arXiv subjects

Wenbin Chen

Publications and source records attributed to Wenbin Chen.

At least 19 recordsLinked to original sources

arXiv2026

Action-grounded tissue affordance enables anticipatory auto-framing that lowers surgeon cognitive workload during laparoscopic surgery

In laparoscopy, surgeon gaze tracks where the instruments will act; easing this demand through visual attention modeling requires dense labels of those interaction loci. These encode tacit knowledge: experts converge on consensus loci yet struggle to state the rules. Here we show that such labels can be recovered from completed actions in surgical videos, in which recorded instrument trajectories are converted into dense, continuous supervision. DiffeoAfford grounds tissue affordance by attaching instrument tips to the tissue and transporting them through deformation using diffeomorphism-constrained tracking, matching context-informed annotators' accuracy. Trained on these labels and never on gaze, a real-time model aligns with surgeon gaze more closely in space and time than does camera-assistant gaze. The framework also transfers across procedures: on hysterectomy videos, a separately trained predictor reaches 95.16% directional consistency with subsequent camera motion. In 12 paired cholecystectomies (24 procedures), the auto-framing application AffordView, which proactively centers predicted targets in view, lowered surgeon cognitive workload on converging subjective, physiological, and behavioral measures, including a reduced number of verbal instructions to the camera assistant. Deriving supervision from action rather than manual annotation offers a scalable route to anticipatory assistance.

cs.CV↗

arXiv2026

Nonconforming Finite Element Approximation and Energy Lower Bound Estimation for the Gross--Pitaevskii Energy Functional

The ground state of Bose--Einstein condensates can be described as the minimizer of the Gross--Pitaevskii energy functional subject to a mass conservation constraint. In this paper, we study the corresponding discrete optimization problem in nonconforming finite element spaces and establish a priori error estimates for the discrete ground state energy, the discrete eigenvalue, and the discrete ground state. Specifically, we derive explicit convergence rates for the a priori error in the particular case of the $EQ_1^{\mathrm{rot}}$ finite element. Furthermore, we proof that within the $EQ_1^{\mathrm{rot}}$ finite element framework, the discrete ground state energy provides a lower bound estimation to the exact energy. Finally, numerical experiments are presented to validate the theoretical analysis.

math.NA↗

arXiv2025

Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity

We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution.

math.NA↗

arXiv2025

Convergence analysis of Sobolev Gradient flows for the rotating Gross-Pitaevskii energy functional

This paper studies the numerical approximation of the ground state of rotating Bose--Einstein condensates, formulated as the minimization of the Gross--Pitaevskii energy functional under a mass conservation constraint. To solve this problem, we consider three Sobolev gradient flow schemes: the $H_0^1$ scheme, the $a_0$ scheme, and the $a_u$ scheme. Convergence of these schemes in the non-rotating case was established by Chen et al., and the rotating $a_u$ scheme was analyzed in Henning et al. In this work, we prove the global convergence of the $H_0^1$ and $a_0$ schemes in the rotating case, and establish local linear convergence for all three schemes near the ground state. Numerical experiments confirm our theoretical findings.

math.NA↗

arXiv2025

Adaptive time-stepping and maximum-principle preserving Lagrangian schemes for gradient flows

We develop in this paper an adaptive time-stepping approach for gradient flows with distinct treatments for conservative and non-conservative dynamics. For the non-conservative gradient flows in Lagrangian coordinates, we propose a modified formulation augmented by auxiliary terms to guarantee positivity of the determinant, and prove that the corresponding adaptive second-order Backward Difference Formulas (BDF2) scheme preserves energy stability and the maximum principle under the time-step ratio constraint $0<r_n\le r_{\max}\le\frac{3}{2}$. On the other hand, for the conservative Wasserstein gradient flows in Lagrangian coordinates, we propose an adaptive BDF2 scheme which is shown to be energy dissipative, and positivity preserving under the time-step ratio constraint $0<r_n\le r_{\max}\le\frac{3+\sqrt{17}}{2}$ in 1D and $0<r_n\le r_{\max}\le \frac{5}{4}$ in 2D, respectively. We also present ample numerical simulations in 1D and 2D to validate the efficiency and accuracy of the proposed schemes.

