Christoph Hofer

Christoph Hofer, Ulrich Langer

In this paper we consider a new version of the dual-primal isogeometric tearing and interconnecting (IETI-DP) method for solving large-scale linear systems of algebraic equations arising from discontinuous Galerkin (dG) isogeometric analysis of diffusion problems on multipatch domains with non-matching meshes. The dG formulation is used to couple the local problems across patch interfaces. The purpose of this paper is to present this new method and provide numerical examples indicating a polylogarithmic condition number bound for the preconditioned system and showing an incredible robustness with respect to large jumps in the diffusion coefficient across the interfaces.

Christoph Hofer, Ulrich Langer, Ioannis Toulopoulos

We propose a new discontinuous Galerkin Isogeometric Analysis (IgA) technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with non-matching interfaces. Due to an incorrect segmentation procedure, it may happen that the interfaces of adjacent subdomains don't coincide. In this way, gap regions, which are not present in the original physical domain, are created. In this paper, the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains. We apply a multi-patch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem. The last formulation is further modified by deriving interface conditions without the presence of the solution in the gap. Finally, the solution of this modified problem is approximated by a special discontinuous Galerkin IgA technique. The ideas are illustrated on a model diffusion problem with discontinuous diffusion coefficients. We develop a rigorous theoretical framework for the proposed method clarifying the influence of the gap size onto the convergence rate of the method. The theoretical estimates are supported by numerical examples in two- and three-dimensional computational domains.

Christoph Hofer, Ulrich Langer

The dual-primal isogeometric tearing and interconnecting (IETI-DP) method is the adaption of the dual-primal finite element tearing and interconnecting (FETI-DP) method to isogeometric analysis of scalar elliptic boundary value problems like, e.g., diffusion problems with heterogeneous diffusion coefficients. The purpose of this paper is to extent the already existing results on condition number estimates to multi-patch domains, which consist of different geometrical mappings for each patch. Another purpose is to prove a polylogarithmic condition number bound for the preconditioned system with stiffness scaling in case of $C^0$ smoothness across patch interfaces. Numerical experiments validate the presented theory.

Christoph Hofer, Ulrich Langer, Stefan Takacs

In this paper, we investigate inexact variants of dual-primal isogeometric tearing and interconnecting methods for solving large-scale systems of linear equations arising from Galerkin isogeometric discretizations of elliptic boundary value problems. The considered methods are extensions of standard finite element tearing and interconnecting methods to isogeometric analysis. The algorithms are implemented by means of energy minimizing primal subspaces. We discuss the replacement of local sparse direct solvers by iterative methods, particularly, multigrid solvers. We investigate the incorporation of these iterative solvers into different formulations of the algorithm. Finally, we present numerical examples comparing the performance of these inexact versions.

Christoph Hofer, Ulrich Langer, Ioannis Toulopoulos

The Isogeometric Analysis (IgA) of boundary value problems in complex domains often requires a decomposition of the computational domain into patches such that each of which can be parametrized by the so-called geometrical mapping. In this paper, we develop discontinuous Galerkin (dG) IgA techniques for solving elliptic diffusion problems on decompositions that can include non-matching parametrizations of the interfaces, i.e., the interfaces of the adjacent patches may be not identical. The lack of the exact parametrization of the patches leads to the creation of gap and overlapping regions between the patches. This does not allow the immediate use of the classical numerical fluxes that are known in the literature. The unknown normal fluxes of the solution on the non-matching interfaces are approximated by Taylor expansions using the values of the solution computed on the boundary of the patches These approximations are used in order to build up the numerical fluxes of the final dG IgA scheme and to couple the local patch-wise discrete problems. The resulting linear systems are solved by using efficient domainecomposition methods based on the tearing and interconnecting technology. We present numerical results of a series of test problems that validate the theoretical estimates presented.

Christoph Hofer

In this paper, we present the analysis of the discontinuous Galerkin dual-primal isogeometric tearing and interconnecting method (dG-IETI-DP) for the two-dimensional case where we only consider vertex primal variables. The dG-IETI-DP method is a combination of the dual-primal isogeometric tearing and interconnecting method (IETI-DP) with the discontinuous Galerkin (dG) method. We use the dG method only on the interfaces to couple different patches. This enables us to handle non-matching grids on patch interfaces as well as segmentation crimes (gaps and overlaps) between the patches. The purpose of this paper is to derive quasi-optimal bounds for the condition number of the preconditioned system with respect to the maximal ratio $H/h:=\max_k(H_k/h_k)$ of subdomain diameter and meshsize. We show that the constant is independent of $h_k$ and $H_k$, but depends on the ratio of meshsizes of neighbouring patches $h_\ell/h_k$.

Christoph Hofer

In this paper we investigate the parallelization of dual-primal isogeometric tearing and interconnecting (IETI-DP) type methods for solving large-scale continuous and discontinuous Galerkin systems of equations arising from Isogeometric analysis of elliptic boundary value problems. These methods are extensions of the finite element tearing and interconnecting methods to isogeometric analysis. The algorithms are implemented by means of energy minimizing primal subspaces. We discuss how these methods can efficiently be parallelized in a distributed memory setting. Weak and strong scaling studies presented for two and three dimensional problems show an excellent parallel efficiency.

