arXiv subjects

Ruchi Guo

Publications and source records attributed to Ruchi Guo.

At least 19 recordsLinked to original sources

arXiv2025

A Novel Preconditioning Framework for Solving Nonlinear PDEs based on Fenchel-Rockafellar Duality and Transformed Primal-Dual Techniques

A DualTPD method is proposed for solving nonlinear partial differential equations. The method is characterized by three main features. First, decoupling via Fenchel--Rockafellar duality is achieved, so that nonlinear terms are discretized by discontinuous finite element spaces, yielding block-diagonal mass matrices and closed-form updates. Second, improved convergence is obtained by applying transformed primal--dual (TPD) dynamics to the nonlinear saddle-point system, which yields strongly monotone behavior. Third, efficient preconditioners are designed for the elliptic-type Schur complement arising from the separated differential operators, and multigrid solvers are applied effectively. Extensive numerical experiments on elliptic $p$-Laplacian and nonlinear $H(\curl)$ problems are presented, showing significant efficiency gains with global, mesh-independent convergence.

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.

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arXiv2025

Variable-preconditioned transformed primal-dual method for generalized Wasserstein Gradient Flows

We propose a Variable-Preconditioned Transformed Primal-Dual (VPTPD) method for solving generalized Wasserstein gradient flows based on the structure-preserving JKO scheme. This is a nontrivial extension of the TPD method [Chen et al. (2025) SIAM J. Sci. Comput.] incorporating proximal splitting techniques to address the challenges arising from the nonsmoothness of the objective function. Our key contributions include: (i) a semi-implicit-explicit iterative scheme that combines proximal gradient steps with explicit gradient steps to treat the nonsmooth and smooth terms respectively; (ii) variable-dependent preconditioners constructed from the Hessian of a regularized objective to balance iteration count and per-iteration cost; (iii) a proof of existence and uniqueness of bounded solutions for the generalized proximal operator with the chosen preconditioner, along with a convergent and bound-preserving Newton solver; and (iv) an adaptive step-size strategy to improve robustness and accelerate convergence under poor Lipschitz conditions of the energy derivative. Comprehensive numerical experiments spanning from 1D to 3D settings demonstrate that our method achieves superior computational efficiency--achieving up to a 20$\times$ speedup over existing methods-thereby highlighting its broad applicability through several challenging simulations.

math.NA↗

arXiv2025

Inexact projected preconditioned gradient methods with variable metrics: a Lyapunov convergence theory (extended version)

Projected gradient methods are widely used for constrained optimization. A key application is for partial differential equations (PDEs), where the objective functional represents physical energy and the linear constraints enforce conservation laws. However, computing the projections onto constraint sets generally requires solving large-scale ill-conditioned systems. A common strategy is to relax projection accuracy and apply preconditioners, which leads to inexact preconditioned projected gradient descent (IPPGD) methods studied here. Furthermore, variable preconditioners dynamically incorporating updated nonlinear information often enhance convergence rates. However, due to the complex interplay between inexactness and adaptive preconditioners, the theoretical analysis and the dynamic behavior of the IPPGD methods still remain quite open. We propose an effective strategy for constructing the inexact projection operator and develop a gradient-type flow to model the resulting IPPGD methods. Discretization of this flow not only recovers the original IPPGD method but also yields a potentially faster novel method. Furthermore, we apply Lyapunov analysis, designing a delicate Lyapunov function, to prove the exponential convergence at the continuous level and linear convergence at the discrete level under certain assumptions. Finally, we validate our approach through numerical experiments, demonstrating robust performance and computational efficiency.

