Xiaobing Feng

Xiaobing Feng, Steven Wise

In this paper we present PDE and finite element analyses for a system of partial differential equations (PDEs) consisting of the Darcy equation and the Cahn-Hilliard equation, which arises as a diffuse interface model for the two phase Hele-Shaw flow. We propose a fully discrete implicit finite element method for approximating the PDE system, which consists of the implicit Euler method combined with a convex splitting energy strategy for the temporal discretization, the standard finite element discretization for the pressure and a split (or mixed) finite element discretization for the fourth order Cahn-Hilliard equation. It is shown that the proposed numerical method satisfies a mass conservation law in addition to a discrete energy law that mimics the basic energy law for the Darcy-Cahn-Hilliard phase field model and holds uniformly in the phase field parameter $ε$. With help of the discrete energy law, we first prove that the fully discrete finite method is unconditionally energy stable and uniquely solvable at each time step. We then show that, using the compactness method, the finite element solution has an accumulation point that is a weak solution of the PDE system. As a result, the convergence result also provides a constructive proof of the existence of global-in-time weak solutions to the Darcy-Cahn-Hilliard phase field model in both two and three dimensions. Finally, we propose a nonlinear multigrid iterative algorithm to solve the finite element equations at each time step. Numerical experiments based on the overall solution method of combining the proposed finite element discretization and the nonlinear multigrid solver are presented to validate the theoretical results and to show the effectiveness of the proposed fully discrete finite element method for approximating the Darcy-Cahn-Hilliard phase field model.

Xiaobing Feng, Haijun Wu

This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and sub-optimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}.

Xiaobing Feng, Michael Neilan

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation $\det(D^2u^0)=f$ based on the vanishing moment method which was proposed recently by the authors in \cite{Feng2}. In this approach, the second order fully nonlinear Monge-Ampère equation is approximated by the fourth order quasilinear equation $-εΔ^2 u^ε+ \det{D^2u^ε} =f$. It was proved in \cite{Feng1} that the solution $u^ε$ converges to the unique convex viscosity solution $u^0$ of the Dirichlet problem for the Monge-Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi type mixed finite element methods for approximating the solution $u^ε$ of the regularized fourth order problem, which computes simultaneously $u^\vepsi$ and the moment tensor $σ^\vepsi:=D^2u^ε$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\vepsi$, for the errors $u^ε-u^ε_h$ and $σ^\vepsi-σ_h^\vepsi$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\vepsi$ for the error $u^0-u_h^\vepsi$ and $σ^\vepsi-σ_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve these rates.

Xiaobing Feng, Yukun Li, Yulong Xing

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is based on fully implicit time-stepping and the other uses the energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on $ε^{-1}$ only in some low polynomial orders, instead of exponential orders. Similar to [14], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space $H^1$ and it is larger than the finite element space. This difficult is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved in \cite{Feng_Prohl04}. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods.

Xiaobing Feng, Yulong Xing

Two local discontinuous Galerkin (LDG) methods using some non-standard numerical fluxes are developed for the Helmholtz equation with the first order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are absolutely stable (hence well-posed) with respect to both the wave number and the mesh size. Optimal order (with respect to the mesh size) error estimates are proved for all wave numbers in the preasymptotic regime. To analyze the proposed LDG methods, they are recasted and treated as (non-conforming) mixed finite element methods. The crux of the analysis is to establish a generalized {\em inf-sup} condition, which holds without any mesh constraint, for each LDG method. The generalized {\em inf-sup} conditions then easily infer the desired absolute stability of the proposed LDG methods. In return, the stability results not only guarantee the well-posedness of the LDG methods but also play a crucial role in the derivation of the error estimates. Numerical experiments, which confirm the theoretical results and compare the proposed two LDG methods, are also presented in the paper.

Xiaobing Feng, Michael Neilan

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation $\det(D^2u^0)=f$ based on the vanishing moment method which was developed by the authors in \cite{Feng2,Feng1}. In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation $-εΔ^2 u^ε+ \det{D^2u^ε} =f$ accompanied by appropriate boundary conditions. This new approach allows one to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation, a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution $u^ε$ of the regularized fourth order problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter $\vepsi$, for the error $u^ε-u^ε_h$. Finally, using the Aygris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of $\vepsi$ for the error $u^0-u_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve these rates.

