
of ε in Ωo, we can write khε,mi kL2(Ω o) ≤ C + k˜h ε,m i kL2(Ω o) ≤ C + k˜h ε,m i kH1(Ω). (A.16)
From (A.4), we have (A.5).
From the decomposition (3.16) and the triangular inequality, we have khε,mi kL2(B εj) ≤ C(kp ε ikL2(B εj)+ kqikL2(Bεj)+ k˜h ε,m i kL2(B εj)) (A.17) as well as k∇hε,mi kL2(B εj) ≤ C(k∇p ε ikL2(B εj)+ k∇qikL2(Bεj)+ k∇˜h ε,m i kL2(B εj)). (A.18)
By (3.18), simple calculation gives kpε
ikL2(B
εi) ≤ Cεi| log εi|, and kp
ε ikL2(B εj)≤ Cεj, if i 6= j; (A.19) k∇pε ikL2(B εi) ≤ C, and k∇p ε ikL2(B εj) ≤ Cεj, if i 6= j. (A.20)
Notice that, we can use the estimate of Lemma 2 for ϕ = ˜hε,mi together with the result obtained in (A.4) to get
k˜hε,mi kL2(B εj) ≤ Cε δi i ε δj j . (A.21)
Using the estimates (A.19) in (A.17), as well as the interior elliptic regularity of qitogether
with (A.21), we obtain (A.6) and (A.7), respectively. From (A.18), we have k∇hε,mi kL2(B εj) ≤ C(k∇p ε ikL2(B εj)+ k∇qikL2(Bεj)+ k˜h ε,m i kH1(Ω)). (A.22)
Using the estimates (A.20) in (A.22), as well as the interior elliptic regularity of qi
to-gether with (A.4), we obtain (A.8) and (A.9), respectively.
Now, let us take hε,mi as a test function in the weak form of (3.9) to obtain Z Ω |∇hε,mi |2 = −1 αi Z Bεi(xi) (V · ∇um0 )hε,mi . (A.23)
From the Cauchy-Schwarz inequality and the interior elliptic regularity of the function um0 , there exists a positive constant C independent of ε, i and m such that
Z Ω |∇hε,mi |2 ≤ Cε−2i kV · ∇um0 kL2(B εi)kh ε,m i kL2(B εi) ≤ Cε−1i khε,mi kL2(B εi) ≤ C(| log εi| + ε 2δi−1 i ), (A.24)
where we have used (A.6) to obtain the last inequality in (A.24). From the Poincar´e inequality, we have Ckhε,mi k2 H1(Ω) ≤ Z Ω |∇hε,mi |2. (A.25)
Combining the results obtained in (A.24) and (A.25), we get (A.10).
By taking ˜˜hε,mi as a test function in the weak formulation of (3.28), we get Z Ω |∇h˜˜ε,mi |2 = −1 αi Z Bεi(xi) Tim(x)h˜˜ε,mi . (A.26)
From the Cauchy-Schwarz inequality, there exists a positive constant C independent of ε, i and m such that
Z Ω |∇h˜˜ε,mi |2 ≤ Cε−2 i kT m i (x)kL2(B εi)k ˜ ˜ hε,mi kL2(B εi)≤ Cεik ˜ ˜ hε,mi kL2(B εi), (A.27)
where we have used the fact that Tim(x) = O(kx − xik2) and the estimate kx − xik2L2(B εi) =
O(ε3i) to obtain the last inequality. Now, we set ϕ = h˜˜ε,mi in Lemma 2 and we use the respective estimate to obtain, from (A.27), that
Z Ω |∇˜˜hε,mi |2 ≤ Cεδi+1 i k ˜ ˜ hε,mi kH1(Ω). (A.28)
From the Poincar´e inequality, we have Ck˜˜hε,mi k2
H1(Ω) ≤
Z
Ω
|∇˜˜hε,mi |2. (A.29)
Combining the results obtained in (A.28) and (A.29), we get the result (A.11).
