arXiv subjects

Andreas Rupp

Publications and source records attributed to Andreas Rupp.

At least 37 records · Page 2Linked to original sources

arXiv2025

Numerical Simulation of Beam Network Models

Network models are used as efficient representation of materials with complex, interconnected locally one-dimensional structures. They typically accurately capture the mechanical properties of a material, while substantially reducing computational cost by avoiding full three-dimensional resolution. Applications include the simulation of fiber-based materials, porous media, and biological systems such as vascular networks. This article focuses on two representative problems: a stationary formulation describing the elastic deformation of beam networks, and a time-dependent formulation modeling elastic wave propagation in such materials. We propose a two-level additive domain decomposition method to efficiently solve the linear system associated with the stationary problem, as well as the linear systems that arise at each time step of the time-dependent problem through implicit time discretization. We present a rigorous convergence analysis of the domain decomposition method when used as a preconditioner, quantifying the convergence rate with respect to network connectivity and heterogeneity. The efficiency and robustness of the proposed approach are demonstrated through numerical simulations of the mechanical properties of commercial-grade paperboard.

math.NA↗

arXiv2025

A convergent Fourier spectral Galerkin method for the fractional Camassa-Holm equation

We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent $α\in [1,2]$ with periodic boundary conditions. The semi-discrete scheme preserves both mass and energy invariants of the fCH equation. For the fractional Benjamin-Bona-Mahony reduction, we establish existence and uniqueness of semi-discrete solutions and prove strong convergence to the unique solution in $ C^1([0, T];H^α_{\mathrm{per}}(I))$ for given $T>0$. For the general fCH equation, we demonstrate spectral accuracy in spatial discretization with optimal error estimates $\mathcal{O}(N^{-r})$ for initial data $u_0 \in H^r(I)$ with $r \geq α+ 2$ and exponential convergence $\mathcal{O}(e^{-cN})$ for smooth solutions. Numerical experiments validate orbital stability of solitary waves achieving optimal convergence, confirming theoretical findings.

math.NA↗

arXiv2025

On positivity preservation of hybrid discontinuous Galerkin methods on hypergraphs

Hybrid finite element methods, particularly hybridized discontinuous Galerkin (HDG) methods, are efficient numerical schemes for discretizing the diffusion equation, which encompasses two main physical principles: mass conservation and positivity preservation. While the former has been extensively analyzed in the literature, this paper investigates the latter. We state a theorem that guarantees the positivity of both the bulk and skeleton approximations to the primary unknown (concentration) and provide counterexamples for nonpositive discretizations. The theoretical findings are confirmed by numerical experiments.

math.NA↗

arXiv2025

Cellular Automaton With CNN

Cellular automata (CA) models are widely used to simulate complex systems with emergent behaviors, but identifying hidden parameters that govern their dynamics remains a significant challenge. This study explores the use of Convolutional Neural Networks (CNN) to identify jump parameters in a two-dimensional CA model. We propose a custom CNN architecture trained on CA-generated data to classify jump parameters, which dictates the neighborhood size and movement rules of cells within the CA. Experiments were conducted across varying domain sizes (25 x 25 to 150 x 150) and CA iterations (0 to 50), demonstrating that the accuracy improves with larger domain sizes, as they provide more spatial information for parameter estimation. Interestingly, while initial CA iterations enhance the performance, increasing the number of iterations beyond a certain threshold does not significantly improve accuracy, suggesting that only specific temporal information is relevant for parameter identification. The proposed CNN achieves competitive accuracy (89.31) compared to established architectures like LeNet-5 and AlexNet, while offering significantly faster inference times, making it suitable for real-time applications. This study highlights the potential of CNNs as a powerful tool for fast and accurate parameter estimation in CA models, paving the way for their use in more complex systems and higher-dimensional domains. Future work will explore the identification of multiple hidden parameters and extend the approach to three-dimensional CA models.

cs.LG↗

arXiv2025

Data-Efficient Limited-Angle CT Using Deep Priors and Regularization

Reconstructing an image from its Radon transform is a fundamental computed tomography (CT) task arising in applications such as X-ray scans. In many practical scenarios, a full 180-degree scan is not feasible, or there is a desire to reduce radiation exposure. In these limited-angle settings, the problem becomes ill-posed, and methods designed for full-view data often leave significant artifacts. We propose a very low-data approach to reconstruct the original image from its Radon transform under severe angle limitations. Because the inverse problem is ill-posed, we combine multiple regularization methods, including Total Variation, a sinogram filter, Deep Image Prior, and a patch-level autoencoder. We use a differentiable implementation of the Radon transform, which allows us to use gradient-based techniques to solve the inverse problem. Our method is evaluated on a dataset from the Helsinki Tomography Challenge 2022, where the goal is to reconstruct a binary disk from its limited-angle sinogram. We only use a total of 12 data points--eight for learning a prior and four for hyperparameter selection--and achieve results comparable to the best synthetic data-driven approaches.

cs.CV↗

arXiv2025

A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods

We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is the first step in reliably merging hybrid skeletal formulations and localized orthogonal decomposition to unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.

math.NA↗

arXiv2025

Parameter estimation for cellular automata

Self-organizing complex systems can be modeled using cellular automaton models. However, the parametrization of these models is crucial and significantly determines the resulting structural pattern. In this research, we introduce and successfully apply a sound statistical method to estimate these parameters. The decisive difference to earlier applications of such approaches is that, in our case, both the CA rules and the resulting patterns are discrete. The method is based on constructing Gaussian likelihoods using characteristics of the structures, such as the mean particle size. We show that our approach is robust for the method parameters, domain size of patterns, or CA iterations.

