Recovering a scalar nonlinearity in semilinear convection–diffusion–reaction equations from full-boundary data

Abstract

We consider the recovery of a scalar nonlinearity in a semilinear Robin boundary value problem arising, for example, in steady-state heat transfer with nonlinear sources and reactive transport. Given multiple experiments with boundary excitations and full-boundary measurements of the corresponding states, we study the inverse problem of recovering the nonlinearity on the range of observed state values. For the reduced formulation, we establish problem-specific stability, sequential weak-to-strong continuity, and Fréchet differentiability properties of the parameter-to-state map associated with the nonlinear composition

. These properties allow us to place the inverse problem within the standard Hilbert-space Tikhonov regularization framework and to record the corresponding existence, stability, and convergence properties of regularized minimizers. We then construct iterative regularization schemes of Gauss–Newton and nonlinear conjugate-gradient type. In addition, we introduce an all-at-once (AAO) Tikhonov formulation that treats the states and the nonlinearity as independent variables and penalizes both data misfit and PDE residuals. This formulation avoids elimination of the states and admits an existence analysis under weaker regularity assumptions on the scalar nonlinearity than those used for the differentiable reduced formulation. We derive the corresponding Gauss–Newton AAO system. Finally, we present two-dimensional numerical experiments for a model problem with synthetic boundary data perturbed by noise, where the unknown nonlinearity is recovered from several boundary excitations and the reduced and AAO reconstruction strategies are compared in terms of reconstruction quality and computational cost.