By considering the decomposition (3.23), we can write k∇˜hε,mj kL2(B εi) ≤ C(k∇˜p ε,m j kL2(B εi)+ ε 2 jk∇q m j kL2(B εi)+ k∇ ˜ ˜ hε,mj kL2(B εi)) ≤ C(k∇˜pε,mj kL2(B εi)+ ε 2 jk∇q m j kL2(B εi)+ k ˜ ˜ hε,mj kH1(Ω)). (A.30)
From (3.26), simple calculation gives k∇˜pε,mj kL2(B
εi) ≤ Cεiε
2
j, if i 6= j. (A.31)
The required result (A.12) is obtained by taking into account the interior elliptic regularity of the function qm
j as well as the estimates (A.11) and (A.31) in (A.30).
Lemma 4. For i, j = 1, . . . , N and m = 1, . . . , M , let ϕε,mij and ˜hε,mii be solutions of (3.11) and (3.33), respectively, and let hε,mii be a function as defined in (3.21). Then, there exists a positive constant C independent of ε such that
k˜hε,mii kH1(Ω) ≤ C(εiδi−1+ ε2δi i−2), (A.32) khiiε,mkH1(Ω) ≤ Cεiδi−2(1 + εi+ εδii), (A.33) khijε,mkH1(Ω) ≤ C(εiδi−1+ εδii−2ε δj j ), if i 6= j, (A.34) khiiε,mkL2(Ωo) ≤ C(1 + εiδi−1+ ε2δi i−2), (A.35) k∇hε,mii kL2(B εi) ≤ C(ε −1 i + εi+ εδi −1 i + ε 2δi−2 i ), (A.36) k∇hε,mii kL2(B εj) ≤ C(εj+ ε δi−1 i + ε 2δi−2 i ), if i 6= j, (A.37) kϕε,mii kH1(Ω) ≤ Cεδii−1(1 + ε−1i + εδii−2+ ε2δi i−3), (A.38) kϕε,mij kH1(Ω) ≤ Cεδii−2(εi+ ε δj−1 j + ε 2δj−2 j ), if i 6= j, (A.39)
for any 0 < δi, δj < 1, with i, j = 1, . . . , N and m = 1, . . . , M .
Proof. By taking ˜hε,mii as a test function in the weak formulation of (3.33), we have Z Ω |∇˜hε,mii |2 = −1 αi Z Bεi(xi) [(V (xi) · ∇um0 (xi))(V · ∇qi) + V · ∇˜h ε,m i ]˜h ε,m ii . (A.40)
From the Cauchy-Schwarz inequality, there exists a positive constant C independent of ε, i and m such that
Z Ω |∇˜hε,mii |2 ≤ Cε−2 i (kV · ∇qikL2(B εi)+ kV · ∇˜h ε,m i kL2(B εi))k˜h ε,m ii kL2(B εi) ≤ Cε−2i (kV · ∇qikL2(B εi)+ k˜h ε,m i kH1(Ω))k˜hε,mii kL2(B εi) ≤ C(ε−1i + εδi−2 i )k˜h ε,m ii kL2(B εi), (A.41)
where we have used the interior elliptic regularity of the function qi and Lemma 3, result
(A.4), to obtain the last inequality. By considering the estimate of Lemma 2 for ϕ = ˜hε,mii in (A.41), we get Z Ω |∇˜hε,mii |2 ≤ C(εδi−1 i + ε 2δi−2 i )k˜h ε,m ii kH1(Ω). (A.42)
From the Poincar´e inequality, we have
Ck˜hε,mii k2H1(Ω) ≤
Z
Ω
|∇˜hε,mii |2. (A.43)
Combining the results obtained in (A.42) and (A.43), we get the result (A.32).
By taking hε,mij in the weak formulation of (3.10), we obtain Z Ω |∇hε,mij |2 = −1 αi Z Bεi(xi) (V · ∇hε,mj )hε,mij . (A.44)
From the Cauchy-Schwarz inequality, there exists a positive constant C independent of ε, i and m such that
Z Ω |∇hε,mij |2 ≤ Cε−2 i kV · ∇h ε,m j kL2(B εi)kh ε,m ij kL2(B εi). (A.45)
From (A.45), in the particular case where i = j, we have Z Ω |∇hε,mii |2 ≤ Cε−2i kV · ∇hε,mi kL2(B εi)kh ε,m ii kL2(B εi) ≤ Cεδi−2 i (1 + εi+ εδii)kh ε,m ii kH1(Ω), (A.46)
where we have used Lemma 3, result (A.8), as well as the result of Lemma 2 with ϕ = hε,mii , to obtain the last inequality. In addition, we also have from (A.45), for i 6= j, that
Z Ω |∇hε,mij |2 ≤ C(εδi−1 i + ε δi−2 i ε δj j )kh ε,m ij kH1(Ω), (A.47)
where we have used Lemma 3, result (A.9), as well as the result of Lemma 2 with ϕ = hε,mij . From the Poincar´e inequality, we have
Ckhε,mij k2H1(Ω) ≤
Z
Ω
|∇hε,mij |2. (A.48)
Combining the results obtained in (A.46) and (A.48), for i = j, we obtain the result (A.33). Analogously, we obtain the result (A.34) from (A.47) and (A.48).
