Towards uncertainty quantification for PDEs on networks
Duration: TBD–TBD
Funding scheme: Research Grants Program (Sachbeihilfe)
Funder: German Research Foundation
Budget: 245,708 €
My role: Principal investigator
Predecessor: Uncertainty quantification for PDEs on hypergraphs
Public description (English)
3D printing is becoming ubiquitous in engineering and science. One of the main reasons for such success is its ability to create small structures not producible in any other known way. Typical examples include lightweight but strong materials (resembling, e.g., honeycombs) and artificial tissues. Such materials need to provide specific properties, while production is subject to uncertainties in the printing process. This project grows out of the need for mathematical algorithms to find optimal structures that retain their outstanding properties even in the presence of small errors. Robustness is vital for lightweight materials used to build lighter cars, planes, and rockets that save fuel. Similarly, 3D-printed artificial tissue has to mimic the real human tissue of fire victims to a high degree.
Compared to the most efficient existing approaches, the methodology of this project reduces the cost of an optimization-based product design cycle by orders of magnitude.
The picture shows micro-fiber scaffolds that are one artificial tissue component. It is taken from Mechanical behavior of a soft hydrogel reinforced with three-dimensional printed microfibre scaffolds by Miguel Castilho et al., licensed under CC BY 4.0.
Public description (German) – Allgemeinverständliche Zusammenfassung
Der 3D-Druck wird in Technik und Wissenschaft immer allgegenwärtiger. Einer der Hauptgründe für diesen Erfolg ist seine Fähigkeit, kleine Strukturen zu schaffen, die auf keine andere Weise hergestellt werden können. Typische Beispiele sind leichte, aber stabile Materialien (z. B. Bienenwaben) und künstliches Gewebe. Solche Materialien benötigen spezifische Eigenschaften, und ihre Herstellung ist mit Unsicherheiten im Druckprozess behaftet. Dieses Projekt erwächst aus dem Bedarf an mathematischen Algorithmen, um optimale Strukturen zu finden, die ihre hervorragenden Eigenschaften auch bei kleinen Fehlern beibehalten. Robustheit ist entscheidend für leichte Materialien, die zum Bau leichterer Autos, Flugzeuge und Raketen verwendet werden, um Treibstoff zu sparen. In ähnlicher Weise muss 3D-gedrucktes künstliches Gewebe das echte menschliche Gewebe von Brandopfern in hohem Maße imitieren.
Im Vergleich zu den effizientesten bestehenden Ansätzen senkt die Methodik dieses Projekts die Kosten eines optimierungsbasierten Produktentwicklungszyklus um Größenordnungen.
Das Bild zeigt Mikrofaser-Gerüste, die eine künstliche Gewebekomponente darstellen. Es stammt aus Mechanical behavior of a soft hydrogel reinforced with three-dimensional printed microfibre scaffolds von Miguel Castilho et al. und ist unter der Lizenz CC BY 4.0 veröffentlicht.
Related publications
Krumbiegel, Felix; Rogava, Jemal; Rupp, Andreas; Vashakidze, Zurab
Conservative three-layer schemes for Kirchhoff-type equations Online Forthcoming
Forthcoming.
@online{KrumbiegelRRV26,
title = {Conservative three-layer schemes for Kirchhoff-type equations},
author = {Felix Krumbiegel and Jemal Rogava and Andreas Rupp and Zurab Vashakidze},
url = {https://doi.org/10.48550/arXiv.2609.27707},
doi = {10.48550/arXiv.2609.27707},
year = {2026},
date = {2026-09-23},
abstract = {We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial--boundary value problem. We consider two symmetric three-layer schemes of Crank--Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Grönwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.},
keywords = {},
pubstate = {forthcoming},
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We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial--boundary value problem. We consider two symmetric three-layer schemes of Crank--Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Grönwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.
Ashu, Valery; Peng, Wenqing; Liu, Zhi-Song; Haario, Heikki; Rupp, Andreas; Ashu, Taiwo; Clusius, Petri; Pichelstorfer, Lukas; Fu, Zihao; Boy, Michael
Evaluation of optimisation and Bayesian inference methods for reaction rates in atmospheric chemical mechanisms Online Forthcoming
Forthcoming, visited: 15.09.2026.
