Statistical Topological Gradient and Shape Optimization for Robust Metal--Semiconductor Contact Reconstruction

TL;DR

This paper introduces a statistically robust framework combining topological gradients and shape optimization to accurately reconstruct metal--semiconductor contact regions from boundary measurements, accounting for noise and uncertainty.

Contribution

It develops a novel statistical approach with a central limit theorem for topological gradients, integrating shape optimization and a sensitivity parameter to improve contact reconstruction accuracy.

Findings

Proves stability of topological gradient under measurement noise.

Establishes a central limit theorem for empirical topological gradients.

Demonstrates improved reconstruction accuracy through numerical experiments.

Abstract

We develop a statistically robust framework for reconstructing metal--semiconductor contact regions using topological gradients. The inverse problem is formulated as the identification of an unknown contact region from boundary measurements governed by an elliptic model with piecewise coefficients. Deterministic stability of the topological gradient with respect to measurement noise is established, and the analysis is extended to a statistical setting with multiple independent observations. A central limit theorem in a separable Hilbert space is proved for the empirical topological gradient, yielding optimal $n^{-1/2}$ convergence and enabling the construction of confidence intervals and hypothesis tests for contact detection. To further refine the reconstruction, a shape optimization procedure is employed, where the free parameter $\beta$ in the CCBM formulation plays a crucial role in controlling interface sensitivity. While $\beta$ affects both topological and shape reconstructions, its influence is particularly pronounced in the shape optimization stage, allowing more accurate estimation of the size and geometry of the contact subregion. The proposed approach provides a rigorous criterion for distinguishing true structural features from noise-induced artifacts, and numerical experiments demonstrate the robustness, precision, and enhanced performance of the combined statistical, topological, and $\beta$-informed shape-based reconstruction.