By considering the decomposition (3.29) as well as the fact that pε
iiand qiiare independent
of ε in Ωo, we can write khε,mii kL2(Ω o) ≤ C + k˜h ε,m ii kL2(Ω o) ≤ C + k˜h ε,m ii kH1(Ω). (A.49)
From (A.32), we have (A.35).
From the decomposition (3.29) and the triangular inequality, we have khε,mii kL2(B εj) ≤ C(k∇p ε iikL2(B εj)+ k∇qiikL2(Bεj)+ k∇˜h ε,m ii kL2(B εj)) ≤ C(k∇pε iikL2(B εj)+ k∇qiikL2(Bεj)+ k˜h ε,m ii kH1(Ω)). (A.50)
By (3.31), simple calculation gives k∇pε iikL2(B εi) ≤ Cε −1 i and k∇p ε iikL2(B εj) ≤ Cεj, if i 6= j. (A.51)
Using the estimates (A.51) in (A.50), as well as the interior elliptic regularity of qii
to-gether with (A.32), we obtain (A.36) and (A.37), respectively.
By taking ϕε,mij as a test function in the weak formulation of (3.11), we have Z Ω |∇ϕε,mij |2 = −1 αi Z Bεi(xi) (V · ∇hε,mjj )ϕε,mij . (A.52)
From the Cauchy-Schwarz inequality, there exists a positive constant C independent of ε, i and m such that
Z Ω |∇ϕε,mij |2 ≤ Cε−2 i kV · ∇h ε,m jj kL2(B εi)kϕ ε,m ij kL2(B εi). (A.53)
From (A.53), in the particular case where i = j, we have Z Ω |∇ϕε,mii |2 ≤ Cε−2 i kV · ∇h ε,m ii kL2(B εi)kϕ ε,m ii kL2(B εi) ≤ Cεδi−1 i (1 + ε −2 i + ε δi−2 i + ε 2δi−3 i )kϕ ε,m ii kH1(Ω), (A.54)
where we have used (A.36) and the estimate of Lemma 2 with ϕ = ϕε,mii to obtain the last inequality. In addition, we also have from (A.53), for i 6= j, that
Z Ω |∇ϕε,mij |2 ≤ Cεδi−2 i (εi+ ε δj−1 j + ε 2δj−2 j )kϕ ε,m ij kL2(B εi), (A.55)
where we have used (A.37) and the estimate of Lemma 2 with ϕ = ϕε,mij . From the Poincar´e inequality, we have
Ckϕε,mij k2 H1(Ω) ≤
Z
Ω
|∇ϕε,mij |2. (A.56)
Combining the results obtained in (A.54) and (A.56), for i = j, we obtain the result (A.38). Analogously, we obtain the result (A.39) from (A.55) and (A.56). Lemma 5. For m = 1, . . . , M , let ˜um
ε be the weak solution of the variational problem to
find ˜um ε ∈ H01(Ω) such that Z Ω ∇˜umε · ∇η + Z Ω χε(V · ∇˜umε )η = Z Ω Φmε η, ∀η ∈ H1 0(Ω). (A.57)
where Φmε is given by (3.13)-(3.15). Then, there exists a positive constant C independent of ε such that k˜umε kH1(Ω) ≤ C(1 + Cε) " N X i=1 N X j=1 ε2δi−1 i (1 + ε −1 i + ε δi−2 i + ε 2δi−3 i )ε 6 j + N X i=1 N X j=1 N X l=1 l6=j εδi i ε δj j (εjε2l + εjε4l + ε δl+2 l + ε δl+3 l + ε 2δl+2 l ) # , (A.58)
for any 0 < δi, δj, δl< 1 with ε = max{εδii}, for i, j, l = 1, . . . , N .
