arXiv subjects

Mahadevan Ganesh

Publications and source records attributed to Mahadevan Ganesh.

5 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

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↗

arXiv2020

An efficient epistemic uncertainty quantification algorithm for a class of stochastic models: A post-processing and domain decomposition framework

Partial differential equations (PDEs) are fundamental for theoretically describing numerous physical processes that are based on some input fields in spatial configurations. Understanding the physical process, in general, requires computational modeling of the PDE. Uncertainty in the computational model manifests through lack of precise knowledge of the input field or configuration. Uncertainty quantification (UQ) in the output physical process is typically carried out by modeling the uncertainty using a random field, governed by an appropriate covariance function. This leads to solving high-dimensional stochastic counterparts of the PDE computational models. Such UQ-PDE models require a large number of simulations of the PDE in conjunction with samples in the high-dimensional probability space, with probability distribution associated with the covariance function. Those UQ computational models having explicit knowledge of the covariance function are known as aleatoric UQ (AUQ) models. The lack of such explicit knowledge leads to epistemic UQ (EUQ) models, which typically require solution of a large number of AUQ models. In this article, using a surrogate, post-processing, and domain decomposition framework with coarse stochastic solution adaptation, we develop an offline/online algorithm for efficiently simulating a class of EUQ-PDE models.

math.NA↗

arXiv2019

An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation

A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium wave propagation model, and avoid an approximation of the RC. In this work we first present and analyze an overlapping decomposition framework that is equivalent to the full-space heterogeneous-homogenous continuous model, governed by the Helmholtz equation with a spatially dependent refractive index and the RC. Our novel overlapping framework allows the user to choose two free boundaries, and gain the advantage of applying established high-order finite and boundary element methods (FEM and BEM) to simulate an equivalent coupled model. The coupled model comprises auxiliary interior bounded heterogeneous and exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be chosen for simulating the exterior problem using a spectrally accurate BEM, and a simple boundary can be used to setup a high-order FEM for the interior problem. Thanks to the spectral accuracy of the exterior computational model, the resulting coupled system in the overlapping region is relatively very small. Using the decomposed equivalent framework, we develop a novel overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space model incorporates the RC exactly. Numerical experiments demonstrate the efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave fields, with the latter induced by a complex heterogeneous medium and a discontinuous refractive index.

math.NA↗