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arxiv: 2606.30340 · v1 · pith:G2CN5ERHnew · submitted 2026-06-29 · 🧮 math.NA · cs.NA· math.AP

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

Hassan Yazdanian , Leila Taghizadeh , Babak Maboudi Afkham This is my paper

Pith reviewed 2026-06-30 04:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords Bayesian inverse problemsPDE-constrained inferencesemiconductor doping reconstructionpushforward prioradjoint methodNo-U-Turn Sampleruncertainty quantificationpiecewise-constant fields
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The pith

A pushforward prior from a latent Matérn Gaussian field through a sigmoid, paired with adjoint NUTS sampling, reconstructs piecewise-constant doping interfaces in semiconductors while quantifying uncertainty.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates a Bayesian inverse problem for recovering doping profiles in pn-junction devices from boundary flux data, where the unknown is a piecewise-constant field with an unknown interface and two plateau values. It builds a pushforward prior by transforming a truncated Karhunen-Loève expansion of a Matérn Gaussian field via a sigmoid to produce differentiable approximations of sharp interfaces. Well-posedness is established by proving the forward Poisson-Boltzmann map is Lipschitz continuous and the posterior is stable in the Hellinger metric. Posterior sampling is performed with the No-U-Turn Sampler using gradients from the adjoint method, and experiments demonstrate one to two orders of magnitude improvement in effective sample size over preconditioned Crank-Nicolson. Reconstructions are shown for both known and jointly inferred plateaus, with the latter exposing posterior correlations that indicate non-identifiability.

Core claim

The central claim is that mapping a latent Gaussian field with Matérn covariance through a sigmoid produces a usable pushforward prior for piecewise-constant doping fields, and that this prior combined with adjoint-computed gradients enables the No-U-Turn Sampler to explore the posterior of the nonlinear PDE-constrained inverse problem more efficiently than dimension-robust alternatives while satisfying the well-posedness conditions of forward-map Lipschitz continuity and Hellinger posterior stability.

What carries the argument

The pushforward prior that applies a sigmoid transformation to a truncated Karhunen-Loève expansion of a Matérn Gaussian random field to generate differentiable approximations of piecewise-constant doping profiles with controllable interface sharpness.

If this is right

  • In the known-plateau setting the method recovers both planar and curved interfaces together with spatially resolved uncertainty maps.
  • When interface geometry and plateau concentrations are inferred jointly, posterior correlations reveal structural non-identifiability.
  • The combination of the proposed prior and NUTS produces effective sample sizes one to two orders of magnitude larger than those obtained with the preconditioned Crank-Nicolson sampler.
  • The Bayesian formulation satisfies well-posedness through Lipschitz continuity of the forward map and Hellinger stability of the posterior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The observed non-identifiability when plateaus are unknown indicates that supplementary measurements or explicit constraints on concentration values would be needed for unique recovery.
  • Because the adjoint supplies gradients at essentially the cost of one forward solve, the sampling approach remains computationally feasible when the number of KL modes or the spatial resolution is increased.
  • The same sigmoid-pushforward construction could be transferred to other PDE inverse problems that require reconstruction of sharp material interfaces while preserving differentiability for gradient-based inference.

Load-bearing premise

The unknown doping field is assumed to be exactly piecewise constant with a single interface separating two constant plateau concentrations.

What would settle it

A numerical experiment on synthetic boundary flux data generated from a known piecewise-constant doping profile in which the NUTS sampler fails to produce effective sample sizes at least ten times larger than those from the pCN sampler, or in which small perturbations of the data yield posteriors whose Hellinger distance exceeds the stability bound proved in the paper.

Figures

Figures reproduced from arXiv: 2606.30340 by Babak Maboudi Afkham, Hassan Yazdanian, Leila Taghizadeh.

