arXiv subjects

Bangti Jin

Publications and source records attributed to Bangti Jin.

At least 55 records · Page 3Linked to original sources

arXiv2025

Optimizing Coarse Propagators in Parareal Algorithms

The parareal algorithm represents an important class of parallel-in-time algorithms for solving evolution equations and has been widely applied in practice. To achieve effective speedup, the choice of the coarse propagator in the algorithm is vital. In this work, we investigate the use of {optimized} coarse propagators. Building upon the error estimation framework, we present a systematic procedure for constructing coarse propagators that enjoy desirable stability and consistent order. Additionally, we provide preliminary mathematical guarantees for the resulting parareal algorithm. Numerical experiments on a variety of settings, e.g., linear diffusion model, Allen-Cahn model, and viscous Burgers model, show that the optimizing procedure can significantly improve parallel efficiency when compared with the more ad hoc choice of some conventional and widely used coarse propagators.

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arXiv2025

An Iterative Deep Ritz Method for Monotone Elliptic Problems

In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.

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arXiv2024

The discrete inverse conductivity problem solved by the weights of an interpretable neural network

In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data.

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arXiv2024

Point Source Identification in Subdiffusion from A Posteriori Internal Measurement

In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time and a general second-order elliptic operator in space. We establish the well-posedness of the direct problem in the sense of transposition and improved local regularity. Using classical unique continuation of the subdiffusion model and improved local solution regularity, we prove the uniqueness of simultaneously recovering the locations of point sources, time-dependent strengths and initial condition for both one- and multi-dimensional cases. Moreover, in the one-dimensional case, the elliptic operator can have time-dependent coefficients. These results extend existing studies on point source identification for parabolic type problems. Additionally we present several numerical experiments to show the feasibility of numerical reconstruction.

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arXiv2024

Theory and Fast Learned Solver for $\ell^1$-TV Regularization

The $\ell^1$ and total variation (TV) penalties have been used successfully in many areas, and the combination of the $\ell^1$ and TV penalties can lead to further improved performance. In this work, we investigate the mathematical theory and numerical algorithms for the $\ell^1$-TV model in the context of signal recovery: we derive the sample complexity of the $\ell^1$-TV model for recovering signals with sparsity and gradient sparsity. Also we propose a novel algorithm (PGM-ISTA) for the regularized $\ell^1$-TV problem, and establish its global convergence and parameter selection criteria. Furthermore, we construct a fast learned solver (LPGM-ISTA) by unrolling PGM-ISTA. The results for the experiment on ECG signals show the superior performance of LPGM-ISTA in terms of recovery accuracy and computational efficiency.

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arXiv2024

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

In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two- and three-dimensional cases, up to 10\% noise in the data.

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arXiv2024

Identification of a Spatially-Dependent Variable Order in One-Dimensional Subdiffusion

In this work we investigate an inverse problem of identifying a spatially variable order in the one-dimensional subdiffusion model from the boundary flux measurement. The model involves a generalized Caputo derivative in time, and arises in the mathematical modeling of anomalous diffusion in heterogeneous media. We prove the unique recovery of a monotone piecewise constant variable order and its range for known and unknown media, respectively. The analysis is based on a delicate asymptotic expansion of the Laplace transform of the data as $p\to0$, which is of independent interest.

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arXiv2024

On an Inverse Problem of the Generalized Bathtub Model of Network Trip Flows

In this work, we investigate the generalized bathtub model, a nonlocal transport equation for describing network trip flows served by privately operated vehicles inside a road network. First, we establish the well-posedness of the mathematical model for both classical and weak solutions. Then we consider an inverse source problem of the model with model parameters embodying particular traffic situations. We establish a conditional Lipschitz stability of the inverse problem under suitable a priori regularity assumption on the problem data, using a Volterra integral formulation of the problem. Inspired by the analysis, we develop an easy-to-implement numerical method for reconstructing the flow rates, and provide the error analysis of the method. Further we present several numerical experiments to complement the theoretical analysis.

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arXiv2024

Point Source Identification Using Singularity Enriched Neural Networks

The inverse problem of recovering point sources represents an important class of applied inverse problems. However, there is still a lack of neural network-based methods for point source identification, mainly due to the inherent solution singularity. In this work, we develop a novel algorithm to identify point sources, utilizing a neural network combined with a singularity enrichment technique. We employ the fundamental solution and neural networks to represent the singular and regular parts, respectively, and then minimize an empirical loss involving the intensities and locations of the unknown point sources, as well as the parameters of the neural network. Moreover, by combining the conditional stability argument of the inverse problem with the generalization error of the empirical loss, we conduct a rigorous error analysis of the algorithm. We demonstrate the effectiveness of the method with several challenging experiments.

