Diffusion and ecological problems : mathematical models

Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

In many parts of the world, groundwater resources are under increasing threat from growing demands, wasteful use, and contamination. To face the challenge, good planning and management practices are needed. A key to the management of groundwater is the ability to model the movement of fluids and contaminants. The purpose of this book is to construct conceptual and mathematical models that can provide the information required for making decisions associated with the management of groundwater resources, and the remediation of contaminated aquifers. The basic approach of this book is to accurately describe the underlying physics of groundwater flow and solute transport in heterogeneous porous media, starting at the microscopic level, and to rigorously derive their mathematical representations at the macroscopic levels.

Modeling groundwater flow and contaminant transport

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Where do inertial particles go in fluid flows

An inverse Gaussian plume approach for estimating atmospheric pollutant emissions from multiple point sources

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.

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

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 which actually changes the design. To coordinate the three iterative processes, we use an exact penalty function of doubly augmented Lagrangian type. The main issue here is how to derive a design space preconditioner for the approximated reduced gradient which ensures a consistent reduction of the employed penalty function as well as significant design corrections. Some numerical experiments for an alternating approach where any combination and sequencing of steps are used to improve feasibility and optimality done on a variant of the Bratu problem are presented.

Reduced quasi-Newton method for simultaneous design and 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.

Properties of an augmented Lagrangian for design optimization

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)

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

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.

/pdf/identification-of-point-sources-in-two-dimensional-advection-2ilchtyj9q.pdf

Adel Hamdi

Papers

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

Reduced quasi-Newton method for simultaneous design and optimization

Properties of an augmented Lagrangian for design optimization

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

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