Adjoint-Based Bayesian Uncertainty Quantification for PDE-Constrained Inverse Problems with Application to Semiconductor Imaging

We formulate a Bayesian framework for reconstructing doping profiles in pn-junction semiconductor devices from boundary flux measurements.

The unknown doping field is modeled as a piecewise-constant function characterized by an unknown interface and two plateau concentrations, leading to a nonlinear ill-posed inverse problem governed by a Poisson-Boltzmann-type equation.

To represent this structure while enabling efficient gradient-based inference, we introduce a pushforward prior constructed by mapping a latent Gaussian field with Matérn-type covariance through a sigmoid transformation.

The latent field is parameterized by a truncated Karhunen-Loève expansion, while the two piecewise-constant levels are represented by scalar plateau parameters.

The prior yields differentiable approximations of piecewise-constant fields with controllable interface sharpness.

We establish well-posedness of the Bayesian formulation by proving Lipschitz continuity of the forward map and Hellinger stability of the posterior.

We then sample the posterior using the No-U-Turn Sampler (NUTS) with gradients computed by the adjoint method.

Numerical experiments show that the combination of the proposed prior and NUTS provides more efficient posterior exploration than the dimension-robust preconditioned Crank-Nicolson (pCN) sampler, yielding one to two orders of magnitude larger effective sample sizes.

In the known-plateau setting, the method reconstructs both planar and curved interfaces and provides spatially resolved uncertainty quantification (UQ).

When the interface geometry and plateau concentrations are inferred jointly, posterior correlations reveal structural non-identifiability.

These results demonstrate the effectiveness of combining pushforward priors with adjoint-gradient-based sampling for reliable UQ in nonlinear partial differential equation-constrained inverse problems with sharp interfaces.