Stephan Schmidt

Marc Herrmann, Roland Herzog, Stephan Schmidt et al.

The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart--Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-$L^2$ and TV-$L^1$, can be implemented in low and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.

Jorgen S. Dokken, Simon W. Funke, August Johansson et al.

An important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration. Common strategies are to employ mesh-deformation or re-meshing, where one or the other typically lacks robustness or is computationally expensive. This paper proposes a different approach, in which the computational domain is represented by multiple, independent meshes. A Nitsche based finite element method is used to weakly enforce continuity over the non-matching mesh interfaces. The optimization is preformed using an iterative gradient method, in which the shape-sensitivities are obtained by employing the Hadamard formulas and the adjoint approach. An optimize-then-discretize approach is chosen due to its independence of the FEM framework. Since the individual meshes may be moved freely, re-meshing or mesh deformations can be entirely avoided in cases where the geometry changes consists of rigid motions or scaling. By this free movement, we obtain robust and computational cheap mesh adaptation for optimization problems even for large domain changes. For general geometry changes, the method can be combined with mesh-deformation or re-meshing techniques to reduce the amount of deformation required. We demonstrate the capabilities of the method on several examples, including the optimal placement of heat emitting wires in a cable to minimize the chance of overheating, the drag minimization in Stokes flow, and the orientation of 25 objects in a Stokes flow.

Ryan Balshaw, P. Stephan Heyns, Daniel N. Wilke et al.

Latent variable models (LVMs) are commonly used to capture the underlying dependencies, patterns, and hidden structure in observed data. Source duplication is a by-product of the data hankelisation pre-processing step common to single channel LVM applications, which hinders practical LVM utilisation. In this article, a Python package titled spectrally-regularised-LVMs is presented. The proposed package addresses the source duplication issue via the addition of a novel spectral regularisation term. This package provides a framework for spectral regularisation in single channel LVM applications, thereby making it easier to investigate and utilise LVMs with spectral regularisation. This is achieved via the use of symbolic or explicit representations of potential LVM objective functions which are incorporated into a framework that uses spectral regularisation during the LVM parameter estimation process. The objective of this package is to provide a consistent linear LVM optimisation framework which incorporates spectral regularisation and caters to single channel time-series applications.

Manuel Weiß, Lukas Baumgärtner, Roland Herzog et al.

We introduce a new paradigm for geometry denoising using prior knowledge about the surface normal vector. This prior knowledge comes in the form of a set of preferred normal vectors, which we refer to as label vectors. A segmentation problem is naturally embedded in the denoising process. The segmentation is based on the similarity of the normal vector to the elements of the set of label vectors. Regularization is achieved by a total variation term. We formulate a split Bregman (ADMM) approach to solve the resulting optimization problem. The vertex update step is based on second-order shape calculus.

Stephan Schmidt, Daniel N. Wilke, Konstantinos C. Gryllias

In vibration-based condition monitoring, optimal filter design improves fault detection by enhancing weak fault signatures within vibration signals. This process involves optimising a derived objective function from a defined objective. The objectives are often based on proxy health indicators to determine the filter's parameters. However, these indicators can be compromised by irrelevant extraneous signal components and fluctuating operational conditions, affecting the filter's efficacy. Fault detection primarily uses the fault component's prominence in the squared envelope spectrum, quantified by a squared envelope spectrum-based signal-to-noise ratio. New optimal filter objective functions are derived from the proposed generalised envelope spectrum-based signal-to-noise objective for machines operating under variable speed conditions. Instead of optimising proxy health indicators, the optimal filter coefficients of the formulation directly maximise the squared envelope spectrum-based signal-to-noise ratio over targeted frequency bands using standard gradient-based optimisers. Four derived objective functions from the proposed objective effectively outperform five prominent methods in tests on three experimental datasets.

Lukas Baumgärtner, Ronny Bergmann, Roland Herzog et al.

We consider the problem of surface segmentation, where the goal is to partition a surface represented by a triangular mesh. The segmentation is based on the similarity of the normal vector field to a given set of label vectors. We propose a variational approach and compare two different regularizers, both based on a total variation measure. The first regularizer penalizes the total variation of the assignment function directly, while the second regularizer penalizes the total variation in the label space. In order to solve the resulting optimization problems, we use variations of the split Bregman (ADMM) iteration adapted to the problem at hand. While computationally more expensive, the second regularizer yields better results in our experiments, in particular it removes noise more reliably in regions of constant curvature.

Jay Raut, Daniel N. Wilke, Stephan Schmidt

Data normalisation, a common and often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitudebased sparse regression methods. This issue is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core assumption of sparsity is undermined by the interaction between data scaling and measurement noise. The resulting discovered models can be dense, uninterpretable, and physically incorrect. To address this critical vulnerability, we introduce the Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm that is inherently robust to data scaling. STCV replaces conventional magnitude-based thresholding with a dimensionless statistical metric, the Coefficient Presence (CP), which assesses the statistical validity and consistency of candidate terms in the model library. This shift from magnitude to statistical significance makes the discovery process invariant to arbitrary data scaling. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate that STCV consistently and significantly outperforms standard Sequential Thresholding Least Squares (STLSQ) and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets. The results show that STCV-based methods can successfully identify the correct, sparse physical laws even when other methods fail. By mitigating the distorting effects of normalisation, STCV makes sparse system identification a more reliable and automated tool for real-world applications, thereby enhancing model interpretability and trustworthiness.

Lukas Baumgärtner, Ronny Bergmann, Roland Herzog et al.

We propose a novel formulation for the second-order total generalized variation (TGV) of the normal vector on an oriented, triangular mesh embedded in $\R^3$. The normal vector is considered as a manifold-valued function, taking values on the unit sphere. Our formulation extends previous discrete TGV models for piecewise constant scalar data that utilize a Raviart-Thomas function space. To extend this formulation to the manifold setting, a tailor-made tangential Raviart-Thomas type finite element space is constructed in this work. The new regularizer is compared to existing methods in mesh denoising experiments.

Stephan Schmidt

A two step mesh deformation approach for large nodal deformations, typically arising from non-parametric shape optimization, fluid-structure interaction or computer graphics, is considered. Two major difficulties, collapsed cells and an undesirable parameterization, are overcome by considering a special form of ray tracing paired with a centroid Voronoi reparameterization. The ray direction is computed by solving an Eikonal equation. With respect to the Hadamard form of the shape derivative, both steps are within the kernel of the objective and have no negative impact on the minimizer. The paper concludes with applications in 2D and 3D fluid dynamics and automatic code generation and manages to solve these problems without any remeshing. The methodology is available as a FEniCS shape optimization add-on at http://www.mathematik.uni-wuerzburg.de/~schmidt/femorph.