arXiv subjects

Stuart C. Hawkins

Publications and source records attributed to Stuart C. Hawkins.

4 recordsLinked to original sources

arXiv2026

A surrogate-Bayesian algorithm for scatterer shape identification from phaseless data

This work addresses the reconstruction of a scatterer's shape from phaseless far field-intensity data arising from multiple incident waves interacting with the scatterer. We formulate the reconstruction as a statistical inverse scattering problem and adopt a Bayesian inference framework, which can readily be used to compute statistical moments for quantification of uncertainties in the shape reconstruction that arise from noise in the data due to measurement constraints. The shape of the scatterer is represented by a spline-based prior, with Bayesian parameters defined at the spline's knots. To efficiently evaluate the Bayesian likelihood across thousands of sampling points, we develop the intensity property inspired neural network (IPINN) surrogate. This surrogate incorporates the Helmholtz equation in the unbounded domain, exterior to each sampled scatterer, along with the radiation condition at infinity, enabling fast and accurate simulation of the acoustic far-field intensity. Importantly, the IPINN surrogate is trained independently of the observed data and requires only a single incident wave for training. We demonstrate that this surrogate approach yields a speed-up of several orders of magnitude. The resulting IPINN-Bayesian framework offers an efficient solution for shape reconstruction in unbounded domains with multiple incident wave boundary conditions, while exactly enforcing the radiation condition. Numerical experiments confirm the efficiency and effectiveness of the proposed algorithm.

math.NA↗

arXiv2025

Far-Field Sensitivity to Local Boundary Perturbations in 2D Wave Scattering

We numerically investigate the sensitivity of the scattered wave field to perturbations in the shape of a scattering body illuminated by an incident plane wave. This study is motivated by recent work on the inverse problem of reconstructing a scatterer shape from measurements of the scattered wave at large distances from the scatterer. For this purpose we consider star-shaped scatterers represented using cubic splines, and our approach is based on a Nystr\"om method-based discretisation of the shape derivative. Using the singular value decomposition, we identify fundamental geometric modes that most strongly influence the scattered wave, providing insight into the most visible boundary features in scattering data.

math.NA↗

arXiv2025

Machine learning on manifolds for inverse scattering: Lipschitz stability analysis

Establishing Lipschitz stability estimates is crucial for ensuring the mathematical robustness of neural network (NN) approximations in machine learning (ML)-based parameter estimation, particularly in physics-informed settings. In this work, we derive such estimates for the inverse of a nonlinear map defined on a manifold that captures both unknown parameters and the nonlinear physical processes they influence. Our analysis is based on finite-dimensional, learnable representations of the manifold and provides Lipschitz stability estimates on the manifold-based subspaces, for a class of inverse maps associated with parameter dependent linear compact operators. Such operators model scattered and far-field data that can be used to detect structures such as cracks. We apply our theoretical ML manifold framework to inverse Helmholtz problems in unbounded regions exterior to cracks, addressing the scattered-field data-driven inverse problem while ensuring injectivity conditions on the manifold, a requirement for the Lipschitz stability. Our method accurately recovers crack-defining parameters without requiring prior knowledge of inputs such as incident wave types or external forces on the crack. Numerical experiments using NN approximations confirm the accuracy, efficiency, and robustness of the proposed approach.

math.NA↗

arXiv2024

An all-frequency stable integral system for Maxwell's equations in 3-D penetrable media: continuous and discrete model analysis

We introduce a new system of surface integral equations for Maxwell's transmission problem in three dimensions. This system has two remarkable features, both of which we prove. First, it is well-posed at all frequencies. Second, the underlying linear operator has a uniformly bounded inverse as the frequency approaches zero, ensuring that there is no low-frequency breakdown. The system is derived from a formulation we introduced in our previous work, which required additional integral constraints to ensure well -posedness across all frequencies. In this study, we eliminate those constraints and demonstrate that our new self adjoint, constraints-free linear system expressed in the desirable form of an identity plus a compact weakly-singular operator is stable for all frequencies. Furthermore, we propose and analyze a fully discrete numerical method for these systems and provide a proof of spectrally accurate convergence for the computational method. We also computationally demonstrate the high-order accuracy of the algorithm using benchmark scatterers with curved surfaces.

math.NA↗