arXiv subjects

Qingguo Hong

Publications and source records attributed to Qingguo Hong.

At least 19 recordsLinked to original sources

arXiv2026

A family of mixed-mixed strongly conservative finite element methods for Biot's model of consolidation

We consider a four-field formulation of Biot's quasi-static model of consolidation with the (effective) elastic stress, the solid displacement, the fluid flux and the fluid pressure as unknown physical quantities of interest. The weak form of this system composed of the momentum and mass balance equations complemented by the stress-strain relation and Darcy's law yields a two-fold perturbed saddle-point problem. Its well-posedness is proven for a natural choice of inifinite-dimensional Hilbert spaces, where the displacement-flux pair is sought in a novel Hilbert space. Next, based on recent works on $H(\rm div \, \rm div)$-conforming finite elements, we propose a family of mixed-mixed finite element methods that preserve the angular momentum as well as the fluid mass balance pointwise, i.e., in a strong sense. We prove the well-posedness of the arising discrete problem and establish optimal a priori error estimates for the related family of fully conservative mixed-mixed finite element methods.

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arXiv2026

A correction adaptive two-grid finite element method for nonselfadjoint or indefinite elliptic problems

We propose, analyze, and numerically validate a correction adaptive two-grid finite element method (CAT-GFEM) for nonselfadjoint or indefinite elliptic problems. In contrast to the adaptive two-grid finite element method (ATGFEM) of Li and Zhang [SIAM J. Sci. Comput., 43 (2021), pp. A908-A928], which is restricted to symmetric positive-definite problems, the proposed method introduces an additional correction step that solves a small-scale discrete residual problem on the coarse mesh. This step entails negligible additional computational cost and allows us to show that the L2-norm error of the corrected discrete solution is a higher-order of the energy-norm error of the discrete solution. Using this result, we prove a contraction property for a suitable sum of quasi-errors on two successive adaptive meshes and establish convergence of the method. Numerical experiments illustrate the improved effectiveness and robustness of our method in comparison with ATGFEM.

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arXiv2024

Greedy Algorithm for Neural Networks for Indefinite Elliptic Problems

The paper presents a priori error analysis of the shallow neural network approximation to the solution to the indefinite elliptic equation and and cutting-edge implementation of the Orthogonal Greedy Algorithm (OGA) tailored to overcome the challenges of indefinite elliptic problems, which is a domain where conventional approaches often struggle due to nontraditional difficulties due to the lack of coerciveness. A rigorous a priori error analysis that shows the neural networks ability to approximate indefinite problems is confirmed numerically by OGA methods. We also present a discretization error analysis of the relevant numerical quadrature. In particular, massive numerical implementations are conducted to justify the theory, some of which showcase the OGAs superior performance in comparison to the traditional finite element method. This advancement illustrates the potential of neural networks enhanced by OGA to solve intricate computational problems more efficiently, thereby marking a significant leap forward in the application of machine learning techniques to mathematical problem-solving.

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arXiv2023

Gauss Newton method for solving variational problems of PDEs with neural network discretizaitons

The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by parameterizing a set of functions. Various approaches, such as the deep Ritz method and physics-informed neural networks, have been developed for numerical solutions. Training algorithms, including gradient descent and greedy algorithms, have been proposed to solve the resulting optimization problems. In this paper, we focus on the variational formulation of the problem and propose a Gauss- Newton method for computing the numerical solution. We provide a comprehensive analysis of the superlinear convergence properties of this method, along with a discussion on semi-regular zeros of the vanishing gradient. Numerical examples are presented to demonstrate the efficiency of the proposed Gauss-Newton method.

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arXiv2023

A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration

In this paper, we develop a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second-order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity numerical method adopts the backward Euler scheme and conforming finite elements for the temporal and spatial discretization, respectively. Under the assumption that the time step size is sufficiently small and time steps are not very large, we show that the singular value distribution of the high-fidelity solution matrix $U$ is close to that of a rank one matrix. We select the eigenfunction associated with the principal eigenvalue of the matrix $U^\top U$ as the basis of the Proper Orthogonal Decomposition (POD) method to obtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency of the new method.