Christoph Hofer, Stefan Takacs

Isogeometric Analysis (IgA) is a framework for setting up spline-based discretizations of partial differential equations, which has been introduced around a decade ago and has gained much attention since then. If large spline degrees are considered, one obtains the approximation power of a high-order method, but the number of degrees of freedom behaves like for a low-order method. One important ingredient to use a discretization with large spline degree, is a robust and preferably parallelizable solver. While numerical evidence shows that multigrid solvers with standard smoothers (like Gauss Seidel) does not perform well if the spline degree is increased, the multigrid solvers proposed by the authors and their co-workers proved to behave optimal both in the grid size and the spline degree. In the present paper, the authors want to show that those solvers are parallelizable and that they scale well in a parallel environment.

Christoph Hofer, Ioannis Toulopoulos

In the Isogeometric Analysis (IGA) framework, the computational domain has very often a multipatch representation. The multipatch domain can be obtained by a volume segmentation of a boundary represented domain, e.g., provided by a Computer Aided Design (CAD) model. Typically, small gap and overlapping regions can appear at the patch interfaces of such multipatch representations. In the current work we consider multipatch representations having only small overlapping regions between the patches. We develop a Discontinuous Galerkin (DG)- IGA method which can be immediately applied to these representations. Our method appropriately connects the fluxes of the one face of the overlapping region with the flux of the opposite face. We provide a theoretical justification of our approach by splitting the whole error into two components: the first is related to the incorrect representation of the patches (consistency error) and the second to the approximation properties of the IGA space. We show bounds for both components of the error. We verify the theoretical error estimates in a series of numerical examples.

Christoph Hofer, Ulrich Langer, Martin Neumüller

We propose and investigate new robust preconditioners for space-time Isogeometric Analysis of parabolic evolution problems. These preconditioners are based on a time parallel multigrid method. We consider a decomposition of the space-time cylinder into time-slabs which are coupled via a discontinuous Galerkin technique. The time-slabs provide the structure for the time-parallel multigrid solver. The most important part of the multigrid method is the smoother. We utilize the special structure of the involved operator to decouple its application into several spatial problems by means of generalized eigenvalue or Schur decompositions. Some of these problems have a symmetric saddle point structure, for which we present robust preconditions. Finally, we present numerical experiments confirming the robustness of our space-time IgA solver.

Sebastian Zeng, Florian Graf, Christoph Hofer et al.

The problem of (point) forecasting $ \textit{univariate} $ time series is considered. Most approaches, ranging from traditional statistical methods to recent learning-based techniques with neural networks, directly operate on raw time series observations. As an extension, we study whether $\textit{local topological properties}$, as captured via persistent homology, can serve as a reliable signal that provides complementary information for learning to forecast. To this end, we propose $\textit{topological attention}$, which allows attending to local topological features within a time horizon of historical data. Our approach easily integrates into existing end-to-end trainable forecasting models, such as $\texttt{N-BEATS}$, and in combination with the latter exhibits state-of-the-art performance on the large-scale M4 benchmark dataset of 100,000 diverse time series from different domains. Ablation experiments, as well as a comparison to a broad range of forecasting methods in a setting where only a single time series is available for training, corroborate the beneficial nature of including local topological information through an attention mechanism.

Christoph Hofer, Roland Kwitt, Mandar Dixit et al.

We study the problem of learning representations with controllable connectivity properties. This is beneficial in situations when the imposed structure can be leveraged upstream. In particular, we control the connectivity of an autoencoder's latent space via a novel type of loss, operating on information from persistent homology. Under mild conditions, this loss is differentiable and we present a theoretical analysis of the properties induced by the loss. We choose one-class learning as our upstream task and demonstrate that the imposed structure enables informed parameter selection for modeling the in-class distribution via kernel density estimators. Evaluated on computer vision data, these one-class models exhibit competitive performance and, in a low sample size regime, outperform other methods by a large margin. Notably, our results indicate that a single autoencoder, trained on auxiliary (unlabeled) data, yields a mapping into latent space that can be reused across datasets for one-class learning.

Christoph Hofer, Roland Kwitt, Marc Niethammer et al.

Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information, typically in the form of summary representations of topological features. However, such topological signatures often come with an unusual structure (e.g., multisets of intervals) that is highly impractical for most machine learning techniques. While many strategies have been proposed to map these topological signatures into machine learning compatible representations, they suffer from being agnostic to the target learning task. In contrast, we propose a technique that enables us to input topological signatures to deep neural networks and learn a task-optimal representation during training. Our approach is realized as a novel input layer with favorable theoretical properties. Classification experiments on 2D object shapes and social network graphs demonstrate the versatility of the approach and, in case of the latter, we even outperform the state-of-the-art by a large margin.