Jürgen Dölz

Jürgen Dölz, Helmut Harbrecht, Stefan Kurz et al.

We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract the problems arising due to the dense matrices produced by boundary element methods. By solving Laplace and Helmholtz problems via a single layer approach we show, through a series of numerical examples suitable for easy comparison with other numerical schemes, that one can indeed achieve extremely high rates of convergence of the pointwise potential through the utilisation of higher order B-spline-based ansatz functions.

Jürgen Dölz, Helmut Harbrecht, Michael D. Peters

We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an $\mathcal{H}$-matrix, in particular if the correlation length is rather short or the correlation kernel is non-smooth. Since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the $\mathcal{H}$-matrix format, we can solve the correspondent $\mathcal{H}$-matrix equation in essentially linear time by using the $\mathcal{H}$-matrix arithmetic. Numerical experiments for three-dimensional finite element discretizations for several correlation lengths and different smoothness are provided. They validate the presented method and demonstrate that the computation times do not increase for non-smooth or shortly correlated data.

Zeger Bontinck, Jacopo Corno, Herbert De Gersem et al.

In this communication the advantages and drawbacks of the isogeometric analysis (IGA) are reviewed in the context of electromagnetic simulations. IGA extends the set of polynomial basis functions, commonly employed by the classical Finite Element Method (FEM). While identical to FEM with Nédélec's basis functions in the lowest order case, it is based on B-spline and Non-Uniform Rational B-spline basis functions. The main benefit of this is the exact representation of the geometry in the language of computer aided design (CAD) tools. This simplifies the meshing as the computational mesh is implicitly created by the engineer using the CAD tool. The curl- and div-conforming spline function spaces are recapitulated and the available software is discussed. Finally, several non-academic benchmark examples in two and three dimensions are shown which are used in optimization and uncertainty quantification workflows.

Jürgen Dölz, Michael Multerer, Michele Palma

Robustness of neural networks is commonly quantified via local or global Lipschitz constants. However, Lipschitz continuity can be overly coarse or overly restrictive as global robustness measure, failing to capture nuanced, data-dependent behavior. We propose a data-driven, architecture-agnostic framework based on the discrete modulus of continuity (DMOC), a non linear generalization of Lipschitz continuity that provides a finer notion of robustness. Unlike many existing approaches, DMOC does not require access to model internals and instead evaluates regularity relative to the data distribution. This shifts the focus from the model to the data, which provide a data-driven baseline of regularity against which the network's robustness is assessed. We establish convergence results for DMOC-induced seminorms with explicit data-driven rates in terms of the separation distance, and introduce a scalable minibatch algorithm that reduces the quadratic cost of exact computation, enabling application to large-scale data sets such as ImageNet. Empirically, DMOC serves as an architecture independent diagnostic: it distinguishes trained from untrained networks, reveals underfitting and overfitting regimes, and yields, as a special case, tight Lipschitz estimates comparable to state-of-the-art method such as ECLipsE and ECLipsE-fast.

Jürgen Dölz, Helmut Harbrecht, Michael D. Multerer

The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices, resulting in an almost linear cost. However, the computational efficiency of the algorithm is based on a recursive scheme which makes the error analysis quite involved. In this article, we propose a new algorithmic framework for the multiplication of hierarchical matrices. It improves currently known implementations by reducing the multiplication of hierarchical matrices towards finding a suitable low-rank approximation of sums of matrix-products. We propose several compression schemes to address this task. As a consequence, we are able to compute the best-approximation of hierarchical matrix products. A cost analysis shows that, under reasonable assumptions on the low-rank approximation method, the cost of the framework is almost linear with respect to the size of the matrix. Numerical experiments show that the new approach produces indeed the best-approximation of the product of hierarchical matrices for a given tolerance. They also show that the new multiplication can accomplish this task in less computation time than the established multiplication algorithm without error control.

Maximilian Nolte, Riccardo Torchio, Sebastian Schöps et al.

This contribution investigates the connection between isogeometric analysis and integral equation methods for full-wave electromagnetic problems up to the low-frequency limit. The proposed spline-based integral equation method allows for an exact representation of the model geometry described in terms of non-uniform rational B-splines without meshing. This is particularly useful when high accuracy is required or when meshing is cumbersome for instance during optimization of electric components. The augmented electric field integral equation is adopted and the deflation method is applied, so the low-frequency breakdown is avoided. The extension to higher-order basis functions is analyzed and the convergence rate is discussed. Numerical experiments on academic and realistic test cases demonstrate the high accuracy of the proposed approach.

Boian Balouchev, Jürgen Dölz, Maximilian Nolte et al.

We discuss the advantages of a spline-based freeform shape optimization approach using the example of a multi-tapered coaxial balun connected to a spiral antenna. The underlying simulation model is given in terms of a recently proposed isogeometric integral equation formulation, which can be interpreted as a high-order generalization of the partial element equivalent circuit method. We demonstrate a significant improvement in the optimized design, i.e., a reduction in the magnitude of the scattering parameter over a wide frequency range.

Jürgen Dölz, Jolanda Weygandt

Spectral clustering is a popular unsupervised learning technique which is able to partition unlabelled data into disjoint clusters of distinct shapes. However, the data under consideration are often experimental data, implying that the data is subject to measurement errors and measurements may even be lost or invalid. These uncertainties in the corrupted input data induce corresponding uncertainties in the resulting clusters, and the clusterings thus become unreliable. Modelling the uncertainties as random processes, we discuss a mathematical framework based on random set theory for the computational Monte Carlo approximation of statistically expected clusterings in case of corrupted, i.e., perturbed, incomplete, and possibly even additional, data. We propose several computationally accessible quantities of interest and analyze their consistency in the infinite data point and infinite Monte Carlo sample limit. Numerical experiments are provided to illustrate and compare the proposed quantities.