Paul DuChateau

Structural identification of an unknown source term in a heat equation

The identification of an unknown state-dependent source term in a reaction-diffusion equation is considered. Integral identities are derived which relate changes in the source term to corresponding changes in the measured output. The identities are used to show that the measured boundary output determines the source term uniquely in an appropriate function class and to show that a source term that minimizes an output least squares functional based on this measured output must also solve the inverse problem. The set of outputs generated by polygonal source functions is shown to be dense in the set of all admissible outputs. Results from some numerical experiments are discussed.

An Inverse Problem for a Nonlinear Diffusion Equation

In this paper we consider the determination of an unknown diffusion coefficient in a nonlinear diffusion equation from overspecified data measured at the boundary. This inverse problem is reformulated as an “auxiliary inverse problem,” where we seek a member of a class of admissible coefficients which minimizes a given error functional. It is shown that this auxiliary problem has at least one solution in a specified admissible class. Finally, the auxiliary problem is approximated by an associated identification problem and some numerical results are presented.

In situ measures of methanotroph activity in upland soils: A reaction‐diffusion model and field observation of water stress

[1] Laboratory assays of methanotroph activity in upland (i.e., well-drained, oxic) ecosystems alter soil physical structure and weaken inference about environmental controls of their natural behavior. To overcome these limitations, we developed a chamber-based approach to quantify methanotroph activity in situ on the basis of measures of soil diffusivity (from additions of an inert tracer gas to the chamber headspace), methane concentration change, and analysis of results with a reaction-diffusion model. The analytic solution to this model predicts that methane consumption rates are equally sensitive to changes in methanotroph activity and diffusivity, but that doubling either of these parameters leads to only a p 2 increase in consumption. With a series of simulations, we generate guidelines for field deployments and show that the approach is robust to plausible departures from assumptions. We applied the approach on a dry grassland in north central Colorado. Our model closely fit measured changes in methane concentrations, indicating that we had accurately characterized the biophysical processes underlying methane uptake. Field patterns showed that, over a 7-week period, soil moisture fell from 38% to 15% water-filled pore spaces, and diffusivity doubled as the larger soil pores drained of water. However, methane uptake rates fell by � 40%, following a 90% decrease in methanotroph activity, suggesting that the decline in methanotroph activity resulted from water stress to methanotrophs. We anticipate that future application of this approach over longer timescales and on more diverse field sites has potential to provide important insights into the ecology of methanotrophs in upland soils.

Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation

In this paper we consider an inverse problem in which we seek to deter- mine an unknown source or reaction term in a reaction-diffusion equation from overspecified data measured on the boundary of the spatial region where the equation applies. The analysis is based on the observation that the overspecified data depends monotonically on the unknown source term in the equation. Here we use this monotonicity to establish a type of unicity result for the inverse problem. The attractiveness of the monotonicity methods illustrated here lies in their versatility. Surveying the rapidly expanding literature devoted to the topic of inverse problems, one finds a considerable variety of ad hoc approaches. Methods which apply to a general class of problems seem to be few. Monotonicity methods, however, apply to a variety of inverse problems involving partial differential equations of parabolic or elliptic type. This paper is organized as follows. In Section 1 we formulate the direct initial boundary value problem (IBVP) and state assumptions on the data and a priori assumptions on the unknown source term. Several properties of the solution to the direct problem are derived. In Section 2 the principal monotonicity estimates are established and in Section 3 these are applied to the inverse problem to prove a type of unicity result for the solution to the inverse problem. In particular, it is a Corollary of Theorem 3.2 that in the class of analytic source terms, the solution of the inverse problem is unique. The problem of identifying an unknown source term in a heat equation 155

Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems

This paper considers the coefficient-to-data mappings associated with unknown coefficient inverse problems for nonlinear parabolic partial differential equations. Integral identities are derived th...

Determining Unknown Coefficients in a Nonlinear Heat Conduction Problem

This paper considers the problem of determining the positive unknown functions

Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient

a( u ),b( u )

Monotonicity and Uniqueness Results in Identifying an Unknown Coefficient in a Nonlinear Diffusion Equation

and the unknown function

An inverse problem for an unknown source in a heat equation

u = u( {x,t} )

Determination of unknown physical properties in heat conduction problems

in the nonlinear diffusion equation