Xiaobing Feng, Michael Neilan, Stefan Schnake

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate $W^{2,p}$ strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the $W^{2,p}$ strong solution in a discrete $W^{2,p}$-norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon-Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. As a byproduct of our analysis, we also establish broken $W^{1,p}$-norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.

Xiaobing Feng, Haijun Wu

This paper develops and analyzes an interior penalty discontinuous Galerkin (IPDG) method using piecewise linear polynomials for the indefinite time harmonic Maxwell equations with the impedance boundary condition in the three dimensional space. The main novelties of the proposed IPDG method include the following: first, the method penalizes not only the jumps of the tangential component of the electric field across the element faces but also the jumps of the tangential component of its vorticity field; second, the penalty parameters are taken as complex numbers of negative imaginary parts. For the differential problem, we prove that the sesquilinear form associated with the Maxwell problem satisfies a generalized weak stability (i.e., inf-sup condition) for star-shaped domains.Such a generalized weak stability readily infers wave-number explicit a priori estimates for the solution of the Maxwell problem, which plays an important role in the error analysis for the IPDG method. For the proposed IPDG method, we show that the discrete sesquilinear form satisfies a coercivity for all positive mesh size $h$ and wave number $k$ and for general domains including non-star-shaped ones. In turn, the coercivity easily yields the well-posedness and stability estimates (i.e., a priori estimates) for the discrete problem without imposing any mesh constraint. Based on these discrete stability estimates, by adapting a nonstandard error estimate technique of Fung and Wu (2009), we derive both the energy-norm and the $L^2$-norm error estimates for the IPDG method in all mesh parameter regimes including pre-asymptotic regime (i.e., $k^2 h\gtrsim 1$). Numerical experiments are also presented to gauge the theoretical results and to numerically examine the pollution effect (with respect to $k$) in the error bounds.

Xiaobing Feng, Michael Neilan

This paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear Monge-Ampére equation and a transport equation. The system is the Eulerian formulation in the dual space for the B. J. Hoskins' semigeostrophic flow equations, which are widely used in meteorology to model slowly varying flows constrained by rotation and stratification. To overcome the difficulty caused by the strong nonlinearity, we first formulate (at the differential level) a vanishing moment approximation of the semigeostrophic flow equations, a methodology recently proposed by the authors \cite{Feng1,Feng2}, which involves approximating the fully nonlinear Monge-Ampére equation by a family of fourth order quasilinear equations. We then construct a fully discrete modified characteristic finite element method for the regularized problem. It is shown that under certain mesh and time stepping constraints, the proposed numerical method converges with an optimal order rate of convergence. In particular, the obtained error bounds show explicit dependence on the regularization parameter $\vepsi$. Numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method.

Xiaobing Feng, Thomas Lewis, Michael Neilan

This paper develops a discontinuous Galerkin (DG) finite element differential calculus theory for approximating weak derivatives of Sobolev functions and piecewise Sobolev functions. By introducing numerical one-sided derivatives as building blocks, various first and second order numericaloperators such as the gradient, divergence, Hessian, and Laplacian operator are defined, and their corresponding calculus rules are established. Among the calculus rules are product and chain rules, integration by parts formulas and the divergence theorem. Approximation properties and the relationship between the proposed DG finite element numerical derivatives and some well-known finite difference numerical derivative formulas on Cartesian grids are also established. Efficient implementation of the DG finite element numerical differential operators is also proposed. Besides independent interest in numerical differentiation, the primary motivation and goal of developing the DG finite element differential calculus is to solve partial differential equations. It is shown that several existing finite element, finite difference and DG methods can be rewritten compactly using the proposed DG finite element differential calculus framework. Moreover, new DG methods for linear and nonlinear PDEs are also obtained from the framework.

Xiaobing Feng, Chiu-Yen Kao, Thomas Lewis

This paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations (PDEs) in 1-D. The goal of the paper is to extend the successful framework of monotone, consistent, and stable finite difference methods for first order fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs such as Monge-Ampère and Bellman type equations. New concepts of consistency, generalized monotonicity, and stability are introduced; among them, the generalized monotonicity and consistency, which are easier to verify in practice, are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear Hamilton-Jacobi equations. The main component of the proposed framework is the concept of "numerical operator", and the main idea used to design consistent, monotone and stable finite difference methods is the concept of "numerical moment". These two new concepts play the same roles as the "numerical Hamiltonian" and the "numerical viscosity" play in the finite difference framework for first order fully nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also are proved to be stable and the second class contains Godunov-like methods. Numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper.