@online{AshuPLHRACPB26,
title = {Evaluation of optimisation and Bayesian inference methods for reaction rates in atmospheric chemical mechanisms},
author = {Valery Ashu and Wenqing Peng and Zhi-Song Liu and Heikki Haario and Andreas Rupp and Taiwo Ashu and Petri Clusius and Lukas Pichelstorfer and Zihao Fu and Michael Boy},
url = {https://doi.org/10.48550/arXiv.2609.14569},
doi = {10.48550/arXiv.2609.14569},
year = {2026},
date = {2026-09-15},
urldate = {2026-09-15},
abstract = {Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.
Liu, Zhi-Song; Büttner, Markus; Scarborough, Matthew; Valseth, Eirik; Aizinger, Vadym; Kainz, Bernhard; Rupp, Andreas
Downscaling neural network for coastal simulations Journal Article
In: Environmental Modelling & Software, vol. 205, no. 107147, 2026, ISSN: 1364-8152.
@article{LiuBSVAKR26,
title = {Downscaling neural network for coastal simulations},
author = {Zhi-Song Liu and Markus Büttner and Matthew Scarborough and Eirik Valseth and Vadym Aizinger and Bernhard Kainz and Andreas Rupp},
url = {https://arxiv.org/abs/2408.16553},
doi = {10.1016/j.envsoft.2026.107147},
issn = {1364-8152},
year = {2026},
date = {2026-09-01},
urldate = {2026-09-01},
journal = {Environmental Modelling & Software},
volume = {205},
number = {107147},
abstract = {Learning the fine-scale details of a coastal ocean simulation from a coarse representation is a challenging task. For real-world applications, high-resolution simulations are necessary to advance understanding of many coastal processes, specifically, to predict flooding resulting from tsunamis and storm surges. We propose a Downscaling Neural Network for Coastal Simulation (DNNCS) for spatiotemporal enhancement to learn the high-resolution numerical solution. Given images of coastal simulations produced on low-resolution computational meshes using low polynomial order discontinuous Galerkin discretizations and a coarse temporal resolution, the proposed DNNCS learns to produce high-resolution free surface elevation and velocity visualizations in both time and space. To model the dynamic changes over time and space, we propose grid-aware spatiotemporal attention to project the temporal features to the spatial domain for non-local feature matching. The coordinate information is also utilized via positional encoding. For the final reconstruction, we use the spatiotemporal bilinear operation to interpolate the missing frames and then expand the feature maps to the frequency domain for residual mapping. Besides data-driven losses, the proposed physics-informed loss guarantees gradient consistency and momentum changes, leading to a 24% reduction in root-mean-square error compared to the model trained with only data-driven losses. Our DNNCS is trained on a coastal simulation dataset and uses it for model optimization and evaluation. Our method shows superior downscaling quality and fast computation compared to the state-of-the-art methods.},
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pubstate = {published},
tppubtype = {article}
}
Learning the fine-scale details of a coastal ocean simulation from a coarse representation is a challenging task. For real-world applications, high-resolution simulations are necessary to advance understanding of many coastal processes, specifically, to predict flooding resulting from tsunamis and storm surges. We propose a Downscaling Neural Network for Coastal Simulation (DNNCS) for spatiotemporal enhancement to learn the high-resolution numerical solution. Given images of coastal simulations produced on low-resolution computational meshes using low polynomial order discontinuous Galerkin discretizations and a coarse temporal resolution, the proposed DNNCS learns to produce high-resolution free surface elevation and velocity visualizations in both time and space. To model the dynamic changes over time and space, we propose grid-aware spatiotemporal attention to project the temporal features to the spatial domain for non-local feature matching. The coordinate information is also utilized via positional encoding. For the final reconstruction, we use the spatiotemporal bilinear operation to interpolate the missing frames and then expand the feature maps to the frequency domain for residual mapping. Besides data-driven losses, the proposed physics-informed loss guarantees gradient consistency and momentum changes, leading to a 24% reduction in root-mean-square error compared to the model trained with only data-driven losses. Our DNNCS is trained on a coastal simulation dataset and uses it for model optimization and evaluation. Our method shows superior downscaling quality and fast computation compared to the state-of-the-art methods.