Figure 1
Figure 1. Figure 1: A two-dimensional pn-junction with a piecewise-constant doping function [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dependency chart for finite-dimensional parameters in the semiconductor inverse [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representative realizations from the pushforward Whittle–Mat´ern prior for the doping [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reconstruction results for the known-plateau setting with a straight interface (Ex [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trace plots of five representative KL coefficients for the reconstruction problem with [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstruction results for the known-plateau setting with a curved interface (Ex [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Joint reconstruction results. (a) Posterior mean and (b) posterior standard deviation [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Trace plots for representative KL coefficients in the joint reconstruction experiment. [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
read the original abstract

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 Mat\'ern-type covariance through a sigmoid transformation. The latent field is parameterized by a truncated Karhunen-Lo\`eve 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper formulates a Bayesian inverse problem for reconstructing piecewise-constant doping profiles (unknown interface plus two plateau values) in pn-junction devices from boundary flux data governed by a nonlinear Poisson-Boltzmann equation. A pushforward prior is constructed by applying a sigmoid transformation to a latent Matérn Gaussian random field parameterized via truncated KL expansion, yielding differentiable approximations to sharp interfaces. Well-posedness is established by proving Lipschitz continuity of the forward map and Hellinger stability of the posterior; posterior sampling is performed with adjoint-enabled NUTS, which is reported to produce 1–2 orders of magnitude higher effective sample sizes than pCN. Numerical results include interface reconstruction, spatially resolved UQ, and detection of structural non-identifiability when plateaus and geometry are inferred jointly.

Significance. If the Lipschitz and Hellinger arguments hold and the reported ESS gains are reproducible, the work supplies a concrete, gradient-enabled workflow for Bayesian UQ on nonlinear PDE-constrained problems that possess sharp-interface structure. The explicit reporting of non-identifiability and the use of a structurally informed pushforward prior are positive contributions that can be reused in related imaging or materials inverse problems.

minor comments (3)
  1. The abstract and introduction should state the specific KL truncation rank and Matérn parameters employed in the numerical examples so that the efficiency comparison can be reproduced without consulting the full text.
  2. Notation for the plateau scalars and the sigmoid sharpness parameter should be introduced once and used consistently; occasional reuse of the same symbol for different quantities appears in the provided abstract.
  3. A short paragraph comparing the computational cost of the adjoint gradient evaluation versus finite-difference checks would strengthen the claim that the method is practical for higher-dimensional KL expansions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the pushforward prior construction, adjoint-enabled NUTS sampling, and the explicit treatment of non-identifiability. The recommendation for minor revision is appreciated. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central claims rest on explicit modeling choices (piecewise-constant doping via pushforward of Matérn field through sigmoid) and direct mathematical proofs (Lipschitz continuity of the forward map from doping to boundary flux, Hellinger stability of the posterior) that invoke standard PDE estimates for the nonlinear Poisson-Boltzmann equation under bounded coefficients. Efficiency comparisons are obtained from numerical experiments with NUTS versus pCN, not by construction from fitted parameters. The truncated KL parameterization and plateau scalars are stated modeling decisions whose consequences (including reported non-identifiability) are exhibited rather than assumed away. No load-bearing step reduces by the paper's own equations or self-citation to a tautological input; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 1 invented entities

The central claims rest on the piecewise-constant doping model, the differentiability properties of the sigmoid pushforward, the Lipschitz and Hellinger stability proofs, and the numerical comparison of samplers; these elements are introduced or asserted within the paper rather than taken from independent external benchmarks.

free parameters (3)
  • plateau concentrations
    Two scalar parameters representing the constant doping levels in each region of the piecewise-constant model.
  • KL truncation level
    Number of terms retained in the truncated Karhunen-Loève expansion of the latent Gaussian field.
  • Matérn covariance parameters
    Parameters controlling correlation length and smoothness of the latent field before the sigmoid transformation.
axioms (2)
  • domain assumption The forward map defined by the Poisson-Boltzmann-type equation is Lipschitz continuous with respect to the doping field.
    Invoked to establish well-posedness of the Bayesian formulation.
  • domain assumption The posterior measure is stable in the Hellinger metric under perturbations of the data.
    Proven as part of the well-posedness result.
invented entities (1)
  • pushforward prior via sigmoid transformation of latent Matérn field no independent evidence
    purpose: To produce differentiable approximations to piecewise-constant fields with controllable interface sharpness for use inside gradient-based sampling.
    New construction introduced to represent the structured doping field while remaining compatible with NUTS.

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