Xiaobing Feng, Haijun Wu

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+\De\bigl(\eps \De u-\eps^{-1} f(u)\bigr)=0$. It is shown that the {\it a posteriori} error bounds depends on $\eps^{-1}$ only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

Xiaobing Feng, Yukun Li, Andreas Prohl

This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for planar curves of graphs. To circumvent the difficulty caused by the low spatial regularity of the SPDE solution, a regularization procedure is first proposed to approximate the SPDE, and an error estimate for the regularized problem is derived. A semi-discrete finite element method, and a space-time fully discrete method are then proposed to approximate the solution of the regularized SPDE problem. Strong convergence with rates are established for both, semi- and fully discrete methods. Computational experiments are provided to study the interplay of the geometric evolution and gradient type-noises.

Xiaobing Feng, Thomas Lewis

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.

Xiaobing Feng, Michael Neilan

The vanishing moment method was introduced by the authors in [37] as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations (PDEs), in particular, using Galerkin-type numerical methods such as finite element methods, spectral methods, and discontinuous Galerkin methods, a task which has not been practicable in the past. The crux of the vanishing moment method is the simple idea of approximating a fully nonlinear second order PDE by a family (parametrized by a small parameter $\vepsi$) of quasilinear higher order (in particular, fourth order) PDEs. The primary objectives of this book are to present a detailed convergent analysis for the method in the radial symmetric case and to carry out a comprehensive finite element numerical analysis for the vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract methodological and convergence analysis frameworks of conforming finite element methods and mixed finite element methods are first developed for fully nonlinear second order PDEs in general settings. The abstract frameworks are then applied to three prototypical nonlinear equations, namely, the Monge-Ampère equation, the equation of prescribed Gauss curvature, and the infinity-Laplacian equation. Numerical experiments are also presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology.

Xiaobing Feng, Michael Neilan

In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator $Δ^2$, the bi-wave operator $\Box^2$ is not an elliptic operator, so the energy space for the bi-wave equation is much larger than the energy space for the biharmonic equation. This then makes it possible to construct low order conforming finite elements for the bi-wave equation. However, the existence and construction of such finite elements strongly depends on the mesh. In the paper, we first characterize mesh conditions which allow and not allow construction of low order conforming finite elements for approximating the bi-wave equation. We then construct a cubic and a quartic conforming finite element. It is proved that both elements have the desired approximation properties, and give optimal order error estimates in the energy norm, suboptimal (and optimal in some cases) order error estimates in the $H^1$ and $L^2$ norm. Finally, numerical experiments are presented to guage the efficiency of the proposed finite element methods and to validate the theoretical error bounds.

Xiaobing Feng, Thomas Lewis

This paper is concerned with developing accurate and efficient discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs which are merely continuous functions by definition. In order to capture discontinuities of the first order derivative $u_x$ of the solution $u$, two independent functions $q_1$ and $q_2$ are introduced to approximate one-sided derivatives of $u$. Similarly, to capture the discontinuities of the second order derivative $u_{xx}$, four independent functions $p_{1}$, $p_{2}$, $p_{3}$, and $p_{4}$ are used to approximate one-sided derivatives of $q_1$ and $q_2$. The proposed LDG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a given fully nonlinear problem into a mostly linear system of equations where the given nonlinear differential operator must be replaced by a numerical operator which allows multiple value inputs of the first and second order derivatives $u_x$ and $u_{xx}$. An easy to verify criterion for constructing "good" numerical operators is also proposed. It consists of a consistency and a generalized monotonicity. To ensure such a generalized monotonicity, the crux of the construction is to introduce the numerical moment in the numerical operator. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes.

John W. Barrett, Xiaobing Feng, Andreas Prohl

Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\leq p<\infty$, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a $BV$-solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms $\cA\cdot\nab\bv$ and $\cA\wedge\nab\bv$; which pair a divergence-$L^1$, or a divergence-measure, tensor field $\cA$, and a $BV$-vector field $\bv$. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the $p$-harmonic map heat flow, and the convergence of the proposed numerical method is also established.

Xiaobing Feng, Yukun Li, Yi Zhang

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-side Lipschitz condition. The semilinear SPDEs considered in this paper is a direct generalization of the SODEs considered in [13]. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a special discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the $H^1$-seminorm of the numerical solution must be established, which are difficult and delicate. A judicious combination of the properties of the drift and diffusion terms and a nontrivial technique borrowed from [16] is used in this paper to achieve the goal. Finally, stability estimates for the second and higher order moments of the $L^2$-norm of the numerical solution is also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the $H^1$-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.

