arXiv subjects

Qimeng Quan

Publications and source records attributed to Qimeng Quan.

11 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.

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arXiv2025

Numerical Analysis of Unsupervised Learning Approaches for Parameter Identification in PDEs

Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve PDE parameter identifications. These approaches employ neural networks as ansatz functions to approximate the parameters and / or the states, and have demonstrated impressive empirical performance. In this paper, we provide a comprehensive survey on these unsupervised learning techniques on one model problem, diffusion coefficient identification, from the classical numerical analysis perspective, and outline a general framework for deriving rigorous error bounds on the discrete approximations obtained using the Galerkin finite element method, hybrid method and deep neural networks. Throughout we highlight the crucial role of conditional stability estimates in the error analysis.

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arXiv2025

Conditional Stability and Numerical Reconstruction of a Parabolic Inverse Source Problem Using Carleman Estimates

In this work we develop a new numerical approach for recovering a spatially dependent source component in a standard parabolic equation from partial interior measurements. We establish novel conditional Lipschitz stability and H\"{o}lder stability for the inverse problem with and without boundary conditions, respectively, using suitable Carleman estimates. Then we propose a numerical approach for solving the inverse problem using conforming finite element approximations in both time and space. Moreover, by utilizing the conditional stability estimates, we prove rigorous error bounds on the discrete approximation. We present several numerical experiments to illustrate the effectiveness of the approach.

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arXiv2024

Stochastic Convergence Analysis of Inverse Potential Problem

In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a least squares formulation with an $H^1(\Omega)$ penalty on the potential in order to obtain a numerical reconstruction, and the Galerkin finite element method for the spatial discretization. Under mild regularity assumptions on the problem data, we provide a stochastic $L^2(\Omega)$ convergence analysis on the regularized solution and the finite element approximation in a high probability sense. The obtained error bounds depend explicitly on the regularization parameter $\gamma$, the number $n$ of observation points and the mesh size $h$. These estimates provide a useful guideline for choosing relevant algorithmic parameters. Furthermore, we develop a monotonically convergent adaptive algorithm for determining a suitable regularization parameter in the absence of \textit{a priori} knowledge. Numerical experiments are also provided to complement the theoretical results.

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arXiv2024

Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion

In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order $\alpha \in (0,1)$ in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several pointwise-in-time regularity estimates that are useful for numerical analysis. Then we develop a high-order time stepping scheme for solving quasilinear subdiffusion, based on convolution quadrature generated by second-order backward differentiation formula with correction at the first step. Further, we establish that the convergence order of the scheme is $O(\tau^{1+\alpha-\epsilon})$ without imposing any additional assumption on the regularity of the solution. The analysis relies on refined Sobolev regularity of the nonlinear perturbation remainder and smoothing properties of discrete solution operators. Several numerical experiments in two space dimensions show the sharpness of the error estimate.

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arXiv2024

Numerical Recovery of the Diffusion Coefficient in Diffusion Equations from Terminal Measurement

In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting decay behavior of the observation in time, we establish a novel H{\"o}lder type stability estimate for a large terminal time $T$. This is achieved by novel decay estimates of the (fractional) time derivative of the solution. To numerically recover the diffusion coefficient, we employ the standard output least-squares formulation with an $H^1(\Omega)$-seminorm penalty, and discretize the regularized problem by the Galerkin finite element method with continuous piecewise linear finite elements in space and backward Euler convolution quadrature in time. Further, we provide an error analysis of discrete approximations, and prove a convergence rate that matches the stability estimate. The derived $L^2(\Omega)$ error bound depends explicitly on the noise level, regularization parameter and discretization parameter(s), which gives a useful guideline of the \textsl{a priori} choice of discretization parameters with respect to the noise level in practical implementation. The error analysis is achieved using the conditional stability argument and discrete maximum-norm resolvent estimates. Several numerical experiments are also given to illustrate and complement the theoretical analysis.

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arXiv2023

Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks

In this work we develop a novel approach using deep neural networks to reconstruct the conductivity distribution in elliptic problems from one measurement of the solution over the whole domain. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective, with deep neural networks as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the deep neural network approximations of the conductivity for both continuous and empirical losses, including rigorous error estimates that are explicit in terms of the noise level, various penalty parameters and neural network architectural parameters (depth, width and parameter bound). We also provide multiple numerical experiments in two- and multi-dimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.

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arXiv2023

Hybrid Neural-Network FEM Approximation of Diffusion Coefficient in Elliptic and Parabolic Problems

In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks. The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and neural networks (NNs) act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias / approximation properties of NNs. We derive \textsl{a priori} error estimates in the standard $L^2(Ω)$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size, and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.

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arXiv2022

Convergence Rate Analysis of Galerkin Approximation of Inverse Potential Problem

In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an $H^1$-seminorm penalty), which is then discretized by the Galerkin finite element method with continuous piecewise linear finite elements in space (and also backward Euler method in time for parabolic problems). We present a detailed analysis of the discrete scheme, and provide convergence rates in a weighted $L^2(Ω)$ for discrete approximations with respect to the exact potential. The error bounds are explicitly dependent on the noise level, regularization parameter and discretization parameter(s). Under suitable conditions, we also derive error estimates in the standard $L^2(Ω)$ and interior $L^2$ norms. The analysis employs sharp a priori error estimates and nonstandard test functions. Several numerical experiments are given to complement the theoretical analysis.

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arXiv2021

Analysis of Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Conditions

Deep Ritz methods (DRM) have been proven numerically to be efficient in solving partial differential equations. In this paper, we present a convergence rate in $H^{1}$ norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition, where the error depends on the depth and width in the deep neural networks and the number of samples explicitly. Further we can properly choose the depth and width in the deep neural networks in terms of the number of training samples. The main idea of the proof is to decompose the total error of DRM into three parts, that is approximation error, statistical error and the error caused by the boundary penalty. We bound the approximation error in $H^{1}$ norm with $\mathrm{ReLU}^{2}$ networks and control the statistical error via Rademacher complexity. In particular, we derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with $\mathrm{ReLU}^{2}$ network, which is of immense independent interest. We also analysis the error inducing by the boundary penalty method and give a prior rule for tuning the penalty parameter.

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arXiv2020

Lowest-degree piecewise polynomial de Rham complex on general quadrilateral grids

This paper is devoted to the construction of finite elements on grids that consist of general quadrilaterals not limited in parallelograms. Two finite elements defined as Ciarlet's triple are established for the $H^1$ and $H(\rm rot)$ elliptic problems, respectively. An $\mathcal{O}(h)$ order convergence rate in energy norm for both of them and an $\mathcal{O}(h^2)$ order convergence in $L^2$ norm for the $H^1$ scheme are proved under the asymptotic-parallelogram assumption on the grids. Further, the two finite element spaces on general quadrilateral grids, together with the space of piecewise constant functions, formulate a discretized de Rham complex. The finite element spaces consist of piecewise polynomial functions, and, thus, are nonconforming on general quadrilateral grids. Indeed, a rigorous analysis is given in this paper that it is impossible to construct a practically useful finite element defined as Ciarlet's triple that can formulate a finite element space which consists of continuous piecewise polynomial functions on a grid that may include arbitrary quadrilaterals.

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