Hauck, Moritz; Holten, Joseph; Målqvist, Axel; Rupp, Andreas; Swoboda, Lucia
A hybridizable discontinuous Galerkin method for wave propagation in elastic beam networks Online Forthcoming
Forthcoming, visited: 24.08.2026.
@online{HauckHMRS26,
title = {A hybridizable discontinuous Galerkin method for wave propagation in elastic beam networks},
author = {Moritz Hauck and Joseph Holten and Axel Målqvist and Andreas Rupp and Lucia Swoboda},
url = {https://doi.org/10.48550/arXiv.2608.21091},
doi = {10.48550/arXiv.2608.21091},
year = {2026},
date = {2026-08-24},
urldate = {2026-08-24},
abstract = {This paper studies the numerical solution of elastic wave propagation on networks, modeled by elastodynamic equations posed on each edge, coupled at the nodes through suitable transmission conditions. We propose and analyze a hybridizable discontinuous Galerkin method that exploits the network structure to reduce the global problem at each time step to a linear system whose size depends only on the number of network nodes and not on the polynomial degree of the discretization. Combining it with an energy-conservative implicit time discretization, we derive a priori error estimates of optimal order in space and time. The implicit time discretization avoids the severe CFL restriction caused by the large variation in fiber segment lengths. To efficiently solve the resulting, typically ill-conditioned global system, we introduce a two-level overlapping additive Schwarz preconditioner. Under suitable assumptions on the network, we establish uniform convergence of the resulting preconditioned conjugate gradient method. Numerical experiments confirm the theoretical findings.},
keywords = {},
pubstate = {forthcoming},
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This paper studies the numerical solution of elastic wave propagation on networks, modeled by elastodynamic equations posed on each edge, coupled at the nodes through suitable transmission conditions. We propose and analyze a hybridizable discontinuous Galerkin method that exploits the network structure to reduce the global problem at each time step to a linear system whose size depends only on the number of network nodes and not on the polynomial degree of the discretization. Combining it with an energy-conservative implicit time discretization, we derive a priori error estimates of optimal order in space and time. The implicit time discretization avoids the severe CFL restriction caused by the large variation in fiber segment lengths. To efficiently solve the resulting, typically ill-conditioned global system, we introduce a two-level overlapping additive Schwarz preconditioner. Under suitable assumptions on the network, we establish uniform convergence of the resulting preconditioned conjugate gradient method. Numerical experiments confirm the theoretical findings.
Barrenechea, Gabriel Raúl; Lederer, Philip Lukas; Rupp, Andreas
A bound-preserving and conservative enriched Galerkin method for elliptic problems Journal Article
In: SIAM Journal on Scientific Computing, vol. 48, no. 3, pp. A1761-A1784, 2026, ISSN: 1095-7197.
@article{BarrenecheaLR26,
title = {A bound-preserving and conservative enriched Galerkin method for elliptic problems},
author = {Gabriel Raúl Barrenechea and Philip Lukas Lederer and Andreas Rupp},
url = {https://epubs.siam.org/eprint/RFIMYFEFKUCBYFXW22EI/full
https://docs.rupp.ink/publications/BarrenecheaLR26.pdf},
doi = {10.1137/25M1780493},
issn = {1095-7197},
year = {2026},
date = {2026-06-17},
urldate = {2026-06-17},
journal = {SIAM Journal on Scientific Computing},
volume = {48},
number = {3},
pages = {A1761-A1784},
abstract = {We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial overpenalization of the discrete solution’s jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in overpenalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial overpenalization of the discrete solution’s jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in overpenalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.
Dwivedi, Mukul; Rupp, Andreas
A convergent hybridizable discontinuous Galerkin method for Einstein–scalar equations Online Forthcoming
Forthcoming, visited: 06.04.2026.