Xiaobing Feng, Cody Lorton

This paper develops and analyzes an efficient Monte Carlo interior penalty discontinuous Galerkin (MCIP-DG) method for elastic wave scattering in random media. The method is constructed based on a multi-modes expansion of the solution of the governing random partial differential equations. It is proved that the mode functions satisfy a three-term recurrence system of partial differential equations (PDEs) which are nearly deterministic in the sense that the randomness only appears in the right-hand side source terms, not in the coefficients of the PDEs. Moreover, the same differential operator applies to all mode functions. A proven unconditionally stable and optimally convergent IP-DG method is used to discretize the deterministic PDE operator, an efficient numerical algorithm is proposed based on combining the Monte Carlo method and the IP-DG method with the $LU$ direct linear solver. It is shown that the algorithm converges optimally with respect to both the mesh size $h$ and the sampling number $M$, and practically its total computational complexity is only amount to solving very few deterministic elastic Helmholtz equations using the $LU$ direct linear solver. Numerically experiments are also presented to demonstrate the performance and key features of the proposed MCIP-DG method.

Shuoming Zhang, Qiuchu Yu, Yangyu Zhang et al.

LLM-based agents are increasingly used to generate GPU kernels, but they often know what optimizations to try without knowing when those optimizations are sound. We introduce KLineage, which learns this missing "when" knowledge from expert kernels: instead of relying on forward rollouts, KLineage walks expert implementations backward through validation-gated simplifications and reverses each accepted step into a reusable optimization skill. Each skill records not only the optimization intent, but also where it applies in code, what conditions made it valid, what effect it had, and what failures its assumptions avoid. A downstream LLM materializes these skills on new code surfaces under the same compile/correctness/profile gate. On five expert workloads across two NVIDIA architectures, these lineage-derived skills serve as an effective optimization curriculum, exceeding recent memory-based LLM-kernel baselines in both final kernel quality and optimization efficiency under the same fixed budget. We additionally use a separate 22-instance held-out check as a sanity test against source-case memorization.

Xiaobing Feng, Yukun Li, Yi Zhang

This paper develops and analyzes some fully discrete mixed finite element methods for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn-Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele-Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element methods. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn-Hilliard equation, the Hölder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele-Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.

Xiaobing Feng, Thomas Lewis

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin (IP-DG) methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative $u_{xx}$ of the solution $u$, three independent functions $p_1, p_2$ and $p_3$ are introduced to represent numerical derivatives using various one-sided limits. The proposed DG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative $u_{xx}$. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes.In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator, which is consistent with the differential operator and satisfies certain monotonicity (called g-monotonicity) properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG framework. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.

Xiaobing Feng, Michael Neilan

This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with "guaranteed" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.

Xiaobing Feng, Stefan Schnake

This paper develops an enhanced finite element method for approximating a class of variational problems which exhibit the \textit{Lavrentiev gap phenomenon} in the sense that the minimum values of the energy functional have a nontrivial gap when the functional is minimized on spaces $W^{1,1}$ and $W^{1,\infty}$. To remedy the standard finite element method, which fails to converge for such variational problems, a simple and effective cut-off procedure is utilized to design the (enhanced finite element) discrete energy functional. In essence the proposed discrete energy functional curbs the gap phenomenon by capping the derivatives of its input on a scale of $O(h^{-α})$ (where $h$ denotes the mesh size) for some positive constant $α$. A sufficient condition is proposed for determining the problem-dependent parameter $\a$. Extensive 1-D and 2-D numerical experiment results are provided to show the convergence behavior and the performance of the proposed enhanced finite element method.

Xiaobing Feng, Thomas Lewis, Stefan Schnake

This paper is concerned with continuous and discrete approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A $C^1$ finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the $H^2$ norm are shown. Lastly, numerical tests are given to show the validity of the method.

Xiaobing Feng, Junshan Lin, Cody Lorton

This paper develops an efficient Monte Carlo interior penalty discontinuous Galerkin method for electromagnetic wave propagation in random media. This method is based on a multi-modes expansion of the solution to the time-harmonic random Maxwell equations. It is shown that each mode function satisfies a Maxwell system with random sources defined recursively. An unconditionally stable IP-DG method is employed to discretize the nearly deterministic Maxwell system and the Monte Carlo method combined with an efficient acceleration strategy is proposed for computing the mode functions and the statistics of the electromagnetic wave. A complete error analysis is established for the proposed multi-modes Monte Carlo IP-DG method. It is proved that the proposed method converges with an optimal order for each of three levels of approximations. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed numerical method and approach.