Proof. By taking η = ˜umε as a test function in (A.57), we have Z Ω |∇˜umε |2+ Z Ω χε(V · ∇˜umε )˜u m ε = − Z Ω Φmε u˜mε (A.59)
and from the definitions of χεand Φmε given by (3.6) and (3.13)-(3.15), respectively, (A.59)
can be written as Z Ω |∇˜umε |2 = − N X i=1 N X j=1 N X l=1 l6=j αjαl Z Bεi(xi) (V · ∇hε,mjl )˜umε − N X i=1 N X j=1 N X l=1 αjαl2 Z Bεi(xi) (V · ∇ϕε,mjl )˜umε − N X i=1 Z Bεi(xi) (V · ∇˜umε )˜umε . (A.60)
From the Cauchy-Schwarz inequality, we obtain Z Ω |∇˜umε |2 ≤ C N X i=1 N X j=1 N X l=1 l6=j ε2jε2lkV · ∇hε,mjl kL2(B εi)k˜u m ε kL2(B εi) +C N X i=1 N X j=1 ε6jkV ·∇ϕε,mjj kL2(B εi)k˜u m ε kL2(B εi)+C N X i=1 N X j=1 N X l=1 l6=j ε2jε4lkV ·∇ϕε,mjl kL2(B εi)k˜u m ε kL2(B εi) + C N X i=1 kV · ∇˜umε kL2(B εi)k˜u m ε kL2(B εi) (A.61)
and taking into account the estimates kV ·∇hε,mjl kL2(B εi) ≤ kh ε,m jl kH1(Ω), kV ·∇ϕε,m jl kL2(B εi)≤ kϕε,mjl kH1(Ω) and kV · ∇˜umε kL2(B εi) ≤ k˜u m ε kH1(Ω), we get Z Ω |∇˜umε |2 ≤ C N X i=1 N X j=1 N X l=1 l6=j ε2jε2lkhε,mjl kH1(Ω)k˜umε kL2(B εi) + C N X i=1 N X j=1 ε6jkϕε,mjj kH1(Ω)k˜umε kL2(B εi) + C N X i=1 N X j=1 N X l=1 l6=j ε2jε4lkϕε,mjl kH1(Ω)k˜umε kL2(B εi)+ C N X i=1 k˜umε kL2(B εi)k˜u m ε kH1(Ω). (A.62)
By considering the estimate of Lemma 2 for ϕ = ˜um
ε in (A.62), we get k˜umε kL2(B εi) ≤
Cεδi
i k˜umε kH1(Ω). We use such estimate in (A.62) to obtain
Z Ω |∇˜umε |2 ≤ C N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 2 lkh ε,m jl kH1(Ω)k˜umε kH1(Ω) + C N X i=1 N X j=1 εδi i ε 6 jkϕ ε,m jj kH1(Ω)k˜umε kH1(Ω) + C N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 4 lkϕ ε,m jl kH1(Ω)k˜umε kH1(Ω)+ C N X i=1 εδi i k˜u m ε k 2 H1(Ω). (A.63)
We define the quantity ε := max{εδi
i }, for i = 1, . . . , N , and then the last inequality above
can be rewritten as Z Ω |∇˜umε |2 ≤ C N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 2 lkh ε,m jl kH1(Ω)k˜umε kH1(Ω) + C N X i=1 N X j=1 εδi i ε 6 jkϕ ε,m jj kH1(Ω)k˜umε kH1(Ω) + C N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 4 lkϕ ε,m jl kH1(Ω)k˜umε kH1(Ω)+ Cεk˜umε k2H1(Ω). (A.64)
From the Poincar´e inequality, we obtain Ck˜umε k2
H1(Ω) ≤
Z
Ω
|∇˜umε |2. (A.65)
Combining the results obtained in (A.64) and (A.65), we get
k˜umε kH1(Ω) ≤ C 1 1 − Cε !" N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 2 lkh ε,m jl kH1(Ω) + N X i=1 N X j=1 εδi i ε 6 jkϕ ε,m jj kH1(Ω)+ N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 4 lkϕ ε,m jl kH1(Ω) # (A.66)
and taking into account that 1 1 − Cε = 1 + Cε + O(ε 2), we obtain k˜umε kH1(Ω) ≤ C(1 + Cε) " N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 2 lkh ε,m jl kH1(Ω) + N X i=1 N X j=1 εδi i ε 6 jkϕ ε,m jj kH1(Ω)+ N X i=1 N X j=1 N X l=1 l6=j εδi i ε 2 jε 4 lkϕ ε,m jl kH1(Ω) # . (A.67)
The required result (A.58), for any 0 < δi, δj, δl < 1 with i, j, l = 1, . . . , N and m =
1, . . . , M , is finally obtained by considering Lemma 4, results (A.34), (A.38) and (A.39). A.2. Estimation for the remainder. We will successively prove that |Em
` (ε)| = o(|ε|4)
for ` = 1, . . . , 17, where |ε| := ε1 + · · · + εN. For simplicity, we use the symbol C to
denote any constant independent of ε. Remainder Em(ε) can be trivially bounded by using similar arguments as in [16], leading to |Em(ε)| = o(|ε|4). The estimations of the other remainders are more involved and will be presented in details in what follows.
