arXiv subjects

Bangti Jin

Publications and source records attributed to Bangti Jin.

At least 19 recordsLinked to original sources

arXiv2026

Error Analysis of the Inverse Conductivity Problem with Scattered Measurements

In this work, we investigate the inverse problem of recovering the conductivity coefficient in an elliptic equation from noisy measurements collected at finitely many deterministic scattered points in the domain $\Omega$, and corrupted by random noise. Inspired by the regularity analysis, we propose a numerical scheme based on the regularized least-squares formulation with a $W^{1,4}(\Omega)$ penalty, and discretize the regularized problem using the Galerkin finite element method with continuous piecewise linear elements. Under suitable assumptions on the problem data, we provide an error analysis of the regularized solution and its Galerkin approximation. We establish $L^2(\Omega)$ error bounds in a high-probability sense, which depend explicitly on the regularization parameter $\gamma$, the number $n$ of data points and the mesh size $h$. We also present numerical experiments to illustrate the theoretical findings.

math.NA↗

arXiv2026

On the extreme eigenvalues of the Gram Matrix in Physics-Informed Neural Networks for the Poisson Equation

The smallest and largest eigenvalues of the Gram matrix induced by the differential neural tangent kernel (DNTK) play a pivotal role in the analysis of over-parameterized PINNs trained by gradient type algorithms. However, a theoretical analysis of the extreme eigenvalues remains completely absent due to the challenge posed by the presence of multiple differential operators. In this work, we provide explicit lower and upper bounds for the extreme eigenvalues of the infinite DNTK matrix for the Poisson equation with the Dirichlet boundary condition for two-layer RePU neural networks without the bias term. The setting is fairly general with respect to the sampling points and input dimension $d$: $\delta$-separated and additionally $d\geq 3$ when deriving the lower bound of the smallest eigenvalue. These results extend that for the neural tangent kernel, and to the best of our knowledge, represent the first results on the spectrum of the DNTK.

math.NA↗

arXiv2026

Stability and Reconstruction of a Nonlinearity in a Parabolic Equation from Partial Boundary Data

In this work, we investigate the inverse problem of determining a semilinear term in a nonlinear parabolic equation from one single boundary flux measurement taken on an arbitrary subset of the boundary. More precisely, we address both uniqueness and stability issues of the inverse problem and establish new H\"older-type stability estimates. The H\"older exponent depends explicitly on the measurement configuration as well as on regularity properties of the semilinear term. The analysis relies on a novel approach based on the derivation of a suitable integral identity involving solutions of the associated adjoint equation. This allows reformulating the inverse problem as an inverse source problem with a sign-changing source term. The main results are obtained by combining fundamental properties of parabolic equations, including maximum principle and appropriate energy estimates. Finally, we complement the theoretical analysis with an iterative reconstruction algorithm inspired by inverse source problems, and illustrate its accuracy on several numerical experiments.

math.AP↗

arXiv2026

Stability of Electrical Impedance Tomography with Anisotropies and its Application to the Deep Calde\'on Method

In this work, we establish new conditional Lipschitz stability results for electrical impedance tomography (EIT) with anisotropies, of recovering the conductivity in a conformal class of a known anisotropic conductivity in both two- and multi-dimensional cases. Then we employ the stability theory to understand the property of the deep Calder\'on method, one deep learning-based technique for image reconstruction in EIT that has shown promising empirical results, but still lacks theoretical underpinnings. Specifically, we relate the stability theory to the robustness of the method with the proper choice of the training data, and present numerical results in two-dimension to complement the theoretical analysis.

math.AP↗

arXiv2026

The Differential Neural Tangent Kernel and Its Positivity

The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime. Recently, the theoretical framework has been extended to physics-informed neural networks (PINNs) for solving linear PDEs, one highly popular class of neural PDE solvers. In the analysis, the positivity of the associated NTK plays a fundamental role. However, establishing the positivity of the NTK for PINNs is highly challenging, due to the presence of multiple differential operators. In this work, we propose a new theoretical framework, called Differential Neural Tangent Kernel (DNTK), for analyzing PINNs through the lens of the NTK, and establish the positivity of the infinite width DNTK for both shallow and deep neural networks for a wide class of activation functions, including RePU and smooth but non-polynomial activations, for all linear differential operators. These theoretical results lay the foundation for the analysis of gradient type algorithms for training PINNs.

cs.LG↗

arXiv2026

Numerical Analysis of Space-Time Dependent Source Identification in Subdiffusion Equations

In this work, we propose an easy-to-implement fixed-point algorithm for reconstructing a space-time dependent source in a subdiffusion model from lateral boundary measurements. The numerical scheme combines a Galerkin finite element method for spatial discretization with a finite difference method for temporal discretization. We establish the linear convergence of the fixed-point iteration and derive an error bound that depends explicitly on the discretization parameters and the noise level. The error analysis relies on stability properties of the continuous inverse problem and technical estimates for the associated direct problem with limited-regularity data. Numerical experiments are presented to support and complement the theoretical analysis.

