Adel Hamdi

Identification of a point source in a linear advection–dispersion–reaction equation: application to a pollution source problem

We consider the problem of identification of a pollution source in a river. The mathematical model is a one-dimensional linear advection?dispersion?reaction equation with the right-hand side spatially supported at a point (the source) and a time-dependent intensity, both unknown. Assuming that the source becomes inactive after the time T*, we prove that it can be identified by recording the evolution of the concentration at two points, one of which is strategic.

Properties of an augmented Lagrangian for design optimization

We consider the task of design optimization, where the constraint is a state equation that can only be solved by a typically rather slowly converging fixed point solver. This process can be augmented by a corresponding adjoint solver, and based on the resulting approximate reduced derivatives, also an optimization iteration that updates the design variables simultaneously. To coordinate the three iterative processes, we use an exact penalty function of a doubly augmented Lagrangian type that should be consistently reduced. Some numerical experiments on a variant of the Bratu problem are presented.

The recovery of a time-dependent point source in a linear transport equation: application to surface water pollution

The aim of this paper is to localize the position of a point source and recover the history of its time-dependent intensity function that is both unknown and constitutes the right-hand side of a 1D linear transport equation. Assuming that the source intensity function vanishes before reaching the final control time, we prove that recording the state with respect to the time at two observation points framing the source region leads to the identification of the source position and the recovery of its intensity function in a unique manner. Note that at least one of the two observation points should be strategic. We establish an identification method that determines quasi-explicitly the source position and transforms the task of recovering its intensity function into solving directly a well-conditioned linear system. Some numerical experiments done on a variant of the water pollution BOD model are presented.

Identification of point sources in two-dimensional advection-diffusion-reaction equation: application to pollution sources in a river. Stationary case

We consider the problem of determining pollution sources in a river by using boundary measurements. The mathematical model is a two-dimensional advection-diffusion-reaction equation in the stationary case. Identifiability and a local Lipschitz stability results are established. A cost function transforming our inverse problem into an optimization one is proposed. This cost function represents the difference between the two solutions computed from the prescribed and measured data respectively. This representation is achieved by using values of these two solutions inside the domain. Numerical results are performed for a rectangular domain. These results are compared to those obtained by using a classical least squares regularized method.

Inverse source problem in an advection–dispersion–reaction system: application to water pollution

In this paper, we consider the problem of identifying an unknown source F( x, t)= λ(t)δ(x − S) in the following system: (∂t − D∂xx + V∂ x + R)u(x, t) = F( x, t), 0 <x<� , 0 <t < T (∂t − D∂xx + V∂ x + R)v(x, t) = Ru(x, t), 0 <x<� , 0 <t < T from measured data [{v(a, t), ∂xv(a, t)}, {v(b, t), ∂xv(b, t)}] for appropriate points a and b. Assuming that the source F became inactive after the time T ∗ (i.e .λ (t)= 0f ort T ∗ ), we prove an identifiability result and propose an identification method. (Some figures in this article are in colour only in the electronic version)

Inverse source problem in a one-dimensional evolution linear transport equation with spatially varying coefficients: application to surface water pollution

Abstract This paper deals with the identification of a time-dependent point source occurring in the right-hand side of a one-dimensional evolution linear advection–dispersion–reaction equation. The originality of this study consists in considering the general case of transport equations with spatially varying dispersion, velocity and reaction coefficients which enables to extend the applicability of the obtained results to various areas of science and engineering. We derive a main condition on the involved spatially varying coefficients that yields identifiability of the sought source, provided its time-dependent intensity function vanishes before reaching the final monitoring time, from recording the generated state at two observation points framing the source region. Then, we establish an identification method that uses those records to determine the elements defining the sought source. Some numerical experiments on a variant of the surface water pollution model are presented.

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

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

Identification of a time-varying point source in a system of two coupled linear diffusion-advection- reaction equations: application to surface water pollution

This paper deals with the identification of a point source (localization of its position and recovering the history of its time-varying intensity function) that constitutes the right-hand side of the first equation in a system of two coupled 1D linear transport equations. Assuming that the source intensity function vanishes before reaching the final control time, we prove the identifiability of the sought point source from recording the state relative to the second coupled transport equation at two observation points framing the source region. Note that at least one of the two observation points should be strategic. We establish an identification method that uses these records to identify the source position as the root of a continuous and strictly monotonic function. Whereas the source intensity function is recovered using a recursive formula without any need of an iterative process. Some numerical experiments on a variant of the surface water pollution BOD–OD coupled model are presented.

Detection and identification of multiple unknown time-dependent point sources occurring in 1D evolution transport equations

This paper addresses the nonlinear inverse source problem of identifying multiple unknown time-dependent point sources occurring in a 1D evolution advection–diffusion–reaction equation. We prove that time records of the associated state taken only upstream and downstream all involved source positions do not uniquely identify the unknown elements defining the sought time-dependent point sources if their number is bigger or equal to 2. Then, based on some impulse response techniques we establish an identifiability theorem that under some reasonable assumptions on the time state recording positions yields uniqueness of those unknown elements. This theorem together with a data assimilation result led to develop a source detection-identification method that goes throughout the monitored domain to detect the presence of each unknown active source occurring between two state recording points. Once a source is detected, the established identification procedure localizes its position, determines the total amount loaded by this source and identifies the historic of its time-dependent source intensity function. Some numerical experiments on a variant of the surface water biological oxygen demand pollution model are presented.

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

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.