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arXiv2022

On the Activation Function Dependence of the Spectral Bias of Neural Networks

Neural networks are universal function approximators which are known to generalize well despite being dramatically overparameterized. We study this phenomenon from the point of view of the spectral bias of neural networks. Our contributions are two-fold. First, we provide a theoretical explanation for the spectral bias of ReLU neural networks by leveraging connections with the theory of finite element methods. Second, based upon this theory we predict that switching the activation function to a piecewise linear B-spline, namely the Hat function, will remove this spectral bias, which we verify empirically in a variety of settings. Our empirical studies also show that neural networks with the Hat activation function are trained significantly faster using stochastic gradient descent and ADAM. Combined with previous work showing that the Hat activation function also improves generalization accuracy on image classification tasks, this indicates that using the Hat activation provides significant advantages over the ReLU on certain problems.

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arXiv2022

A sharp Korn's inequality for piecewise $H^1$ space and its application

In this paper, we revisit Korn's inequality for the piecewise $H^1$ space based on general polygonal or polyhedral decompositions of the domain. Our Korn's inequality is expressed with minimal jump terms. These minimal jump terms are identified by characterizing the restriction of rigid body mode to edge/face of the partitions. Such minimal jump conditions are shown to be sharp for achieving the Korn's inequality as well. The sharpness of our result and explicitly given minimal conditions can be used to test whether any given finite element spaces satisfy Korn's inequality, immediately as well as to build or modify nonconforming finite elements for Korn's inequality to hold.

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arXiv2022

An efficient iterative method for dynamical Ginzburg-Landau equations

In this paper, we propose a new finite element approach to simulate the time-dependent Ginzburg-Landau equations under the temporal gauge, and design an efficient preconditioner for the Newton iteration of the resulting discrete system. The new approach solves the magnetic potential in H(curl) space by the lowest order of the second kind Nedelec element. This approach offers a simple way to deal with the boundary condition, and leads to a stable and reliable performance when dealing with the superconductor with reentrant corners. The comparison in numerical simulations verifies the efficiency of the proposed preconditioner, which can significantly speed up the simulation in large-scale computations.

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arXiv2022

Generalized Korn's Inequalities for Piecewise $H^2$ Vector Fields

The purpose of this paper is to construct a new class of discrete generalized Korn's inequalities for piecewise H2 vector fields in three-dimensional space. The resulting Korn's inequalities are different from the standard Korn's inequalities, as they involve the trace-free symmetric gradient operator, in place of the usual symmetric gradient operator. It is anticipated that the new generalized Korn's inequalities will be useful for the analysis of a broad range of finite element methods, including mixed finite element methods and discontinuous Galerkin methods.

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arXiv2021

New Discontinuous Galerkin Algorithms and Analysis for Linear Elasticity with Symmetric Stress Tensor

This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor $\dsig$ and displacement $\du$ inside each element, and the modifications of these two variables $\hsig$ and $\hu$ on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as $H^1$-based and the other known as $H({\rm div})$-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.

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arXiv2021

Greedy Training Algorithms for Neural Networks and Applications to PDEs

Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or empirically, to converge to the underlying PDE solution with arbitrarily high accuracy. The primary difficulty lies in solving the highly non-convex optimization problems resulting from the neural network discretization, which are difficult to treat both theoretically and practically. It is our goal in this work to take a step toward remedying this. For this purpose, we develop a novel greedy training algorithm for shallow neural networks. Our method is applicable to both the variational formulation of the PDE and also to the residual minimization formulation pioneered by physics informed neural networks (PINNs). We analyze the method and obtain a priori error bounds when solving PDEs from the function class defined by shallow networks, which rigorously establishes the convergence of the method as the network size increases. Finally, we test the algorithm on several benchmark examples, including high dimensional PDEs, to confirm the theoretical convergence rate. Although the method is expensive relative to traditional approaches such as finite element methods, we view this work as a proof of concept for neural network-based methods, which shows that numerical methods based upon neural networks can be shown to rigorously converge.