math.NA↗

arXiv2026

Stability Estimates for the Inverse Problem of Reconstructing Point sources in Parabolic Equations

In this work, we investigate the stability issue of the inverse problem of determining the locations and time-dependent amplitudes of point sources in a parabolic equation with a non-self adjoint elliptic operator from boundary observations. We derive different stability estimates for determining the locations and the amplitudes of the sources in the space, the plane as well as in dimension one. The analysis employs a novel approach that combines several different arguments, including the improved regularity of the solutions, the application of Carleman estimates, time extension of solutions, and construction of explicit solutions to the adjoint equations. Further we provide numerical reconstructions to complement the theoretical findings.

math.AP↗

arXiv2026

Identification of a Point Source in the Heat Equation from Sparse Boundary Measurements

In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and piecewise constant in time amplitude when the domain is the unit ball in $\mathbb{R}^d$ ($d\geq2$), and the unique recovery of the location and compactly supported amplitude when the domain is simply connected, smooth and bounded in $\mathbb{R}^2$, under mild conditions on the observational points. The proof combines distinct analytical tools, including the representation of the flux data via Laplacian eigenfunctions on the unit ball, a detailed analysis of the properties of the heat and Poisson kernels, as well as methods drawn from complex analysis. Further we present several numerical experiments to illustrate the feasibility of the recovery from sparse boundary data.

math.AP↗

arXiv2026

Unique Determination of Variable Order in Subdiffusion from a Single Measurement

We study the inverse problem of recovering a spatially dependent variable order in a time-fractional diffusion model from the boundary flux measurement generated by a single boundary excitation. It arises in the identification of heterogeneous media in anomalous diffusion processes. In this work, we establish several new uniqueness results for the inverse problem in the case of piecewise constant variable orders, without any monotonicity condition. The analysis follows a new approach that combines properties of harmonic functions, a linearization technique in the Laplace domain, and tools from complex, asymptotic, and geometrical analysis. In addition, we weaken the regularity assumptions on the problem data and extend the analysis of previous contributions to higher-dimensional settings.

math.AP↗

arXiv2026

On the contraction rate of the posterior distribution for nonlinear PDE parameter identification

In this work, we investigate the estimation of a parameter $f$ in PDEs using Bayesian procedures, and focus on posterior distributions constructed using Gaussian process priors, and its variational approximation. We establish contraction rates for the posterior distribution and the variational approximation in the regime of low-regularity parameters. The main novelty of the study lies in relaxing the condition that the ground truth parameter must lie in the reproducing kernel Hilbert space of the Gaussian process prior, which is commonly imposed in existing studies on posterior contraction rate analysis [14,40,44]. The analysis relies on a delicate approximation argument that suitably balances various error sources. We illustrate the general theory on three nonlinear inverse problems for PDEs.

math.ST↗

arXiv2026

Stochastic Gradient Descent for Nonlinear Inverse Problems in Banach Spaces

Stochastic gradient descent (SGD) and its variants are widely used and highly effective optimization methods in machine learning, especially for neural network training. By using a single datum or a small subset of the data, selected randomly at each iteration, SGD scales well to problem size and has been shown to be effective for solving large-scale inverse problems. In this work, we investigate SGD for solving nonlinear inverse problems in Banach spaces through the lens of iterative regularization. Under general assumptions, we prove almost sure convergence of the iterates to the minimum distance solution and show the regularizing property in expectation under an a priori stopping rule. Further, we establish convergence rates under the conditional stability assumptions for both exact and noisy data. Numerical experiments on Schlieren tomography and electrical impedance tomography are presented to show distinct features of the method.

math.NA↗

arXiv2025

Shallow neural network yields regularization for ill-posed inverse problems

In this paper, we develop a regularization theory for neural network approximations of general ill-posed operator equations with noisy data. Within the framework of iterative regularization, we introduce two expanding neural network methods (ENNs) under different a priori assumptions on the exact solution. Instead of prescribing a fixed architecture, ENNs adaptively select the number of neurons through an a posteriori stopping rule, so that the selected network size serves as a regularization parameter balancing approximation accuracy and stability with respect to data noise. We prove the regularization properties of the proposed ENNs and establish quantitative relationships between the selected network size and the noise level. Within the framework of variational regularization, we propose a neural network-based Tikhonov scheme and derive both convergence and convergence-rate results under mild assumptions. The resulting estimates account for the noise level, the network size, and the underlying smoothness expressed through general variational source conditions, thereby allowing greater flexibility than existing results. Numerical experiments demonstrate the effectiveness and robustness of the proposed algorithms. In particular, they show that, for highly noisy data, relatively small network architectures can already produce stable reconstructions, whereas excessively large architectures may degrade stability due to overfitting.

math.NA↗

arXiv2025

A Stable Iterative Direct Sampling Method for Elliptic Inverse Problems with Partial Cauchy Data

