Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales

Abstract

Abstract. We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established by Cannon and we therefore focus on the stable solution in the presence of data noise. For this, we utilize a reformulation of the inverse problem as a linear ill-posed operator equation with perturbed data and operators. We are able to explicitly characterize the mapping properties of the corresponding operators which allow us to apply regularization in Hilbert scales. We can then prove convergence and convergence rates of the regularized reconstructions under very mild assumptions on the exact parameter. These are, in fact, already needed for the analysis of the forward problem and no additional source conditions are required. Numerical tests are presented to illustrate the theoretical statements.

Parameter identification in a semilinear hyperbolic system

“…In order to verify the approximate source condition (5.4), it thus suffices to consider the condition z = K * w for the linear problem. From the explicit respresentation (6.4) of the operator K this can be translated directly to a smoothness condition on z in terms of weighted Sobolev spaces and some boundary conditions; we refer to [11] for a detailed derivation in a similar context. Remark 6.2.…”

Identification of nonlinear heat conduction laws

“…Via the transformation U = A(u), the quasilinear problem ( 1)-( 2) can be transformed into a Neumann problem for the Poisson equation, and solvability follows from standard results for linear elliptic equations [14]; see also [13] for details concerning this particular problem. We thus obtain Theorem 1 Let (A1)-(A3) hold.…”

“…Following [4], see also [13,22], the parameter function a(u) can be uniquely determined from temperature measurements g = u| γ on a boundary curve γ. We present an alternative proof of this uniqueness result below which allows us to treat also perturbations in the data f , j, and in g, and to obtain a stability result for the inverse problem.…”

“…Following [4], see also [13,22], the parameter function a(u) can be uniquely determined from temperature measurements g = u| γ on a boundary curve γ.…”

On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms

“…The identification of a = a(u) in the quasilinear elliptic equation −div(a(u)∇u) = 0 has been investigated by Cannon [4], who gave a constructive proof for the determination of the coefficient from knowledge of a single measurement of u along a curve on ∂Ω. A stable numerical method for the problem has been proposed in [12]. Simultaneous identification of two parameters a and c in the elliptic equation −div(a(u)∇u) + c(x)u = 0 has been considered in [13].…”