math.OC↗

arXiv2025

An Enriched Immersed Finite Element Method for 3D Interface Problems

We introduce an enriched immersed finite element method for addressing interface problems characterized by general non-homogeneous jump conditions. Unlike many existing unfitted mesh methods, our approach incorporates a homogenization concept. The IFE trial function set is composed of two components: the standard homogeneous IFE space and additional enrichment IFE functions. These enrichment functions are directly determined by the jump data, without adding extra degrees of freedom to the system. Meanwhile, the homogeneous IFE space is isomorphic to the standard finite element space on the same mesh. This isomorphism remains stable regardless of interface location relative to the mesh, ensuring optimal $\mathcal{O}(h^2)$ conditioning that is independent of the interface location and facilitates an immediate development of a multigrid fast solver; namely the iteration numbers are independent of not only the mesh size but also the relative interface location. Theoretical analysis and extensive numerical experiments are carried out in the efforts to demonstrate these features.

math.NA↗

arXiv2024

Virtual element methods based on boundary triangulation:fitted and unfitted meshes

One remarkable feature of virtual element methods (VEMs) is their great flexibility and robustness when used on almost arbitrary polytopal meshes. This very feature makes it widely used in both fitted and unfitted mesh methods. Despite extensive numerical studies, a rigorous analysis of robust optimal convergence has remained open for highly anisotropic 3D polyhedral meshes. In this work, we consider the VEMs in \cite{2023CaoChenGuo,2017ChenWeiWen} that introduce a boundary triangulation satisfying the maximum angle condition. We close this theoretical gap regarding optimal convergence on polyhedral meshes in the lowest-order case for the following three types of meshes: (1) elements only contain non-shrinking inscribed balls but \textit{are not necessarily star convex} to those balls; (2) elements are cut arbitrarily from a background Cartesian mesh, which can extremely shrink; (3) elements contain different materials on which the virtual spaces involve discontinuous coefficients. The first two widely appear in generating fitted meshes for interface and fracture problems, while the third one is used for unfitted mesh on interface problems. In addition, the present work also generalizes the maximum angle condition from simplicial meshes to polyhedral meshes.

math.NA↗

arXiv2024

Solver-in-the-Loop joint operator learning: fractional Laplace-Beltrami features for interface reconstruction

In this work, we propose a joint operator learning method for reconstructing images of conductivity coefficients from boundary data. Inspired by the idea of employing partial differential equation (PDE) solvers as preconditioners for this inverse problem, we investigate a ``solver-in-the-loop'' training mechanism. It allows the interaction of learnable parameters integrated in a PDE solver module and those in neural networks for reconstructing images. Specifically, we employ a fractional Laplace-Beltrami operator with a learnable fractional order, which transforms boundary data into high-dimensional features. These features then serve as input to a neural network, significantly improving reconstruction accuracy. For this purpose, a Learning-Automated FEM (LA-FEM) package, facilitating this ``solver-in-the-loop'' property, is developed with PyTorch as a backend. The new LA-FEM module conveniently allows the auto-differentiation regarding an objective function to freely propagate through the PDE solver from the forward problem and the coupled neural networks for the inverse problem.

math.NA↗

arXiv2023

Transformed Primal-Dual Methods with Variable-Preconditioners

This paper introduces a novel Transformed Primal-Dual with variable-metric/preconditioner (TPDv) algorithm, designed to efficiently solve affine constrained optimization problems common in nonlinear partial differential equations (PDEs). Diverging from traditional methods, TPDv iteratively updates time-evolving preconditioning operators, enhancing adaptability. The algorithm is derived and analyzed, demonstrating global linear convergence rates under mild assumptions. Numerical experiments on challenging nonlinear PDEs, including the Darcy-Forchheimer model and a nonlinear electromagnetic problem, showcase the algorithm's superiority over existing methods in terms of iteration numbers and computational efficiency. The paper concludes with a comprehensive convergence analysis.

math.NA↗

arXiv2023

Anisotropic analysis of VEM for time-harmonic Maxwell equations in inhomogeneous media with low regularity