@online{DwivediR26b,
title = {A convergent hybridizable discontinuous Galerkin method for Einstein–scalar equations},
author = {Mukul Dwivedi and Andreas Rupp},
url = {https://doi.org/10.48550/arXiv.2604.04613},
doi = {10.48550/arXiv.2604.04613},
year = {2026},
date = {2026-04-06},
urldate = {2026-04-06},
abstract = {We propose and analyze a hybridized discontinuous Galerkin (HDG) method for the spherically symmetric Einstein–scalar system in Bondi gauge. After rewriting the model as a local first-order PDE–ODE system by introducing suitable scaled variables, we construct a semidiscrete scheme in which the element unknowns are computed locally and the coupling is carried by traces on the mesh skeleton. In the present radial setting, these traces can be eliminated recursively, so that only the main evolution variable is advanced in time, while the metric variables are recovered from discrete constraint relations. We prove local semidiscrete well-posedness, derive a global L2 stability estimate, establish an optimal order L2 error bound for the main evolution variable for polynomial degree (larger than or equal to 1), and obtain reconstruction error estimates for the metric variables and the associated mass functional. Numerical experiments verify the predicted spatial convergence rate and illustrate qualitative features of the Einstein–scalar dynamics, including large-data collapse profiles and smooth-pulse evolution.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
We propose and analyze a hybridized discontinuous Galerkin (HDG) method for the spherically symmetric Einstein–scalar system in Bondi gauge. After rewriting the model as a local first-order PDE–ODE system by introducing suitable scaled variables, we construct a semidiscrete scheme in which the element unknowns are computed locally and the coupling is carried by traces on the mesh skeleton. In the present radial setting, these traces can be eliminated recursively, so that only the main evolution variable is advanced in time, while the metric variables are recovered from discrete constraint relations. We prove local semidiscrete well-posedness, derive a global L2 stability estimate, establish an optimal order L2 error bound for the main evolution variable for polynomial degree (larger than or equal to 1), and obtain reconstruction error estimates for the metric variables and the associated mass functional. Numerical experiments verify the predicted spatial convergence rate and illustrate qualitative features of the Einstein–scalar dynamics, including large-data collapse profiles and smooth-pulse evolution.
Dwivedi, Mukul; Railo, Jesse; Rupp, Andreas
A Galerkin finite element method for the fractional Calderón problem Online Forthcoming
Forthcoming, visited: 27.03.2026.
@online{DwivediRR26,
title = {A Galerkin finite element method for the fractional Calderón problem},
author = {Mukul Dwivedi and Jesse Railo and Andreas Rupp},
url = {https://doi.org/10.48550/arXiv.2603.26287},
doi = {10.48550/arXiv.2603.26287},
year = {2026},
date = {2026-03-27},
urldate = {2026-03-27},
abstract = {We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation [Ghosh et al., 2020], we propose a decomposition strategy and a Galerkin–Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive a priori error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.},
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We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation [Ghosh et al., 2020], we propose a decomposition strategy and a Galerkin–Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive a priori error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.
Dwivedi, Mukul; Rupp, Andreas
A hybridizable discontinuous Galerkin method for the Ostrovsky equation Online Forthcoming
Forthcoming, visited: 14.02.2026.
@online{DwivediR26,
title = {A hybridizable discontinuous Galerkin method for the Ostrovsky equation},
author = {Mukul Dwivedi and Andreas Rupp},
url = {https://doi.org/10.48550/arXiv.2602.13786},
doi = {10.48550/arXiv.2602.13786},
year = {2026},
date = {2026-02-14},
urldate = {2026-02-14},
abstract = {This paper develops the hybridizable discontinuous Galerkin (HDG) method for the Ostrovsky equation, a nonlinear dispersive wave equation featuring both third-order dispersion and a nonlocal antiderivative term with Coriolis effect. On a bounded interval, the nonlocal operator ( partial^{-1} x ) is localized through an auxiliary variable ( v ) satisfying ( v_x = u ) together with an additional boundary constraint that ensures uniqueness. We employ a mixed first-order formulation to decompose the dispersive operator and to localize the nonlocal term, and we couple the resulting semi-discrete HDG scheme with a θ-time stepping method for θ∈[1/2,1]. We prove L2-stability for suitable stabilization parameters and derive an a priori L2(Ω) error estimate for smooth solutions that explicitly accounts for the nonlinear convective flux.
