Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations

Adel Hamdi; Imed Mahfoudhi

doi:10.1515/jiip-2014-0051

Published by De Gruyter March 20, 2016

Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations

Adel Hamdi and Imed Mahfoudhi

From the journal Journal of Inverse and Ill-posed Problems

https://doi.org/10.1515/jiip-2014-0051

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Abstract

The paper deals with the nonlinear inverse source problem of identifying an unknown time-dependent point source occurring in a two-dimensional evolution advection-dispersion-reaction equation with spatially varying velocity field and dispersion tensor. The originality of this study consists in establishing a constructive identifiability theorem that leads to develop an identification method using only significant boundary observations and operating other than the classic optimization approach. To this end, we derive two dispersion-current functions that have the main property to be of orthogonal gradients which yield identifiability of the elements defining the involved unknown source from some boundary observations related to the associated state. Provided the velocity field fulfills the so-called no-slipping condition, the required boundary observations are reduced to only recording the state on the outflow boundary and its flux on the inflow boundary of the monitored domain. We establish an identification method that uses those boundary records (1) to localize the sought source position as the unique solution of a nonlinear system defined by the two dispersion-current functions, (2) to give lower and upper bounds of the total amount loaded by the unknown time-dependent source intensity function, (3) to transform the identification of this latest into solving a deconvolution problem. Some numerical experiments on a variant of the surface water BOD pollution model are presented.

Keywords: Nonlinear inverse source problem; boundary null controllability; optimization,advection-dispersion-reaction equation; application to surface water pollution

MSC 2010: 35R30; 76B75; 93B05

A Appendix

We establish the proof of the result announced in Proposition 4. This proof is inspired by the results obtained in [21].

Proof.

For all ε>0 and ξ0∈L2⁢(Ω), let Jε be the functional defined from J introduced in (2.31) as follows: Jε⁢(ξ0)=J⁢(ξ0)+ε⁢∥ξ0∥L2⁢(Ω). We start by proving the coercivity and the strictly convexity of Jε. Then, we conclude about the existence and the unicity of the minimizer for J. We employ the two linear operators G and K introduced in (4.10)–(4.11).

Step 1 (coercivity). Let (ξ0j)j≥1 be a sequence of L2⁢(Ω) such that limj→∞⁡∥ξ0j∥L2⁢(Ω)=∞, ξ~0j=ξ0j/∥ξ0j∥L2⁢(Ω) for all j≥1 and Q:L2⁢(Ω)→ℝ that to a given ξ0 associates

Q⁢(ξ0)=12⁢〈G⁢(ξ0),G⁢(ξ0)〉L2⁢(Γout×(T*,T)).

Then, we have

limj→∞⁡Jε⁢(ξ0j)∥ξ0j∥L2⁢(Ω)=limj→∞⁡(12⁢∥ξ0j∥L2⁢(Ω)⁢Q⁢(ξ~0j)+ε-〈e-12⁢ψ⁢φ0,K⁢(ξ~0j)〉L2⁢(Ω)).

That leads to distinguish two cases.

Case 1. We have lim⁡infj→∞⁡Q⁢(ξ~0j)>0 which implies that lim⁡infj→∞⁡Jε⁢(ξ0j)=∞.

Case 2. We have lim⁡infj→∞⁡Q⁢(ξ~0j)=0. Then, since the sequence (ξ~0j)j≥1 is bounded, we can extract a subsequence (indexed also by j for simplicity) that converges weakly in L2⁢(Ω) to ξ~0. Thus, we have Q⁢(ξ~0)=0 which implies in view of the unique continuation theorem from [13] that ξ~0=0. That makes

lim⁡infj→∞⁡Jε⁢(ξ0j)∥ξ0j∥L2⁢(Ω)=ε

and thus, the coercivity of Jε follows.

Step 2 (convexity). The strict convexity of the functional Jε for all ε≥0 is clearly implied by the strict convexity of Q. Besides, due to the linearity of ξ the solution of the adjoint problem (2.30) with respect to ξ0, we have for all ξ~0 in L2⁢(Ω) and β in (0,1),

(A.1)Q⁢(β⁢ξ0+(1-β)⁢ξ~0)=12⁢〈β⁢G⁢(ξ0)+(1-β)⁢G⁢(ξ~0),β⁢G⁢(ξ0)+(1-β)⁢G⁢(ξ~0)〉L2⁢(Γout×(T*,T))=β2⁢Q⁢(ξ0)+(1-β)2⁢Q⁢(ξ~0)+β⁢(1-β)⁢〈G⁢(ξ0),G⁢(ξ~0)〉L2⁢(Γout×(T*,T))=β⁢Q⁢(ξ0)+(1-β)⁢Q⁢(ξ~0)-12⁢β⁢(1-β)⁢〈G⁢(ξ0-ξ~0),G⁢(ξ0-ξ~0)〉L2⁢(Γout×(T*,T))≤β⁢Q⁢(ξ0)+(1-β)⁢Q⁢(ξ~0).

Note that the third equality in (A.1) is obtained from

〈G⁢(ξ0),G⁢(ξ~0)〉L2⁢(Γout×(T*,T))=Q⁢(ξ0)+Q⁢(ξ~0)-12⁢〈G⁢(ξ0-ξ~0),G⁢(ξ0-ξ~0)〉L2⁢(Γout×(T*,T)).

Now, suppose there exists ξ0 and ξ~0 in L2⁢(Ω) such that the inequality in (A.1) holds as an equality. Then, we get

(A.2)〈G⁢(ξ0-ξ~0),G⁢(ξ0-ξ~0)〉L2⁢(Γout×(T*,T))=0⟹ξ0=ξ~0.

The implication in (A.2) is obtained using the unique continuation theorem from [13]. That proves the strict convexity of the functional Jε for all ε≥0.

Therefore, for all ε>0 the functional Jε admits a unique minimizer ξ0ε^. Besides, notice that the mapping

ξ0ε^↦N⁢(ξ0ε^):=∥D⁢∇⁡ξε^⋅ν∥L2⁢(Γout×(T*,T))

defines a norm of ξ0ε^, where ξε^ is the solution of the adjoint problem (2.30) with the initial data ξε^⁢(⋅,T)=ξ0ε^. Then, since Jε⁢(0)=0, we have Jε⁢(ξ0ε^)≤0 which implies that

(A.3)12⁢∥D⁢∇⁡ξε^⋅ν∥L2⁢(Γout×(T*,T))2≤〈e-12⁢ψ⁢φ0,ξε^⁢(⋅,T*)〉L2⁢(Ω).

In addition, using a classic observability result [20, 23] on the adjoint problem (2.30) with the initial data ξε^⁢(⋅,T)=ξ0ε^, we get

(A.4)∥ξε^⁢(⋅,T*)∥L2⁢(Ω)≤C⁢∥D⁢∇⁡ξ^ε⋅ν∥L2⁢(Γout×(T*,T)),

where C is a positive real number independent of ε. Hence, (A.3) and (A.4) imply that the norm N⁢(ξ0ε^) is uniformly bounded (independently of ε) and thus, the sequence made by the minimizers (ξ0ε^)ε>0 of the functionals Jε converges, when ε tends to 0, to ξ^0 that is a local minimizer of the functional J. Furthermore, in view of the strict convexity of J, ξ^0 is the unique minimizer of J. ∎

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Received: 2014-7-20

Revised: 2015-4-29

Accepted: 2016-2-1

Published Online: 2016-3-20

Published in Print: 2016-12-1

Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations

Abstract

A Appendix

Proof.

References

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