math.NA↗

arXiv2024

Two Kinds of Learning Algorithms for Continuous-Time VWAP Targeting Execution

The optimal execution problem has always been a continuously focused research issue, and many reinforcement learning (RL) algorithms have been studied. In this article, we consider the execution problem of targeting the volume weighted average price (VWAP) and propose a relaxed stochastic optimization problem with an entropy regularizer to encourage more exploration. We derive the explicit formula of the optimal policy, which is Gaussian distributed, with its mean value being the solution to the original problem. Extending the framework of continuous RL to processes with jumps, we provide some theoretical proofs for RL algorithms. First, minimizing the martingale loss function leads to the optimal parameter estimates in the mean-square sense, and the second algorithm is to use the martingale orthogonality condition. In addition to the RL algorithm, we also propose another learning algorithm: adaptive dynamic programming (ADP) algorithm, and verify the performance of both in two different environments across different random seeds. Convergence of all algorithms has been verified in different environments, and shows a larger advantage in the environment with stronger price impact. ADP is a good choice when the agent fully understands the environment and can estimate the parameters well. On the other hand, RL algorithms do not require any model assumptions or parameter estimation, and are able to learn directly from interactions with the environment.

math.OC↗

arXiv2024

A new flow dynamic approach for Wasserstein gradient flows

We develop in this paper a new regularized flow dynamic approach to construct efficient numerical schemes for Wasserstein gradient flows in Lagrangian coordinates. Instead of approximating the Wasserstein distance which needs to solve constrained minimization problems, we reformulate the problem using the Benamou-Brenier's flow dynamic approach, leading to algorithms which only need to solve unconstrained minimization problem in $L^2$ distance. Our schemes automatically inherit some essential properties of Wasserstein gradient systems such as positivity-preserving, mass conservative and energy dissipation. We present ample numerical simulations of Porous-Medium equations, Keller-Segel equations and Aggregation equations to validate the accuracy and stability of the proposed schemes. Compared to numerical schemes in Eulerian coordinates, our new schemes can capture sharp interfaces for various Wasserstein gradient flows using relatively smaller number of unknowns.

math.NA↗

arXiv2024

Convergence analysis of a second order numerical scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn-Hilliard-Navier-Stokes (CHNS) system, combined with logarithmic Flory-Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard $\ell^\infty (0, T; \ell^2) \cap \ell^2 (0, T; H_h^2)$ error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the $\ell^\infty (0, T; H_h^1) \cap \ell^2 (0, T; H_h^3)$ error analysis for the phase variable and the $\ell^\infty (0, T; \ell^2)$ analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result.

math.NA↗

arXiv2024

EPIC: a provable accelerated Eigensolver based on Preconditioning and Implicit Convexity

This paper is concerned with the extraction of the smallest eigenvalue and the corresponding eigenvector of a symmetric positive definite matrix pencil. We reveal implicit convexity of the eigenvalue problem in Euclidean space. A provable accelerated eigensolver based on preconditioning and implicit convexity (EPIC) is proposed. Theoretical analysis shows the acceleration of EPIC with the rate of convergence resembling the expected rate of convergence of the well-known locally optimal preconditioned conjugate gradient (LOPCG). A complete proof of the expected rate of convergence of LOPCG is elusive so far. Numerical results confirm our theoretical findings of EPIC.

math.NA↗

arXiv2023

Investigation on the Compressibility Characteristics of Low Mach Number Laminar Flow in Rotating Channel