Numerical examples illustrate the convergence properties and demonstrate the scheme's capability to handle smooth and non-smooth solutions, including solitary wave propagation and peaked solitary wave (peakon) propagation in the zero dispersive limit regime.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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This paper develops the hybridizable discontinuous Galerkin (HDG) method for the Ostrovsky equation, a nonlinear dispersive wave equation featuring both third-order dispersion and a nonlocal antiderivative term with Coriolis effect. On a bounded interval, the nonlocal operator ( partial^{-1} x ) is localized through an auxiliary variable ( v ) satisfying ( v_x = u ) together with an additional boundary constraint that ensures uniqueness. We employ a mixed first-order formulation to decompose the dispersive operator and to localize the nonlocal term, and we couple the resulting semi-discrete HDG scheme with a θ-time stepping method for θ∈[1/2,1]. We prove L2-stability for suitable stabilization parameters and derive an a priori L2(Ω) error estimate for smooth solutions that explicitly accounts for the nonlinear convective flux.
Numerical examples illustrate the convergence properties and demonstrate the scheme's capability to handle smooth and non-smooth solutions, including solitary wave propagation and peaked solitary wave (peakon) propagation in the zero dispersive limit regime.
Numerical examples illustrate the convergence properties and demonstrate the scheme's capability to handle smooth and non-smooth solutions, including solitary wave propagation and peaked solitary wave (peakon) propagation in the zero dispersive limit regime.
Burman, Erik; Knöller, Marvin; Oksanen, Lauri; Rupp, Andreas
Convergence analysis for the recovery of the friction threshold in a scalar Tresca model Online Forthcoming
Forthcoming, visited: 10.02.2026.
@online{BurmanKOR26,
title = {Convergence analysis for the recovery of the friction threshold in a scalar Tresca model},
author = {Erik Burman and Marvin Knöller and Lauri Oksanen and Andreas Rupp},
url = {https://doi.org/10.48550/arXiv.2602.08967},
doi = {10.48550/arXiv.2602.08967},
year = {2026},
date = {2026-02-10},
urldate = {2026-02-10},
abstract = {We consider a scalar valued elliptic partial differential equation on a sufficiently smooth domain Ω, subject to a regularized Tresca friction-type boundary condition on a subset Γ of ∂Ω. The friction threshold, a positive function appearing in this boundary condition, is assumed to be unknown and serves as the coefficient to be recovered in our inverse problem. Assuming that (i) the friction threshold lies in a finite dimensional space with known basis functions, (ii) the right hand sides of the partial differential equation are known, and (iii) the solution to the partial differential equation on some small open subset ω⊂Ω is available, we develop an iterative computational method for the recovery of the friction threshold. This algorithm is simple to implement and is based on piecewise linear finite elements. We show that the proposed algorithm converges in second order to a function ah and, moreover, that ah converges in second order in the finite element's mesh size h to the true (unknown) friction threshold. We highlight our theoretical results by simulations that confirm our rates numerically.},
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We consider a scalar valued elliptic partial differential equation on a sufficiently smooth domain Ω, subject to a regularized Tresca friction-type boundary condition on a subset Γ of ∂Ω. The friction threshold, a positive function appearing in this boundary condition, is assumed to be unknown and serves as the coefficient to be recovered in our inverse problem. Assuming that (i) the friction threshold lies in a finite dimensional space with known basis functions, (ii) the right hand sides of the partial differential equation are known, and (iii) the solution to the partial differential equation on some small open subset ω⊂Ω is available, we develop an iterative computational method for the recovery of the friction threshold. This algorithm is simple to implement and is based on piecewise linear finite elements. We show that the proposed algorithm converges in second order to a function ah and, moreover, that ah converges in second order in the finite element's mesh size h to the true (unknown) friction threshold. We highlight our theoretical results by simulations that confirm our rates numerically.
Dwivedi, Mukul; Gutendorf, Ruben; Rupp, Andreas
A hybridizable discontinuous Galerkin method for the non-local Camassa–Holm–Kadomtsev–Petviashvili equation Online Forthcoming
Forthcoming, visited: 21.01.2026.