Xiaobing Feng, Cody Lorton

In this paper we propose and analyze an interior penalty discontinuous Galerkin (IP-DG) method using piecewise linear polynomials for the elastic Helmholtz equations with the first order absorbing boundary condition. It is proved that the sesquilinear form for the problem satisfies a generalized weak coercivity property, which immediately infers a stability estimate for the solution of the differential problem in all frequency regimes. It is also proved that the proposed IP-DG method is unconditionally stable with respect to both frequency $ω$ and mesh size $h$. Sub-optimal order (with respect to $h$) error estimates in the broken $H^1$-norm and in the $L^2$-norm are obtained in all mesh regimes. These estimate improve to optimal order when the mesh size $h$ is restricted to the pre-asymptotic regime (i.e., $ω^βh =O(1)$ for some $1\leq β<2$). The novelties of the proposed IP-DG method include: first, the method penalizes not only the jumps of the function values across the element edges but also the jumps of the normal derivatives; second, the penalty parameters are taken as complex numbers with positive imaginary parts. In order to establish the desired unconditional stability estimate for the numerical solution, the main idea is to exploit a (simple) property of linear functions to overcome the main difficulty caused by non-Hermitian nature and strong indefiniteness of the Helmholtz-type problem. The error estimate is then derived using a nonstandard technique adapted from \cite{Feng_Wu09}. Numerical experiments are also presented to validate the theoretical results and to numerically examine the pollution effect (with respect to $ω$) in the error bounds.

Junyu Li, Ye Zhang, Wen Shu et al.

Multiple instance learning (MIL) has been successfully applied for whole slide images (WSIs) analysis in computational pathology, enabling a wide range of prediction tasks from tumor subtyping to inferring genetic mutations and multi-omics biomarkers. However, existing MIL methods predominantly focus on single-task learning, resulting in not only overall low efficiency but also the overlook of inter-task relatedness. To address these issues, we proposed an adapted architecture of Multi-gate Mixture-of-experts with Multi-proxy for Multiple instance learning (M4), and applied this framework for simultaneous prediction of multiple genetic mutations from WSIs. The proposed M4 model has two main innovations: (1) utilizing a mixture of experts with multiple gating strategies for multi-genetic mutation prediction on a single pathological slide; (2) constructing multi-proxy expert network and gate network for comprehensive and effective modeling of pathological image information. Our model achieved significant improvements across five tested TCGA datasets in comparison to current state-of-the-art single-task methods. The code is available at:https://github.com/Bigyehahaha/M4.

Xiaobing Feng, Stefan Schnake

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [Feng2013]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.

Shuoming Zhang, Jiacheng Zhao, Qiuchu Yu et al.

This survey has provided a systematic overview of the emerging field of LLM-enabled compilation by addressing several key research questions. We first answered how LLMs are being integrated by proposing a comprehensive, multi-dimensional taxonomy that categorizes works based on their Design Philosophy (Selector, Translator, Generator), LLM Methodology, their operational Level of Code Abstraction, and the specific Task Type they address. In answering what advancements these approaches offer, we identified three primary benefits: the democratization of compiler development, the discovery of novel optimization strategies, and the broadening of the compiler's traditional scope. Finally, in addressing the field's challenges and opportunities, we highlighted the critical hurdles of ensuring correctness and achieving scalability, while identifying the development of hybrid systems as the most promising path forward. By providing these answers, this survey serves as a foundational roadmap for researchers and practitioners, charting the course for a new generation of LLM-powered, intelligent, adaptive and synergistic compilation tools.

Shuoming Zhang, Jiacheng Zhao, Ruiyuan Xu et al.

Content Warning: This paper may contain unsafe or harmful content generated by LLMs that may be offensive to readers. Large Language Models (LLMs) are extensively used as tooling platforms through structured output APIs to ensure syntax compliance so that robust integration with existing softwares like agent systems, could be achieved. However, the feature enabling functionality of grammar-guided structured output presents significant security vulnerabilities. In this work, we reveal a critical control-plane attack surface orthogonal to traditional data-plane vulnerabilities. We introduce Constrained Decoding Attack (CDA), a novel jailbreak class that weaponizes structured output constraints to bypass safety mechanisms. Unlike prior attacks focused on input prompts, CDA operates by embedding malicious intent in schema-level grammar rules (control-plane) while maintaining benign surface prompts (data-plane). We instantiate this with a proof-of-concept Chain Enum Attack, achieves 96.2% attack success rates across proprietary and open-weight LLMs on five safety benchmarks with a single query, including GPT-4o and Gemini-2.0-flash. Our findings identify a critical security blind spot in current LLM architectures and urge a paradigm shift in LLM safety to address control-plane vulnerabilities, as current mechanisms focused solely on data-plane threats leave critical systems exposed.

