Andreas Rupp

Andreas Rupp, Vadym Aizinger, Balthasar Reuter et al.

We formulate a coupled surface/subsurface flow model that relies on hydrostatic equations with free surface in the free flow domain and on the Darcy model in the subsurface part. The model is discretized using the local discontinuous Galerkin method, and a statement of discrete energy stability is proved for the fully non-linear coupled system.

Vesa Kaarnioja, Andreas Rupp, Jay Gopalakrishnan

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 achieves a dimension-independent, faster-than-Monte Carlo convergence rate 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.

Mukul Dwivedi, Jesse Railo, Andreas Rupp

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 \cite{ghosh2020uniqueness}, 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 {\it 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.

Valery Ashu, Zhisong Liu, Heikki Haario et al.

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.

Ilmari Vahteristo, Zhi-Song Liu, Andreas Rupp

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.

Zhi-Song Liu, Roland Maier, Andreas Rupp

Finite element methods typically require a high resolution to satisfactorily approximate micro and even macro patterns of an underlying physical model. This issue can be circumvented by appropriate numerical homogenization or multiscale strategies that are able to obtain reasonable approximations on under-resolved scales. In this paper, we study the implicit neural representation and propose a continuous super-resolution network as a numerical homogenization strategy. It can take coarse finite element data to learn both in-distribution and out-of-distribution high-resolution finite element predictions. Our highlight is the design of a local implicit transformer, which is able to learn multiscale features. We also propose Gabor wavelet-based coordinate encodings which can overcome the bias of neural networks learning low-frequency features. Finally, perception is often preferred over distortion so scientists can recognize the visual pattern for further investigation. However, implicit neural representation is known for its lack of local pattern supervision. We propose to use stochastic cosine similarities to compare the local feature differences between prediction and ground truth. It shows better performance on structural alignments. Our experiments show that our proposed strategy achieves superior performance as an in-distribution and out-of-distribution super-resolution strategy.