We develop a novel iterative direct sampling method (IDSM) for solving linear or nonlinear elliptic inverse problems with partial Cauchy data. It integrates three innovations: a data completion scheme to reconstruct missing boundary information, a heterogeneously regularized Dirichlet-to-Neumann map to enhance the near-orthogonality of probing functions, and a stabilization-correction strategy to ensure the numerical stability. The resulting method is remarkably robust with respect to measurement noise, is flexible with the measurement configuration, enjoys provable stability guarantee, and achieves enhanced resolution for recovering inhomogeneities. Numerical experiments in electrical impedance tomography, diffuse optical tomography, and cardiac electrophysiology show its effectiveness in accurately reconstructing the locations and geometries of inhomogeneities.

math.NA↗

arXiv2025

An Iterative Direct Sampling Method for Reconstructing Moving Inhomogeneities in Parabolic Problems

We propose in this work a novel iterative direct sampling method for imaging moving inhomogeneities in parabolic problems using boundary measurements. It can efficiently identify the locations and shapes of moving inhomogeneities when very limited data are available, even with only one pair of lateral Cauchy data, and enjoys remarkable numerical stability for noisy data and over an extended time horizon. The method is formulated in an abstract framework, and is applicable to linear and nonlinear parabolic problems, including linear, nonlinear, and mixed-type inhomogeneities. Numerical experiments across diverse scenarios show its effectiveness and robustness against the data noise.

math.NA↗

arXiv2025

On the convergence of stochastic variance reduced gradient for linear inverse problems

Stochastic variance reduced gradient (SVRG) is an accelerated version of stochastic gradient descent based on variance reduction, and is promising for solving large-scale inverse problems. In this work, we analyze SVRG and a regularized version that incorporates a priori knowledge of the problem, for solving linear inverse problems in Hilbert spaces. We prove that, with suitable constant step size schedules and regularity conditions, the regularized SVRG can achieve optimal convergence rates in terms of the noise level without any early stopping rules, provided that the truncation level is chosen suitably, and standard SVRG is also optimal for problems with nonsmooth solutions under a priori stopping rules. The analysis is based on an explicit error recursion and suitable a priori estimates on the inner loop updates with respect to the anchor point. Numerical experiments are provided to complement the theoretical analysis.

math.NA↗

arXiv2025

A Warm-basis Method for Bridging Learning and Iteration: a Case Study in Fluorescence Molecular Tomography

Fluorescence Molecular Tomography (FMT) is a widely used non-invasive optical imaging technology in biomedical research. It usually faces significant accuracy challenges in depth reconstruction, and conventional iterative methods struggle with poor $z$-resolution even with advanced regularization. Supervised learning approaches can improve recovery accuracy but rely on large, high-quality paired training dataset that is often impractical to acquire in practice. This naturally raises the question of how learning-based approaches can be effectively combined with iterative schemes to yield more accurate and stable algorithms. In this work, we present a novel warm-basis iterative projection method (WB-IPM) and establish its theoretical underpinnings. The method is able to achieve significantly more accurate reconstructions than the learning-based and iterative-based methods. In addition, it allows a weaker loss function depending solely on the directional component of the difference between ground truth and neural network output, thereby substantially reducing the training effort. These features are justified by our error analysis as well as simulated and real-data experiments.

math.NA↗

arXiv2025

On the uniqueness of the discrete Calderon problem on multi-dimensional lattices

In this work, we investigate the discrete Calder\'{o}n problem on grid graphs of dimension three or higher, formed by hypercubic structures. The discrete Calder\'{o}n problem is concerned with determining whether the discrete Dirichlet-to-Neumann (DtN) operator, which links boundary potentials to boundary current responses, can uniquely identify the conductivity values on the graph edges. We provide an affirmative answer to the question, thereby extending the classical uniqueness result of Curtis and Morrow for two-dimensional square lattices. The proof employs a novel slicing technique that decomposes the problem into lower-dimensional components. Additionally, we support the theoretical finding with numerical experiments that illustrate the effectiveness of the approach.

math-ph↗

arXiv2025

Solving Inverse Acoustic Obstacle Scattering Problem with Phaseless Far-Field Measurement Using Deep Neural Network Surrogates

In this work, we investigate the use of deep neural networks (DNNs) as surrogates for solving the inverse acoustic scattering problem of recovering a sound-soft obstacle from phaseless far-field measurements. We approximate the forward maps from the obstacle to the far-field data using DNNs, and for star-shaped domains in two and three dimensions, we establish the expression rates for fully connected feedforward neural networks with the ReLU activation for approximating the forward maps. The analysis is based on the weak formulation of the direct problem, and can handle variable coefficients. Numerically we validate the accuracy of the DNN surrogates of the forward maps, and demonstrate the use of DNN surrogates in the Bayesian treatment of the inverse obstacle scattering problem. Numerical experiments indicate that the surrogates are effective in both two- and three-dimensional cases, and can significantly speed up the exploration of the posterior distribution of the shape parameters using Markov chain Monte Carlo.

math.NA↗