Babak Maboudi Afkham

Hassan Yazdanian, Leila Taghizadeh, Babak Maboudi Afkham

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.

Siiri Rautio, Hjørdis Schlüter, Andreas Hauptmann et al.

We present a tailored framework for solving regularized linear inverse problems using quantum optimization methods. By discretizing the solution space and encoding data fidelity and regularization terms into quadratic unconstrained binary optimization (QUBO) models, we formulate both Tikhonov- and sparsity-promoting regularized inverse problems within a unified quantum optimization framework. We further introduce a notion of quantum sensitivity that characterizes the effect of perturbations arising from approximate quantum evolution and discretization. We derive bounds relating these perturbations to stability properties of the underlying inverse problem, thereby establishing a theoretical connection between quantum solution variability and classical notions of ill-posedness. The framework is extended to variational inverse problems through wavelet-based representations and complemented by reduced-order modeling strategies in both parameter and Hamiltonian spaces to mitigate current hardware limitations. Numerical experiments on simulated and physical quantum hardware indicate that the resulting low-energy solution distributions retain information about the underlying inverse problem, while also revealing the limitations imposed by finite-time evolution, discretization, and hardware noise.

Babak Maboudi Afkham, Ana Carpio

This paper introduces an infinite-dimensional Bayesian framework for acoustic seabed tomography, leveraging wave scattering to simultaneously estimate the seabed and its roughness. Tomography is considered an ill-posed problem where multiple seabed configurations can result in similar measurement patterns. We propose a novel approach focusing on the statistical isotropy of the seabed. Utilizing fractional differentiability to identify seabed roughness, the paper presents a robust numerical algorithm to estimate the seabed and quantify uncertainties. Extensive numerical experiments validate the effectiveness of this method, offering a promising avenue for large-scale seabed exploration.

Hjørdis Schlüter, Babak Maboudi Afkham

Acousto-electric tomography (AET) is a hybrid imaging modality that combines electrical impedance tomography with focused ultrasound perturbations to obtain interior power density measurements, which provide additional information that can enhance the stability of conductivity reconstruction. In this work, we study the AET inverse problem within a Bayesian framework and compare statistical reconstruction with analytical approaches. The unknown conductivity is modeled as a random field, and inference is based on the posterior distribution conditioned on the measurements. We consider likelihood constructions based on both L1- and L2-type data misfit norms and establish Bayesian well-posedness for both formulations within the framework of Stuart (2010). Numerical experiments investigate the performance of the Bayesian method from noisy power density measurements using the L1 and L2 likelihood functions and a smooth prior and a piecewise-constant prior for different limited view configurations, including severely limited boundary access. In particular, we demonstrate that small inclusions near the accessible boundary can be reconstructed from AET data corresponding to a single EIT measurement, and we quantify reconstruction uncertainty through posterior statistics.