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arXiv2021

A Priori Analysis of Stable Neural Network Solutions to Numerical PDEs

Methods for solving PDEs using neural networks have recently become a very important topic. We provide an a priori error analysis for such methods which is based on the $\mathcal{K}_1(\mathbb{D})$-norm of the solution. We show that the resulting constrained optimization problem can be efficiently solved using a greedy algorithm, which replaces stochastic gradient descent. Following this, we show that the error arising from discretizing the energy integrals is bounded both in the deterministic case, i.e. when using numerical quadrature, and also in the stochastic case, i.e. when sampling points to approximate the integrals. In the later case, we use a Rademacher complexity analysis, and in the former we use standard numerical quadrature bounds. This extends existing results to methods which use a general dictionary of functions to learn solutions to PDEs and importantly gives a consistent analysis which incorporates the optimization, approximation, and generalization aspects of the problem. In addition, the Rademacher complexity analysis is simplified and generalized, which enables application to a wide range of problems.

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arXiv2021

A new practical framework for the stability analysis of perturbed saddle-point problems and applications

In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babuska's theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuska-Brezzi (LBB) condition, and the other standard assumptions in Brezzi's theory, in a combined abstract norm. The construction suggests to form the latter from individual fitted norms that are composed from proper seminorms. This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biot's equations.

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arXiv2021

An Extended Galerkin analysis in finite element exterior calculus

For the Hodge--Laplace equation in finite element exterior calculus, we introduce several families of discontinuous Galerkin methods in the extended Galerkin framework. For contractible domains, this framework utilizes seven fields and provides a unifying inf-sup analysis with respect to all discretization and penalty parameters. It is shown that the proposed methods can be hybridized as a reduced two-field formulation.

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arXiv2020

Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot's consolidation and multiple-network poroelasticity models

This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity (MPET) equations describing flow in elastic porous media that is permeated by single or multiple fluid networks. Here, the focus is on a three-field formulation of the problem in which the displacement field of the elastic matrix and, additionally, one velocity field and one pressure field for each of the $n \ge 1$ fluid networks are the unknown physical quantities. Generalizing Biot's model of consolidation, which is obtained for $n=1$, the MPET equations for $n\ge1$ exhibit a double saddle point structure. The proposed approach is based on a framework of augmenting and splitting this three-by-three block system in such a way that the resulting block Gauss-Seidel preconditioner defines a fully decoupled iterative scheme for the flux-, pressure-, and displacement fields. In this manner, one obtains an augmented Lagrangian Uzawa-type method, the analysis of which is the main contribution of this work. The parameter-robust uniform linear convergence of this fixed-point iteration is proved by showing that its rate of contraction is strictly less than one independent of all physical and discretization parameters. The theoretical results are confirmed by a series of numerical tests that compare the new fully decoupled scheme to the very popular partially decoupled fixed-stress split iterative method, which decouples only flow--the flux and pressure fields remain coupled in this case--from the mechanics problem. We further test the performance of the block triangular preconditioner defining the new scheme when used to accelerate the GMRES algorithm.

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arXiv2020

An Abstract Stabilization Method with Applications to Nonlinear Incompressible Elasticity

In this paper, we propose and analyze an abstract stabilized mixed finite element framework that can be applied to nonlinear incompressible elasticity problems. In the abstract stabilized framework, we prove that any mixed finite element method that satisfies the discrete inf-sup condition can be modified so that it is stable and optimal convergent as long as the mixed continuous problem is stable. Furthermore, we apply the abstract stabilized framework to nonlinear incompressible elasticity problems and present numerical experiments to verify the theoretical results.

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arXiv2020

Robust block preconditioners for poroelasticity

In this paper we study the linear systems arising from discretized poroelasticity problems. We formulate one block preconditioner for the two-filed Biot model and several preconditioners for the classical three-filed Biot model under the unified relationship framework between well-posedness and preconditioners. By the unified theory, we show all the considered preconditioners are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.

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arXiv2019

An Extended Galerkin Analysis for Elliptic Problems

A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different discretization variables, $u_h, \bm{p}_h, \check u_h$ and $\check p_h$, where $u_h$ and $\bm{p}_h$ are for approximation of $u$ and $\bm{p}=-α\nabla u$ inside each element, and $ \check u_h$ and $\check p_h$ are for approximation of residual of $u$ and $\bm{p} \cdot \bm{n}$ on the boundary of each element. The resulting 4-field discretization is proved to satisfy inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, most existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.

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