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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 $α\in (0,1)$ 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 $O(τ^{1+α-ε})$ 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

Early Stopping of Untrained Convolutional Neural Networks

In recent years, new regularization methods based on (deep) neural networks have shown very promising empirical performance for the numerical solution of ill-posed problems, e.g., in medical imaging and imaging science. Due to the nonlinearity of neural networks, these methods often lack satisfactory theoretical justification. In this work, we rigorously discuss the convergence of a successful unsupervised approach that utilizes untrained convolutional neural networks to represent solutions to linear ill-posed problems. Untrained neural networks are particularly appealing for many applications because they do not require paired training data. The regularization property of the approach relies solely on the architecture of the neural network instead. Due to the vast over-parameterization of the employed neural network, suitable early stopping is essential for the success of the method. We establish that the classical discrepancy principle is an adequate method for early stopping of two-layer untrained convolutional neural networks learned by gradient descent, and furthermore, it yields an approximation with minimax optimal convergence rates. Numerical results are also presented to illustrate the theoretical findings.

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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{ö}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 $H^1(Ω)$-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 $L^2(Ω)$ 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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arXiv2024

Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System

In this work, we investigate a neural network based solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order optimality system of the optimal control problem, and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme, and provide $L^2(Ω)$ error bounds on the state, control and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.

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arXiv2024

Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

Physics-Informed Neural Networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN (SEPINN), by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three-dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches.

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arXiv2024

Solving Inverse Obstacle Scattering Problem with Latent Surface Representations

We propose a novel iterative numerical method to solve the three-dimensional inverse obstacle scattering problem of recovering the shape of the obstacle from far-field measurements. To address the inherent ill-posed nature of the inverse problem, we advocate the use of a trained latent representation of surfaces as the generative prior. This prior enjoys excellent expressivity within the given class of shapes, and meanwhile, the latent dimensionality is low, which greatly facilitates the computation. Thus, the admissible manifold of surfaces is realistic and the resulting optimization problem is less ill-posed. We employ the shape derivative to evolve the latent surface representation, by minimizing the loss, and we provide a local convergence analysis of a gradient descent type algorithm to a stationary point of the loss. We present several numerical examples, including also backscattered and phaseless data, to showcase the effectiveness of the proposed algorithm.

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arXiv2024

An Adaptive Phase-Field Method for Structural Topology Optimization

In this work, we develop an adaptive algorithm for the efficient numerical solution of the minimum compliance problem in topology optimization. The algorithm employs the phase field approximation and continuous density field. The adaptive procedure is driven by two residual type a posteriori error estimators, one for the state variable and the other for the first-order optimality condition of the objective functional. The adaptive algorithm is provably convergent in the sense that the sequence of numerical approximations generated by the adaptive algorithm contains a subsequence convergent to a solution of the continuous first-order optimality system. We provide several numerical simulations to show the distinct features of the algorithm.

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arXiv2024

Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time

In this work we investigate an inverse coefficient problem for the one-dimensional subdiffusion model, which involves a Caputo fractional derivative in time. The inverse problem is to determine two coefficients and multiple parameters (the order, and length of the interval) from one pair of lateral Cauchy data. The lateral Cauchy data are given on disjoint sets in time with a single excitation and the measurement is made on a time sequence located outside the support of the excitation. We prove two uniqueness results for different lateral Cauchy data. The analysis is based on the solution representation, analyticity of the observation and a refined version of inverse Sturm-Liouville theory due to Sini [35]. Our results heavily exploit the memory effect of fractional diffusion for the unique recovery of the coefficients in the model. Several numerical experiments are also presented to complement the analysis.

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arXiv2024

Score-Based Generative Models for PET Image Reconstruction

Score-based generative models have demonstrated highly promising results for medical image reconstruction tasks in magnetic resonance imaging or computed tomography. However, their application to Positron Emission Tomography (PET) is still largely unexplored. PET image reconstruction involves a variety of challenges, including Poisson noise with high variance and a wide dynamic range. To address these challenges, we propose several PET-specific adaptations of score-based generative models. The proposed framework is developed for both 2D and 3D PET. In addition, we provide an extension to guided reconstruction using magnetic resonance images. We validate the approach through extensive 2D and 3D $\textit{in-silico}$ experiments with a model trained on patient-realistic data without lesions, and evaluate on data without lesions as well as out-of-distribution data with lesions. This demonstrates the proposed method's robustness and significant potential for improved PET reconstruction.

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