In high-speed rotating channels, significant compressive effects are observed, resulting in distinct flow characteristics compared to incompressible flows. In this study, we employed a finite volume method based on the simple algorithm to solve for low-speed compressible laminar flow within rotating channels using an orthogonal uniform grid. The governing equations include the full Navier-Stokes equations and the energy equation. Contrary to stationary channel, the alterations in flow within rotating channel are primarily influenced by the compressive effects of centrifugal force and the compressibility of fluid within the flow's normal section. The first effect involves a reduction in the velocity due to centrifugal force, leading to an increasing influence of the Coriolis force compared to inertial forces along the flow direction. This trend in axial changes aligns closely with the increase in rotation speed. The second effect arises from the increase in Mach number and the Coriolis compression, resulting in slight density differences within the cross-section. Strong centrifugal forces generate significant centrifugal additional force (buoyancy force). Consequently, under the same local rotation number, the velocity profiles of the mainstream experience considerable changes. Additionally, higher Mach number significantly impact wall shear stress, with the leading side being notably affected. For instance, at a cross-sectional Ro = 0.6 and Ma = 0.035, the dimensionless shear stress on the leading side decreased by 13%. Furthermore, while an increase in Mach number has minimal impact on the cross-sectional secondary flow structure, changes in mainstream velocity profiles influence secondary flow intensity, resulting in an enhanced velocity peak and a shift towards the trailing side.

physics.flu-dyn↗

arXiv2023

Study of the Effect of a Novel Dimensionless Parameter -- the Centrifugal Work Number(CW), on Spanwise Rotating channel Low-speed Compressible Flow

In the study of rotating channel flow, the key dimensionless parameters typically include the Reynolds number, rotation number, Prandtl number and buoyancy number. Our research focused on comparing the flow characteristics between the enlarged model, analyzed under the rotating similarity theory, and the original channel flow. Significantly different flow behaviors were observed between these two cases. Through theoretical derivation and dimensional analysis, we identified a new significant parameter - the centrifugal work number (CW). This parameter characterizes the ratio of centrifugal work to gas enthalpy in the rotating channel and plays a crucial role in measuring the compressibility of fluids within the rotating channel. Additionally, we utilized large eddy simulation(LES) to validate the impact of the centrifugal work ratio on the flow state of the rotating channel, thus enhancing the similarity theory of rotating channel compressible flow.

physics.flu-dyn↗

arXiv2023

Riemannian Acceleration with Preconditioning for symmetric eigenvalue problems

The analysis of the acceleration behavior of gradient-based eigensolvers with preconditioning presents a substantial theoretical challenge. In this work, we present a novel framework for preconditioning on Riemannian manifolds and introduce a metric, the leading angle, to evaluate preconditioners for symmetric eigenvalue problems. We extend the locally optimal Riemannian accelerated gradient method for Riemannian convex optimization to develop the Riemannian Acceleration with Preconditioning (RAP) method for symmetric eigenvalue problems, thereby providing theoretical evidence to support its acceleration. Our analysis of the Schwarz preconditioner for elliptic eigenvalue problems demonstrates that RAP achieves a convergence rate of $1-C\kappa^{-1/2}$, which is an improvement over the preconditioned steepest descent method's rate of $1-C\kappa^{-1}$. The exponent in $\kappa^{-1/2}$ is sharp, and numerical experiments confirm our theoretical findings.

math.NA↗

arXiv2023

A Uniquely Solvable, Positivity-Preserving and Unconditionally Energy Stable Numerical Scheme for the Functionalized Cahn-Hilliard Equation with Logarithmic Potential

We propose and analyze a first-order finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Huggins potential. The semi-implicit numerical scheme is designed based on a suitable convex-concave decomposition of the FCH free energy. We prove unique solvability of the numerical algorithm and verify its unconditional energy stability without any restriction on the time step size. Thanks to the singular nature of the logarithmic part in the Flory-Huggins potential near the pure states $\pm 1$, we establish the so-called positivity-preserving property for the phase function at a theoretic level. As a consequence, the numerical solutions will never reach the singular values $\pm 1$ in the point-wise sense and the fully discrete scheme is well defined at each time step. Next, we present a detailed optimal rate convergence analysis and derive error estimates in $l^{\infty}(0,T;L_h^2)\cap l^2(0,T;H^3_h)$ under a linear refinement requirement $Δt\leq C_1 h$. To achieve the goal, a higher order asymptotic expansion (up to the second order temporal and spatial accuracy) based on the Fourier projection is utilized to control the discrete maximum norm of solutions to the numerical scheme. We show that if the exact solution to the continuous problem is strictly separated from the pure states $\pm 1$, then the numerical solutions can be kept away from $\pm 1$ by a positive distance that is uniform with respect to the size of the time step and the grid. Finally, a few numerical experiments are presented. Convergence test is performed to demonstrate the accuracy and robustness of the proposed numerical scheme. Pearling bifurcation, meandering instability and spinodal decomposition are observed in the numerical simulations.