It has been extensively studied in the literature that solving Maxwell equations is very sensitive to the mesh structure, space conformity and solution regularity. Roughly speaking, for almost all the methods in the literature, optimal convergence for low-regularity solutions heavily relies on conforming spaces and highly-regular simplicial meshes. This can be a significant limitation for many popular methods based on polytopal meshes in the case of inhomogeneous media, as the discontinuity of electromagnetic parameters can lead to quite low regularity of solutions near media interfaces, and potentially worsened by geometric singularities, making many popular methods based on broken spaces, non-conforming or polytopal meshes particularly challenging to apply. In this article, we present a virtual element method for solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media with quite arbitrary polytopal meshes, and the media interface is allowed to have geometric singularity to cause low regularity. There are two key novelties: (i) the proposed method is theoretically guaranteed to achieve robust optimal convergence for solutions with merely $\mathbf{H}^{\theta}$ regularity, $\theta\in(1/2,1]$; (ii) the polytopal element shape can be highly anisotropic and shrinking, and an explicit formula is established to describe the relationship between the shape regularity and solution regularity. Extensive numerical experiments will be given to demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2022

On the maximum angle conditions for polyhedra with virtual element methods

Finite element methods are well-known to admit robust optimal convergence on simplicial meshes satisfying the maximum angle conditions. But how to generalize this condition to polyhedra is unknown in the literature. In this work, we argue that this generation is possible for virtual element methods (VEMs). In particular, we develop an anisotropic analysis framework for VEMs where the virtual spaces and projection spaces remain abstract and can be problem-adapted, carrying forward the ``virtual'' spirit of VEMs. Three anisotropic cases will be analyzed under this framework: (1) elements only contain non-shrinking inscribed balls but are not necessarily star convex to those balls; (2) elements are cut arbitrarily from a background Cartesian mesh, which can extremely shrink; (3) elements contain different materials on which the virtual spaces involve discontinuous coefficients. The error estimates are guaranteed to be independent of polyhedral element shapes. The present work largely improves the current theoretical results in the literature and also broadens the scope of the application of VEMs.

math.NA↗

arXiv2022

Transformer Meets Boundary Value Inverse Problems

A Transformer-based deep direct sampling method is proposed for electrical impedance tomography, a well-known severely ill-posed nonlinear boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural networks? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data in different frequencies are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions as different input channels. Then, by introducing learnable non-local kernels, the direct sampling is recast to a modified attention mechanism. The new method achieves superior accuracy over its predecessors and contemporary operator learners and shows robustness to noises in benchmarks. This research shall strengthen the insights that, despite being invented for natural language processing tasks, the attention mechanism offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures.

cs.LG↗

arXiv2022

A family of immersed finite element spaces and applications to three dimensional $\mathbf{H}(\text{curl})$ interface problems

Maxwell interface problems are of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in 3D computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on non-conforming approximation spaces, which may cause a loss of accuracy for solving Maxwell equations, and the widely-used penalty techniques in the literature may not help in recovering the optimal convergence. In this article, we provide a remedy by developing N\'ed\'elec-type immersed finite element spaces with a Petrov-Galerkin scheme that is able to produce optimal-convergent solutions. To establish a systematic framework, we analyze all the $H^1$, $\mathbf{H}(\text{curl})$ and $\mathbf{H}(\text{div})$ IFE spaces and form a discrete de Rham complex. Based on these fundamental results, we further develop a fast solver using a modified Hiptmair-Xu preconditioner which works for both the GMRES and CG methods.

math.NA↗

arXiv2022

Immersed Virtual Element Methods for Electromagnetic Interface Problems in Three Dimensions

Finite element methods for electromagnetic problems modeled by Maxwell-type equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to electromagnetic interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a 3D $\mathbf{H}(\mathrm{curl})$ interface problem is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence. To develop a systematic framework, the $H^1$, $\mathbf{H}(\mathrm{curl})$ and $\mathbf{H}(\mathrm{div})$ interface problems and their corresponding problem-orientated immersed virtual element spaces are considered all together. In addition, the de Rham complex will be established based on which the Hiptmair-Xu (HX) preconditioner can be used to develop a fast solver for the $\mathbf{H}(\mathrm{curl})$ interface problem.