Shuoming Zhang, Jiacheng Zhao, Chunwei Xia et al.

Large language models (LLMs) have the potential to revolutionize how we design and implement compilers and code translation tools. However, existing LLMs struggle to handle long and complex programs. We introduce LEGO-Compiler, a novel neural compilation system that leverages LLMs to translate high-level languages into assembly code. Our approach centers on three key innovations: LEGO translation, which decomposes the input program into manageable blocks; breaking down the complex compilation process into smaller, simpler verifiable steps by organizing it as a verifiable LLM workflow by external tests; and a feedback mechanism for self-correction. Supported by formal proofs of translation composability, LEGO-Compiler demonstrates high accuracy on multiple datasets, including over 99% on ExeBench and 97.9% on industrial-grade AnsiBench. Additionally, LEGO-Compiler has also acheived near one order-of-magnitude improvement on compilable code size scalability. This work opens new avenues for applying LLMs to system-level tasks, complementing traditional compiler technologies.

Songjun Cao, Yike Zhang, Xiaobing Feng et al.

Nowadays voice search for points of interest (POI) is becoming increasingly popular. However, speech recognition for local POI has remained to be a challenge due to multi-dialect and massive POI. This paper improves speech recognition accuracy for local POI from two aspects. Firstly, a geographic acoustic model (Geo-AM) is proposed. The Geo-AM deals with multi-dialect problem using dialect-specific input feature and dialect-specific top layer. Secondly, a group of geo-specific language models (Geo-LMs) are integrated into our speech recognition system to improve recognition accuracy of long tail and homophone POI. During decoding, specific language models are selected on demand according to users' geographic location. Experiments show that the proposed Geo-AM achieves 6.5%$\sim$10.1% relative character error rate (CER) reduction on an accent testset and the proposed Geo-AM and Geo-LM totally achieve over 18.7% relative CER reduction on Tencent Map task.

Jiansong Li, Xiao Dong, Guangli Li et al.

The training of deep neural networks (DNNs) is usually memory-hungry due to the limited device memory capacity of DNN accelerators. Characterizing the memory behaviors of DNN training is critical to optimize the device memory pressures. In this work, we pinpoint the memory behaviors of each device memory block of GPU during training by instrumenting the memory allocators of the runtime system. Our results show that the memory access patterns of device memory blocks are stable and follow an iterative fashion. These observations are useful for the future optimization of memory-efficient training from the perspective of raw memory access patterns.

Guangli Li, Xiu Ma, Xueying Wang et al.

The increasing computational cost of deep neural network models limits the applicability of intelligent applications on resource-constrained edge devices. While a number of neural network pruning methods have been proposed to compress the models, prevailing approaches focus only on parametric operators (e.g., convolution), which may miss optimization opportunities. In this paper, we present a novel fusion-catalyzed pruning approach, called FuPruner, which simultaneously optimizes the parametric and non-parametric operators for accelerating neural networks. We introduce an aggressive fusion method to equivalently transform a model, which extends the optimization space of pruning and enables non-parametric operators to be pruned in a similar manner as parametric operators, and a dynamic filter pruning method is applied to decrease the computational cost of models while retaining the accuracy requirement. Moreover, FuPruner provides configurable optimization options for controlling fusion and pruning, allowing much more flexible performance-accuracy trade-offs to be made. Evaluation with state-of-the-art residual neural networks on five representative intelligent edge platforms, Jetson TX2, Jetson Nano, Edge TPU, NCS, and NCS2, demonstrates the effectiveness of our approach, which can accelerate the inference of models on CIFAR-10 and ImageNet datasets.

Guangli Li, Lei Liu, Xueying Wang et al.

Accelerating deep convolutional neural networks has become an active topic and sparked an interest in academia and industry. In this paper, we propose an efficient low-precision quantized Winograd convolution algorithm, called LANCE, which combines the advantages of fast convolution and quantization techniques. By embedding linear quantization operations into the Winograd-domain, the fast convolution can be performed efficiently under low-precision computation on graphics processing units. We test neural network models with LANCE on representative image classification datasets, including SVHN, CIFAR, and ImageNet. The experimental results show that our 8-bit quantized Winograd convolution improves the performance by up to 2.40x over the full-precision convolution with trivial accuracy loss.