@online{DwivediGR26,
title = {A hybridizable discontinuous Galerkin method for the non-local Camassa–Holm–Kadomtsev–Petviashvili equation},
author = {Mukul Dwivedi and Ruben Gutendorf and Andreas Rupp},
doi = {10.48550/arXiv.2601.13800},
year = {2026},
date = {2026-01-21},
urldate = {2026-01-21},
abstract = {This paper develops a hybridizable discontinuous Galerkin method for the two-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation. The method employs Cartesian meshes with tensor-product polynomial spaces, enabling separate treatment of ( x ) and ( y ) derivatives. The non-local operator ( partial_x^{-1} u_y ) is localized through an auxiliary variable ( v ) satisfying ( v_x = u_y ), allowing efficient element-by-element computations. We prove energy stability of the semi-discrete scheme and derive ( mathcal{O}(h^{k+1/2}) ) convergence in space. Numerical experiments validate the theoretical results and demonstrate the method's capability to accurately resolve smooth solutions and peaked solitary waves (peakons).},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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This paper develops a hybridizable discontinuous Galerkin method for the two-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation. The method employs Cartesian meshes with tensor-product polynomial spaces, enabling separate treatment of ( x ) and ( y ) derivatives. The non-local operator ( partial_x^{-1} u_y ) is localized through an auxiliary variable ( v ) satisfying ( v_x = u_y ), allowing efficient element-by-element computations. We prove energy stability of the semi-discrete scheme and derive ( mathcal{O}(h^{k+1/2}) ) convergence in space. Numerical experiments validate the theoretical results and demonstrate the method's capability to accurately resolve smooth solutions and peaked solitary waves (peakons).
Kaarnioja, Vesa; Rupp, Andreas; Gopalakrishnan, Jay
Sufficient conditions for QMC analysis of finite elements for parametric differential equations Online Forthcoming
Forthcoming, visited: 04.11.2025.
@online{KaarniojaRG25,
title = {Sufficient conditions for QMC analysis of finite elements for parametric differential equations},
author = {Vesa Kaarnioja and Andreas Rupp and Jay Gopalakrishnan},
doi = {10.48550/arXiv.2511.01703},
year = {2025},
date = {2025-11-04},
urldate = {2025-11-04},
abstract = {Parametric regularity of discretizations of flux vector fields satisfying a balance law is studied under some assumptions on a random parameter that links the flux with an unknown primal variable (often through a constitutive law). In the primary example of the stationary diffusion equation, the parameter corresponds to the inverse of the diffusivity. The random parameter is modeled here as a Gevrey-regular random field. Specific focus is on random fields expressible as functions of countably infinite sequences of independent random variables, which may be uniformly or normally distributed. Quasi-Monte Carlo (QMC) error bounds for some quantity of interest that depends on the flux are then derived using the parametric regularity. It is shown that the QMC method converges optimally if the quantity of interest depends continuously on the primal variable, its flux, or its gradient. A series of assumptions are introduced with the goal of encompassing a broad class of discretizations by various finite element methods. The assumptions are verified for the diffusion equation discretized using conforming finite elements, mixed methods, and hybridizable discontinuous Galerkin schemes. Numerical experiments confirm the analytical findings, highlighting the role of accurate flux approximation in QMC methods.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
Parametric regularity of discretizations of flux vector fields satisfying a balance law is studied under some assumptions on a random parameter that links the flux with an unknown primal variable (often through a constitutive law). In the primary example of the stationary diffusion equation, the parameter corresponds to the inverse of the diffusivity. The random parameter is modeled here as a Gevrey-regular random field. Specific focus is on random fields expressible as functions of countably infinite sequences of independent random variables, which may be uniformly or normally distributed. Quasi-Monte Carlo (QMC) error bounds for some quantity of interest that depends on the flux are then derived using the parametric regularity. It is shown that the QMC method converges optimally if the quantity of interest depends continuously on the primal variable, its flux, or its gradient. A series of assumptions are introduced with the goal of encompassing a broad class of discretizations by various finite element methods. The assumptions are verified for the diffusion equation discretized using conforming finite elements, mixed methods, and hybridizable discontinuous Galerkin schemes. Numerical experiments confirm the analytical findings, highlighting the role of accurate flux approximation in QMC methods.