arXiv subjects

Kees Vuik

Publications and source records attributed to Kees Vuik.

5 recordsLinked to original sources

arXiv2025

The Dynamical Anatomy of Anderson Acceleration:From Adaptive Momentum to Variable-Mass ODEs

This paper provides a rigorous derivation and analysis of accelerated optimization algorithms through the lens of High-Resolution Ordinary Differential Equations (ODEs). While classical Nesterov acceleration is well-understood via asymptotic vanishing damping, the dynamics of Anderson Acceleration (AA) remain less transparent. This work makes significant theoretical contributions to AA by bridging discrete acceleration algorithms with continuous dynamical systems, while also providing practical algorithmic innovations. Our work addresses fundamental questions about the physical nature of Anderson Acceleration that have remained unanswered since its introduction in 1965. Firstly, we prove that AA can be exactly rewritten as an adaptive momentum method and, in the high-resolution limit, converges to a second-order ODE with Variable Effective Mass. Through a Lyapunov energy analysis, we reveal the specific instability mechanism of standard AA: unchecked growth in effective mass acts as negative damping, physically injecting energy into the system and violating dissipation constraints. Conversely, high-resolution analysis identifies an implicit Hessian-driven damping term that provides stabilization in stiff regimes. Leveraging these dynamical insights, we then propose Energy-Guarded Anderson Acceleration (EG-AA), an algorithm that acts as an inertial governor to enforce thermodynamic consistency. Morevoer, our convergence analysis, formulated via the Acceleration Gain Factor, proves that EG-AA improves upon gradient descent by maximizing the geometric contraction of the linear subspace projection while actively suppressing nonlinear approximation errors. Theoretical bounds confirm that EG-AA is no worse than standard AA, and numerical experiments demonstrate strictly improved convergence stability and rates in ill-conditioned convex composite problems compared to standard Anderson mixing.

math.OC↗

arXiv2020

Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation

Finding fast yet accurate numerical solutions to the Helmholtz equation remains a challenging task. The pollution error (i.e. the discrepancy between the numerical and analytical wave number k) requires the mesh resolution to be kept fine enough to obtain accurate solutions. A recent study showed that the use of Isogeometric Analysis (IgA) for the spatial discretization significantly reduces the pollution error. However, solving the resulting linear systems by means of a direct solver remains computationally expensive when large wave numbers or multiple dimensions are considered. An alternative lies in the use of (preconditioned) Krylov subspace methods. Recently, the use of the exact Complex Shifted Laplacian Preconditioner (CSLP) with a small complex shift has shown to lead to wave number independent convergence while obtaining more accurate numerical solutions using IgA. In this paper, we propose the use of deflation techniques combined with an approximated inverse of the CSLP using a geometric multigrid method. Numerical results obtained for both one- and two-dimensional model problems, including constant and non-constant wave numbers, show scalable convergence with respect to the wave number and approximation order p of the spatial discretization. Furthermore, when kh is kept constant, the proposed approach leads to a significant reduction of the computational time compared to the use of the exact inverse of the CSLP with a small shift.

math.NA↗

arXiv2017

A Stable SPH Discretization of the Elliptic Operator with Heterogeneous Coefficients

Smoothed particle hydrodynamics (SPH) has been extensively used to model high and low Reynolds number flows, free surface flows and collapse of dams, study pore-scale flow and dispersion, elasticity, and thermal problems. In different applications, it is required to have a stable and accurate discretization of the elliptic operator with homogeneous and heterogeneous coefficients. In this paper, the stability and approximation analysis of different SPH discretization schemes (traditional and new) of the diagonal elliptic operator for homogeneous and heterogeneous media are presented. The optimum and new discretization scheme is also proposed. This scheme enhances the Laplace approximation (Brookshaw's scheme (1985) and Schwaiger's scheme (2008)) used in the SPH community for thermal, viscous, and pressure projection problems with an isotropic elliptic operator. The numerical results are illustrated by numerical examples, where the comparison between different versions of the meshless discretization methods are presented.

math.NA↗

arXiv2017

A Modified and Calibrated Drift-Diffusion-Reaction Model for Time-Domain Analysis of Charging Phenomena in Electron-Beam Irradiated Insulators

This paper presents an improved version of the previously proposed self-consistent drift-diffusion-reaction model correcting for non-physical behavior at longer time scales. To this end a novel boundary condition is employed that takes into account the effect of tertiary electrons and a fully dynamic trap-assisted generation-recombination mechanism is implemented. Sensitivity of the model with respect to material parameters is investigated and a calibration procedure is developed that reproduces experimental yield-energy curves for uncharged insulators. Long-time charging and yield variations are analyzed for stationary defocused and focused beams as well as moving beams dynamically scanning composite insulators.

cond-mat.mtrl-sci↗

arXiv2015

Self-consistent drift-diffusion-reaction model for the electron beam interaction with dielectric samples

The charging of insulating samples degrades the quality and complicates the interpretation of images in scanning electron microscopy and is important in other applications, such as particle detectors. In this paper we analyze this nontrivial phenomenon on different time scales employing the drift-diffusion-reaction approach augmented with the trapping rate equations and a realistic semi-empirical source function describing the pulsed nature of the electron beam. We consider both the fast processes following the impact of a single primary electron, the slower dynamics resulting from the continuous bombardment of a sample, and the eventual approach to the steady-state regime.

physics.class-ph↗