Qiang Du, Xiaobing Feng

This paper surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the paper presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The paper also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science.

Xiaobing Feng, Hailiang Liu, Shu Ma

In this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schrödinger equations. The proposed schemes all satisfy both mass conservation and energy conservation. Truncation and dispersion error analyses are provided for each proposed scheme. Efficient fixed-point iterative solvers are also constructed to solve the resulting nonlinear discrete problems. As a byproduct, an efficient one-step implementation of the BDF schemes is obtained as well. Extensive numerical experiments are presented to demonstrate the convergence and the capability of capturing the blow-up phenomenon of the proposed schemes.

Xueying Wang, Lei Liu, Guangli Li et al.

Background subtraction is a significant component of computer vision systems. It is widely used in video surveillance, object tracking, anomaly detection, etc. A new data source for background subtraction appeared as the emergence of low-cost depth sensors like Microsof t Kinect, Asus Xtion PRO, etc. In this paper, we propose a background subtraction approach on depth videos, which is based on convolutional neural networks (CNNs), called BGSNet-D (BackGround Subtraction neural Networks for Depth videos). The method can be used in color unavailable scenarios like poor lighting situations, and can also be applied to combine with existing RGB background subtraction methods. A preprocessing strategy is designed to reduce the influences incurred by noise from depth sensors. The experimental results on the SBM-RGBD dataset show that the proposed method outperforms existing methods on depth data.

Guangli Li, Lei Liu, Xueying Wang et al.

Recently, deep neural networks (DNNs) have been widely applied in mobile intelligent applications. The inference for the DNNs is usually performed in the cloud. However, it leads to a large overhead of transmitting data via wireless network. In this paper, we demonstrate the advantages of the cloud-edge collaborative inference with quantization. By analyzing the characteristics of layers in DNNs, an auto-tuning neural network quantization framework for collaborative inference is proposed. We study the effectiveness of mixed-precision collaborative inference of state-of-the-art DNNs by using ImageNet dataset. The experimental results show that our framework can generate reasonable network partitions and reduce the storage on mobile devices with trivial loss of accuracy.

Xiaobing Feng, Max Jensen

This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Ampère equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the Monge-Ampère equation. A significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built-in property of the new formulation. Moreover, this new approach allows one to tap the wealthy numerical methods, such as semi-Lagrangian schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Ampère type equations. It is proved that the considered numerical methods are monotone, pointwise consistent and uniformly stable. Consequently, its solutions converge uniformly to the unique convex viscosity solution of the Monge-Ampère Dirichlet problem. A super-linearly convergent Howard's algorithm, which is a Newton type method, is utilized as the nonlinear solver to take advantage of the monotonicity of the scheme. Numerical experiments are also presented to gauge the performance of the proposed numerical method and the nonlinear solver.

Xiaobing Feng, Yukun Li, Yi Zhang

This paper studies finite element approximations of the stochastic Allen-Cahn equation with gradient-type multiplicative noises that are white in time and correlated in space. The sharp interface limit as the parameter $ε\rightarrow 0$ of the stochastic equation formally approximates a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow. Both the stochastic Allen-Cahn equation and the stochastic mean curvature flow arise from materials science, fluid mechanics and cell biology applications. Two fully discrete finite element methods which are based on different time-stepping strategies for the nonlinear term are proposed. Strong convergence with sharp rates for both fully discrete finite element methods is proved. This is done with a crucial help of the Hölder continuity in time with respect to the spatial $L^2$-norm and $H^1$-seminorm for the strong solution of the stochastic Allen-Cahn equation, which are key technical lemmas proved in paper. It also relies on the fact that high moments of the strong solution are bounded in various spatial and temporal norms. Numerical experiments are provided to gauge the performance of the proposed fully discrete finite element methods and to study the interplay of the geometric evolution and gradient-type noises.