nlin.CG↗

arXiv2024

Denture reinforcement via topology optimization

We present a computational design method that optimizes the reinforcement of dentures and increases the stiffness of dentures. Our approach optimally places reinforcement in the denture, which modern multi-material three-dimensional printers could implement. The study focuses on reducing denture displacement by identifying regions that require reinforcement (E-glass material) with the help of topology optimization. Our method is applied to a three-dimensional complete lower jaw denture. We compare the displacement results of a non-reinforced denture and a reinforced denture that has two materials. The comparison results indicate that there is a decrease in the displacement in the reinforced denture. Considering node-based displacement distribution, the reinforcement reduces the displacement magnitudes in the reinforced denture compared to the non-reinforced denture. The study guides dental technicians on where to automatically place reinforcement in the fabrication process, helping them save time and reduce material usage.

cs.CE↗

arXiv2024

Quasi-Monte Carlo and discontinuous Galerkin

In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field, and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings.

math.NA↗

arXiv2024

Arbitrary order approximations at constant cost for Timoshenko beam network models

This paper considers the numerical solution of Timoshenko beam network models, comprised of Timoshenko beam equations on each edge of the network, which are coupled at the nodes of the network using rigid joint conditions. Through hybridization, we can equivalently reformulate the problem as a symmetric positive definite system of linear equations posed on the network nodes. This is possible since the nodes, where the beam equations are coupled, are zero-dimensional objects. To discretize the beam network model, we propose a hybridizable discontinuous Galerkin method that can achieve arbitrary orders of convergence under mesh refinement without increasing the size of the global system matrix. As a preconditioner for the typically very poorly conditioned global system matrix, we employ a two-level overlapping additive Schwarz method. We prove uniform convergence of the corresponding preconditioned conjugate gradient method under appropriate connectivity assumptions on the network. Numerical experiments support the theoretical findings of this work.

math.NA↗

arXiv2024

Strongly consistent low-dissipation WENO schemes for finite elements

We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.

math.NA↗

arXiv2024

Recovery of transversely-isotropic elastic material parameters in induction motor rotors

We propose numerical algorithms for recovering parameters in eigenvalue problems for linear elasticity of transversely isotropic materials. Specifically, the algorithms are used to recover the elastic constants of a rotor core. Numerical tests show that in the noiseless setup, two pairs of bending modes are sufficient for recovering one to four parameters accurately. To recover all five parameters that govern the elastic properties of electric engines accurately, we require three pairs of bending modes and one torsional mode. Moreover, we study the stability of the inversion method against multiplicative noise; for tests in which the data contained multiplicative noise of at most $1\%$, we find that all parameters can be recovered with an error less than $10\%$.

math.NA↗

arXiv2024

Homogeneous multigrid for hybrid discretizations: application to HHO methods

We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.

math.NA↗

arXiv2023

Error estimates for completely discrete FEM in energy-type and weaker norms

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion-convection-reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy-type norms and, by generalizing the Aubin-Nitsche technique, optimal error estimates in weaker norms.

math.NA↗

arXiv2023

Homogeneous multigrid method for HDG applied to the Stokes equation

We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart--Thomas, or a hybrid Brezzi--Douglas--Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Numerical experiments underline our analytical findings.

math.NA↗

arXiv2021

Partial differential equations on hypergraphs and networks of surfaces: derivation and hybrid discretizations

We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on hypergraphs as singular limits of PDEs in networks of thin domains (such as fault planes, pipes, etc.), and we observe that (mixed) hybrid formulations offer useful tools to formulate such PDEs. Thus, our numerical framework is based on hybrid finite element methods (in particular, the class of hybrid discontinuous Galerkin methods).

math.NA↗

arXiv2021

Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

The algebraic flux correction (AFC) schemes presented in this work constrain a standard continuous finite element discretization of a nonlinear hyperbolic problem to satisfy relevant maximum principles and entropy stability conditions. The desired properties are enforced by applying a limiter to antidiffusive fluxes that represent the difference between the high-order baseline scheme and a property-preserving approximation of Lax--Friedrichs type. In the first step of the limiting procedure, the given target fluxes are adjusted in a way that guarantees preservation of local and/or global bounds. In the second step, additional limiting is performed, if necessary,to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes considered in this work are applicable to finite element discretizations of scalar hyperbolic equations and systems alike. The underlying inequality constraints are formulated using Tadmor's entropy stability theory. The proposed limiters impose entropy-conservative or entropy-dissipative bounds on the rate of entropy production by antidiffusive fluxes and Runge--Kutta (RK) time discretizations. Two versions of the fully discrete entropy fix are developed for this purpose. To motivate the use of limiter-based entropy fixes, we prove a finite element version of the Lax--Wendroff theorem and perform numerical studies for standard test problems. In our numerical experiments, entropy-dissipative schemes converge to correct weak solutions of scalar conservation laws, of the Euler equations, and of the shallow water equations.

math.NA↗

arXiv2021

Analysis of algebraic flux correction for semi-discrete advection problems

We present stability and error analysis for algebraic flux correction schemes based on monolithic convex limiting. For a continuous finite element discretization of the time-dependent advection equation, we prove global-in-time existence and the worst-case convergence rate of 1/2 w.r.t. the L2 error of the spatial semi-discretization. Moreover, we address the important issue of stabilization for raw antidiffusive fluxes. Our a priori error analysis reveals that their limited counterparts should satisfy a generalized coercivity condition. We introduce a limiter for enforcing this condition in the process of flux correction. To verify the results of our theoretical studies, we perform numerical experiments for simple one-dimensional test problems. The methods under investigation exhibit the expected behavior in all numerical examples. In particular, the use of stabilized fluxes improves the accuracy of numerical solutions and coercivity enforcement often becomes redundant.

math.NA↗