In order to obtain the estimate of each remainder, we start by using the Cauchy-Schwarz inequality and then we use the appropriate lemmas of Section A.1 as well as additional results related to some functions introduced in Section 3. Proceeding in this way, we obtain |Em 1 (ε)| ≤ Ck˜u m ε kL2(Ω o)ku m 0 − z mk L2(Ω o) ≤ Ck˜u m ε kH1(Ω) = o(|ε|4), (A.68)
for any 2/3 < δ < 1, where we have used Lemma 5;
|E2m(ε)| ≤ C|ε|6 N X i=1 khε,mi kL2(Ωo) " N X j=1 khε,mjj kL2(Ωo)+ N X j=1 N X l=1 l6=j khε,mjl kH1(Ω) # = o(|ε|4), (A.69) for any 0 < δ < 1, where we have used Lemma 3, result (A.5), and Lemma 4, results (A.34) and (A.35);
|Em 3 (ε)| ≤ C|ε| 8 N X i=1 khε,mi kL2(Ω o) " N X j=1 kϕε,mjj kH1(Ω)+ N X j=1 N X l=1 l6=j kϕε,mjl kH1(Ω) # = o(|ε|4), (A.70) for any 0 < δ < 1, where we have used Lemma 3, result (A.5), and Lemma 4, results (A.38) and (A.39);
|Em 4 (ε)| ≤ C|ε| 2k˜um ε kH1(Ω) N X i=1 khε,mi kL2(Ω o) = o(|ε| 4), (A.71)
for any 0 < δ < 1, where we have used Lemma 3, result (A.5), and Lemma 5;
|Em 5 (ε)| ≤ C|ε| 8 " N X i=1 khε,mii kL2(Ω o)+ N X i=1 N X j=1 j6=i khε,mij kH1(Ω) # × " N X l=1 khε,mll kL2(Ω o)+ N X l=1 N X p=1 p6=l khε,mlp kH1(Ω) # = o(|ε|4), (A.72)
for any 0 < δ < 1, where we have used Lemma 4, results (A.34) and (A.35); |Em 6 (ε)| ≤ C|ε| 10 " N X i=1 khε,mii kL2(Ω o)+ N X i=1 N X j=1 j6=i khε,mij kH1(Ω) # × " N X l=1 kϕε,mll kH1(Ω)+ N X l=1 N X p=1 p6=l kϕε,mlp kH1(Ω) # = o(|ε|4), (A.73)
for any 1/6 < δ < 1, where we have used Lemma 4, results (A.34), (A.35), (A.38) and (A.39); |Em 7 (ε)| ≤ C|ε| 4k˜um ε kH1(Ω) " N X i=1 khε,mii kL2(Ω o)+ N X i=1 N X j=1 j6=i khε,mij kH1(Ω) # = o(|ε|4), (A.74)
for any 0 < δ < 1, where we have used Lemma 4, results (A.34)-(A.35), and Lemma 5;
|Em 8 (ε)| ≤ C|ε|12 " N X i=1 kϕε,mii kH1(Ω)+ N X i=1 N X j=1 j6=i kϕε,mij kH1(Ω) # × " N X l=1 kϕε,mll kH1(Ω)+ N X l=1 N X p=1 p6=l kϕε,mlp kH1(Ω) # = o(|ε|4), (A.75)
for any 0 < δ < 1, where we have used Lemma 4, results (A.38) and (A.39);
|Em 9 (ε)| ≤ C|ε| 6k˜um ε kH1(Ω) " N X i=1 kϕε,mii kH1(Ω)+ N X i=1 N X j=1 j6=i kϕε,mij kH1(Ω) # = o(|ε|4), (A.76)
for any 0 < δ < 1, where we have used Lemma 4, results (A.38)-(A.39), and Lemma 5; |Em
10(ε)| ≤ Ck˜u m
ε kH1(Ω)k˜uεmkH1(Ω) = o(|ε|4), (A.77)
for any 0 < δ < 1, where we have used Lemma 5;
|E11m(ε)| ≤ C|ε|2 N X i=1 (ε2ikV · ∇qimkL2(B εi)+ kV · ∇ ˜ ˜ hε,mi kL2(B εi))kv mk L2(B εi) ≤ C|ε|3 N X i=1 (ε3i + k˜˜hiε,mkH1(Ω)) = o(|ε|4), (A.78)