Xiaobing Feng, Lauren Hennings, Michael Neilan

This paper is concerned with finite element approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. A nonstandard (primal) finite element method, which uses finite-dimensional subspaces consisting globally continuous piecewise polynomial functions, is proposed and analyzed. The main novelty of the finite element method is to introduce an interior penalty term, which penalizes the jump of the flux across the interior element edges/faces, to augment a nonsymmetric piecewise defined and PDE-induced bilinear form. Existence, uniqueness and error estimate in a discrete $W^{2,p}$ energy norm are proved for the proposed finite element method. This is achieved by establishing a discrete Calderon-Zygmund-type estimate and mimicking strong solution PDE techniques at the discrete level. Numerical experiments are provided to test the performance of proposed finite element method and to validate the convergence theory.

Shuchang Zhou, Zhihua Zhang, Xiaobing Feng

In this paper we propose and study an optimization problem over a matrix group orbit that we call \emph{Group Orbit Optimization} (GOO). We prove that GOO can be used to induce matrix decomposition techniques such as singular value decomposition (SVD), LU decomposition, QR decomposition, Schur decomposition and Cholesky decomposition, etc. This gives rise to a unified framework for matrix decomposition and allows us to bridge these matrix decomposition methods. Moreover, we generalize GOO for tensor decomposition. As a concrete application of GOO, we devise a new data decomposition method over a special linear group to normalize point cloud data. Experiment results show that our normalization method is able to obtain recovery well from distortions like shearing, rotation and squeezing.

Xiaobing Feng, Zhihao Ge, Yukun Li

This paper concerns with finite element approximations of a quasi-static poroelasticity model in displacement-pressure formulation which describes the dynamics of poro-elastic materials under an applied mechanical force on the boundary. To better describe the multiphysics process of deformation and diffusion for poro-elastic materials, we first present a reformulation of the original model by introducing two pseudo-pressures, one of them is shown to satisfy a diffusion equation, we then propose a time-stepping algorithm which decouples (or couples) the reformulated PDE problem at each time step into two sub-problems, one of which is a generalized Stokes problem for the displacement vector field (of the solid network of the poro-elastic material) along with one pseudo-pressure field and the other is a diffusion problem for the other pseudo-pressure field (of the solvent of the material). In the paper, the Taylor-Hood mixed finite element method combined with the $P_1$-conforming finite element method is used as an example to demonstrate the viability of the proposed multiphysics approach. It is proved that the solutions of the fully discrete finite element methods fulfill a discrete energy law which mimics the differential energy law satisfied by the PDE solution and converges optimally in the energy norm. Moreover, it is showed that the proposed formulation also has a built-in mechanism to overcome so-called "locking phenomenon" associated with the numerical approximations of the poroelasticity model. Numerical experiments are presented to show the performance of the proposed approach and methods and to demonstrate the absence of "locking phenomenon" in our numerical experiments.

Xiaobing Feng, Haijun Wu

This paper develops some interior penalty $hp$-discontinuous Galerkin ($hp$-DG) methods for the Helmholtz equation in two and three dimensions. The proposed $hp$-DG methods are defined using a sesquilinear form which is not only mesh-dependent but also degree-dependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order $p$. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed $hp$-discontinuous Galerkin methods are absolutely stable (hence, well-posed). For each fixed wave number $k$, sub-optimal order error estimates in the broken $H^1$-norm and the $L^2$-norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition $k^3h^2p^{-1}\le C_0$ by utilizing these stability and error estimates and using a stability-error iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}, which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size $h$, the polynomial degree $p$, the wave number $k$, as well as all the penalty parameters for the numerical solutions.

Xiaobing Feng, Yinnian He

The goal of this paper is to develop and analyze some fully discrete finite element methods for a displacement-pressure model modeling swelling dynamics of polymer gels under mechanical constraints. In the model, the swelling dynamics is governed by the solvent permeation and the elastic interaction; the permeation is described by a pressure equation for the solvent, and the elastic interaction is described by displacement equations for the solid network of the gel. By introducing an "elastic pressure" we first present a reformulation of the original model, and then propose a time-stepping scheme which decouples the PDE system at each time step into two sub-problems, one of which is a generalized Stokes problem for the displacement vector field and another is a diffusion problem for a "pseudo-pressure" field. To make such a multiphysical approach feasible, it is vital to discover admissible constraints to resolve the uniqueness issue for both sub-problems. The main advantage of the proposed approach is that it allows one to utilize any convergent Stokes solver together with any convergent diffusion equation solver to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are used as an example to present the ideas and to demonstrate the viability of the proposed multiphysical approach. It is proved that, under a mesh constraint, both the proposed semi-discrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for the numerical solutions of both the semi-discrete and fully discrete methods. Numerical experiments are also presented to show the efficiency of the proposed approach and methods.