math.NA↗

arXiv2021

Immersed Virtual Element Methods for Elliptic Interface Problems in Two Dimensions

This article presents an immersed virtual element method for solving a class of interface problems that combines the advantages of both body-fitted mesh methods and unfitted mesh methods. A background body-fitted mesh is generated initially. On those interface elements, virtual element spaces are constructed as solution spaces to local interface problems, and exact sequences can be established for these new spaces involving discontinuous coefficients. The discontinuous coefficients of interface problems are recast as Hodge star operators that are the key to project immersed virtual functions to classic immersed finite element (IFE) functions for computing numerical solutions. An a priori convergence analysis is established robust with respect to the interface location. The proposed method is capable of handling more complicated interface element configuration and provides better performance than the conventional penalty-type IFE method for the H(curl)-interface problem arising from Maxwell equations. It also brings a connection between various methods such as body-fitted methods, IFE methods, virtual element methods, etc.

math.NA↗

arXiv2021

Learn an index operator by CNN for solving diffusive optical tomography: a deep direct sampling method

In this work, we investigate the diffusive optical tomography (DOT) problem in the case that limited boundary measurements are available. Motivated by the direct sampling method (DSM), we develop a deep direct sampling method (DDSM) to recover the inhomogeneous inclusions buried in a homogeneous background. In this method, we design a convolutional neural network (CNN) to approximate the index functional that mimics the underling mathematical structure. The benefits of the proposed DDSM include fast and easy implementation, capability of incorporating multiple measurements to attain high-quality reconstruction, and advanced robustness against the noise. Numerical experiments show that the reconstruction accuracy is improved without degrading the efficiency, demonstrating its potential for solving the real-world DOT problems.

math.NA↗

arXiv2021

A Virtual Finite Element Method for Two Dimensional Maxwell Interface Problems with a Background Unfitted Mesh

A virtual element method (VEM) with the first order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual H(curl)-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.

math.NA↗

arXiv2020

Solving Two Dimensional H(curl)-elliptic Interface Systems with Optimal Convergence On Unfitted Meshes

In this article, we develop and analyze a finite element method with the first family N\'ed\'elec elements of the lowest degree for solving a Maxwell interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use $\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface elements while keep using the standard N\'ed\'elec functions on all the non-interface elements. We establish a few important properties for the IFE functions including the unisolvence according to the edge degrees of freedom, the exact sequence relating to the $H^1$ IFE functions and the optimal approximation capabilities. In order to achieve the optimal convergence rates, we employ a Petrov-Galerkin method in which the IFE functions are only used as the trial functions and the standard N\'ed\'elec functions are used as the test functions which can eliminate the non-conformity errors. We analyze the inf-sup conditions under certain conditions and show the optimal convergence rates which are also validated by numerical experiments.

math.NA↗

arXiv2020

Construct Deep Neural Networks Based on Direct Sampling Methods for Solving Electrical Impedance Tomography

This work investigates the electrical impedance tomography (EIT) problem when only limited boundary measurements are available, which is known to be challenging due to the extreme ill-posedness. Based on the direct sampling method (DSM), we propose deep direct sampling methods (DDSMs) to locate inhomogeneous inclusions in which two types of deep neural networks (DNNs) are constructed to approximate the index function(functional): fully connected neural network(FNN) and convolutional neural network (CNN). The proposed DDSMs are easy to be implemented, capable of incorporating multiple Cauchy data pairs to achieve high-quality reconstruction and highly robust with respect to large noise. Additionally, the implementation of DDSMs adopts offline-online decomposition, which helps to reduce a lot of computational costs and makes DDSMs as efficient as the conventional DSM. The numerical experiments are presented to demonstrate the efficacy and show the potential benefits of combining DNN with DSM.

math.NA↗