Alexander Haberl

Alex Bespalov, Timo Betcke, Alexander Haberl et al.

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.

Thomas Führer, Alexander Haberl, Dirk Praetorius et al.

We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method (BEM) for elliptic first-kind integral equations. Our adaptive algorithm steers the termination of PCG as well as the local mesh-refinement. Besides convergence with optimal algebraic rates, we also prove almost optimal computational complexity. In particular, we provide an additive Schwarz preconditioner which can be computed in linear complexity and which is optimal in the sense that the condition numbers of the preconditioned systems are uniformly bounded. As model problem serves the 2D or 3D Laplace operator and the associated weakly-singular integral equation with energy space $\widetilde{H}^{-1/2}(Γ)$. The main results also hold for the hyper-singular integral equation with energy space $H^{1/2}(Γ)$.

Hrishikesh Gupta, Stefan Thalhammer, Jean-Baptiste Weibel et al.

Transparent objects are ubiquitous in daily life, making their perception and robotics manipulation important. However, they present a major challenge due to their distinct refractive and reflective properties when it comes to accurately estimating the 6D pose. To solve this, we present ReFlow6D, a novel method for transparent object 6D pose estimation that harnesses the refractive-intermediate representation. Unlike conventional approaches, our method leverages a feature space impervious to changes in RGB image space and independent of depth information. Drawing inspiration from image matting, we model the deformation of the light path through transparent objects, yielding a unique object-specific intermediate representation guided by light refraction that is independent of the environment in which objects are observed. By integrating these intermediate features into the pose estimation network, we show that ReFlow6D achieves precise 6D pose estimation of transparent objects, using only RGB images as input. Our method further introduces a novel transparent object compositing loss, fostering the generation of superior refractive-intermediate features. Empirical evaluations show that our approach significantly outperforms state-of-the-art methods on TOD and Trans32K-6D datasets. Robot grasping experiments further demonstrate that ReFlow6D's pose estimation accuracy effectively translates to real-world robotics task. The source code is available at: https://github.com/StoicGilgamesh/ReFlow6D and https://github.com/StoicGilgamesh/matting_rendering.

Benedikt Kantz, Clemens Staudinger, Christoph Feilmayr et al.

eXplainable Artificial Intelligence (XAI) aims at providing understandable explanations of black box models. In this paper, we evaluate current XAI methods by scoring them based on ground truth simulations and sensitivity analysis. To this end, we used an Electric Arc Furnace (EAF) model to better understand the limits and robustness characteristics of XAI methods such as SHapley Additive exPlanations (SHAP), Local Interpretable Model-agnostic Explanations (LIME), as well as Averaged Local Effects (ALE) or Smooth Gradients (SG) in a highly topical setting. These XAI methods were applied to various types of black-box models and then scored based on their correctness compared to the ground-truth sensitivity of the data-generating processes using a novel scoring evaluation methodology over a range of simulated additive noise. The resulting evaluation shows that the capability of the Machine Learning (ML) models to capture the process accurately is, indeed, coupled with the correctness of the explainability of the underlying data-generating process. We furthermore show the differences between XAI methods in their ability to correctly predict the true sensitivity of the modeled industrial process.

Thomas Führer, Alexander Haberl, Norbert Heuer

We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity, under the condition of an $L_2$ right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a non-closed space. This can be fixed by switching to the Kirchhoff-Love traces from [Führer, Heuer, Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, Math. Comp., 88 (2019)]. Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the DPG method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the non-closedness of a trace space), our analysis applies to any space dimension larger than or equal to two.

Gregor Gantner, Alexander Haberl, Dirk Praetorius et al.

We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of of Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments confirm the theoretical results.

Alex Bespalov, Alexander Haberl, Dirk Praetorius

We prove that for compactly perturbed elliptic problems, where the corresponding bilinear form satisfies a Garding inequality, adaptive mesh-refinement is capable of overcoming the preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. As an important consequence of our analysis, one does not have to deal with the a-priori assumption that the underlying meshes are sufficiently fine. Hence, the overall conclusion of our results is that adaptivity has stabilizing effects and can overcome possibly pessimistic restrictions on the meshes. In particular, our analysis covers adaptive mesh-refinement for the finite element discretization of the Helmholtz equation from where our interest originated.

Michael Feischl, Gregor Gantner, Alexander Haberl et al.

In a recent work, we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms.

Michael Feischl, Gregor Gantner, Alexander Haberl et al.

We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly-singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error estimator for Galerkin IGA boundary element methods, the present work focuses more on collocation and weighted- residual error estimators, which provide reliable upper bounds for the energy error. Our analysis allows piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. We formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments show that the proposed adaptive strategy leads to optimal convergence, and related IGA boundary element methods are superior to standard boundary element methods with piecewise polynomials.