for any 0 < δ < 1, where we have used the interior elliptic regularity of the functions vm and qim as well as the estimate kV · ∇˜˜hε,mi kL2(B
εi) ≤ Ck
˜ ˜
hε,mi kH1(Ω) together with Lemma
3, result (A.11); |Em 12(ε)| ≤ C|ε| 2 N X i=1 N X j=1 j6=i kV · ∇˜hε,mj kL2(B εi)kv mk L2(B εi) = o(|ε| 4), (A.79)
for any 0 < δ < 1, where we have used the interior elliptic regularity of the function vm
as well as Lemma 3, result (A.12);
|Em 13(ε)| ≤ C|ε| 4 N X i=1 N X j=1 kV · ∇qjjkL2(B εi)kv mk L2(B εi)= O(|ε| 6), (A.80)
where we have used the interior elliptic regularity of the functions qjj and vm;
|Em 14(ε)| ≤ C|ε| 4 N X i=1 N X j=1 kV · ∇˜hε,mjj kL2(B εi)kv mk L2(B εi) ≤ C|ε|5 N X j=1 k˜hε,mjj kH1(Ω)= o(|ε|4), (A.81)
for any 1/2 < δ < 1, where we have used the interior elliptic regularity of the function vm as well as the estimate kV · ∇˜hε,mjj kL2(B
εi) ≤ Ck˜h
ε,m
jj kH1(Ω) together with Lemma 4, result
(A.32); |Em 15(ε)| ≤ C|ε| 4 N X i=1 N X j=1 j6=i kV · ∇pε jjkL2(B εi)kv mk L2(B εi) = O(|ε| 6), (A.82)
where we have used the interior elliptic regularity of the function vm and the explicit solution of pε
jj, given by (3.31), to estimate kV · ∇pεjjkL2(B
εi)= O(εi), for i 6= j;
|Em 16(ε)| ≤ C|ε| 4 N X i=1 N X j=1 (khikL2(Ω o)k˜h ε,m j kL2(Ω o)+ khjkL2(Ωo)k˜h ε,m i kL2(Ω o)) ≤ C|ε|4 N X i=1 N X j=1 (k˜hε,mi kH1(Ω)+ k˜hε,mj kH1(Ω)) = o(|ε|4), (A.83)
for any 0 < δ < 1, where we have used the fact that khikL2(Ω
o) = O(1) and Lemma 3,
result (A.4); |Em 17(ε)| ≤ C|ε|4 N X i=1 k˜hε,mi kH1(Ω) N X j=1 k˜hε,mj kH1(Ω) = o(|ε|4), (A.84)
for any 0 < δ < 1, where we have used Lemma 3, result (A.4).
Acknowledgments
This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency) and FAPERJ (Research Foundation of the State of Rio de Janeiro). These supports are gratefully acknowledged. These research results have also received funding from the EU H2020 Programme and from MCTI/RNP-Brazil under the HPC4E Project, grant agreement n◦ 689772. R. Prakash wishes to thank CONICYT for the financial support through FONDECYT INICIACI ´ON n◦ 11180551. He would also acknowledge the support from the Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Concepci´on (Chile). J. Soko lowski acknowledges the partial support of the Agence Nationale de la Recherche (ANR) in France, under Grant ANR-17-CE08-0039 (Project ArchiMatHOS).