math.NA↗

arXiv2022

Error estimates and blow-up analysis of a finite-element approximation for the parabolic-elliptic Keller-Segel system

The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, a new fully discrete scheme for this model was proposed in [46], mass conservation, positivity and energy decay were proved for the proposed scheme, which are important properties of the original system. In this paper, we establish the error estimates of this scheme. Then, based on the error estimates, we derive the finite-time blowup of nonradial numerical solutions under some conditions on the mass and the moment of the initial data.

math.NA↗

arXiv2022

A Review of Machine Learning-based Failure Management in Optical Networks

Failure management plays a significant role in optical networks. It ensures secure operation, mitigates potential risks, and executes proactive protection. Machine learning (ML) is considered to be an extremely powerful technique for performing comprehensive data analysis and complex network management and is widely utilized for failure management in optical networks to revolutionize the conventional manual methods. In this study, the background of failure management is introduced, where typical failure tasks, physical objects, ML algorithms, data source, and extracted information are illustrated in detail. An overview of the applications of ML in failure management is provided in terms of alarm analysis, failure prediction, failure detection, failure localization, and failure identification. Finally, the future directions on ML for failure management are discussed from the perspective of data, model, task, and emerging techniques.

cs.NI↗

arXiv2022

Convergence analysis of the Newton-Schur method for the symmetric elliptic eigenvalue problem

In this paper, we consider the Newton-Schur method in Hilbert space and obtain quadratic convergence. For the symmetric elliptic eigenvalue problem discretized by the standard finite element method and non-overlapping domain decomposition method, we use the Steklov-Poincaré operator to reduce the eigenvalue problem on the domain $Ω$ into the nonlinear eigenvalue subproblem on $Γ$, which is the union of subdomain boundaries. We prove that the convergence rate for the Newton-Schur method is $ε_{N}\leq CH^{2}(1+\ln(H/h))^{2}ε^{2}$, where the constant $C$ is independent of the fine mesh size $h$ and coarse mesh size $H$, and $ε_{N}$ and $ε$ are errors after and before one iteration step respectively. Numerical experiments confirm our theoretical analysis.

math.NA↗

arXiv2022

A Weak Galerkin Mixed Finite Element Method for second order elliptic equations on 2D Curved Domains

This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary condition in this case. It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretization with polynomial order $α>1$. The purpose of this paper is two-fold. First, we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains, which exhibits an $O(h^{1/2})$ convergence for all $α\ge 1$. It is a little surprising to see that even the lowest-order WG-MFEM ($α=1$) experiences a loss of accuracy. This is different from known results for the finite element method (FEM) or the mixed FEM, and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain. Second, we propose a remedy to bring the approximation rate back to optimal by employing two techniques. One is a specially designed boundary correction technique. The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes, which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements. Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all $α$. Numerical results further confirm this conclusion.

math.NA↗

arXiv2021

Error estimate of a decoupled numerical scheme for the Cahn-Hilliard-Stokes-Darcy system

We analyze a fully discrete finite element numerical scheme for the Cahn-Hilliard-Stokes-Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn-Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in~\cite{CHW2017}. In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e.,, in the $\ell^\infty (0, T; H^1) \cap \ell^2 (0, T; H^2)$ norm for the phase variables, as well as in the $\ell^\infty (0, T; H^1) \cap \ell^2 (0, T; H^2)$ norm for the velocity variable. Such an energy norm error estimate leads to a cancellation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete $\ell^2 (0;T; H^3)$ bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo-Nirenberg inequality in the finite element setting.

math.NA↗