TY - JOUR A1 - Alhaddad, Samer A1 - Förstner, Jens A1 - Groth, Stefan A1 - Grünewald, Daniel A1 - Grynko, Yevgen A1 - Hannig, Frank A1 - Kenter, Tobias A1 - Pfreundt, F.J. A1 - Plessl, Christian A1 - Schotte, Merlind A1 - Steinke, Thomas A1 - Teich, J. A1 - Weiser, Martin A1 - Wende, Florian T1 - The HighPerMeshes Framework for Numerical Algorithms on Unstructured Grids JF - Concurrency and Computation: Practice and Experience N2 - Solving PDEs on unstructured grids is a cornerstone of engineering and scientific computing. Heterogeneous parallel platforms, including CPUs, GPUs, and FPGAs, enable energy-efficient and computationally demanding simulations. In this article, we introduce the HPM C++-embedded DSL that bridges the abstraction gap between the mathematical formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different programming models on the other hand. Thus, the HPM DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HPM DSL, and demonstrate its usage with three examples. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters, is presented. A code generator and a matching back end allow the acceleration of HPM code with GPUs. Finally, the achievable performance and scalability are demonstrated for different example problems. Y1 - 2022 U6 - https://doi.org/10.1002/cpe.6616 VL - 34 IS - 14 ER - TY - JOUR A1 - Alhaddad, Samer A1 - Förstner, Jens A1 - Groth, Stefan A1 - Grünewald, Daniel A1 - Grynko, Yevgen A1 - Hannig, Frank A1 - Kenter, Tobias A1 - Pfreundt, Franz-Josef A1 - Plessl, Christian A1 - Schotte, Merlind A1 - Steinke, Thomas A1 - Teich, Jürgen A1 - Weiser, Martin A1 - Wende, Florian T1 - HighPerMeshes - A Domain-Specific Language for Numerical Algorithms on Unstructured Grids JF - Euro-Par 2020: Parallel Processing Workshops. N2 - Solving partial differential equations on unstructured grids is a cornerstone of engineering and scientific computing. Nowadays, heterogeneous parallel platforms with CPUs, GPUs, and FPGAs enable energy-efficient and computationally demanding simulations. We developed the HighPerMeshes C++-embedded Domain-Specific Language (DSL) for bridging the abstraction gap between the mathematical and algorithmic formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different parallel programming and runtime models on the other hand. Thus, the HighPerMeshes DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HighPer-Meshes DSL, and demonstrate its usage with three examples, a Poisson and monodomain problem, respectively, solved by the continuous finite element method, and the discontinuous Galerkin method for Maxwell’s equation. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters is presented. Finally, the achievable performance and scalability are demonstrated for a typical example problem on a multi-core CPU cluster. Y1 - 2021 U6 - https://doi.org/10.1007/978-3-030-71593-9_15 SP - 185 EP - 196 PB - Springer ER - TY - JOUR A1 - Bartels, Tinko A1 - Fisikopoulos, Vissarion A1 - Weiser, Martin T1 - Fast Floating-Point Filters for Robust Predicates JF - BIT Numerical Mathematics N2 - Geometric predicates are at the core of many algorithms, such as the construction of Delaunay triangulations, mesh processing and spatial relation tests. These algorithms have applications in scientific computing, geographic information systems and computer-aided design. With floating-point arithmetic, these geometric predicates can incur round-off errors that may lead to incorrect results and inconsistencies, causing computations to fail. This issue has been addressed using a combination of exact arithmetic for robustness and floating-point filters to mitigate the computational cost of exact computations. The implementation of exact computations and floating-point filters can be a difficult task, and code generation tools have been proposed to address this. We present a new C++ meta-programming framework for the generation of fast, robust predicates for arbitrary geometric predicates based on polynomial expressions. We combine and extend different approaches to filtering, branch reduction, and overflow avoidance that have previously been proposed. We show examples of how this approach produces correct results for data sets that could lead to incorrect predicate results with naive implementations. Our benchmark results demonstrate that our implementation surpasses state-of-the-art implementations. Y1 - 2023 U6 - https://doi.org/10.1007/s10543-023-00975-x VL - 63 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - Error Bounds for Discrete-Continuous Free Flight Trajectory Optimization JF - Journal of Optimization Theory and Applications N2 - Flight planning, the computation of optimal routes in view of flight time and fuel consumption under given weather conditions, is traditionally done by finding globally shortest paths in a predefined airway network. Free flight trajectories, not restricted to a network, have the potential to reduce the costs significantly, and can be computed using locally convergent continuous optimal control methods. Hybrid methods that start with a discrete global search and refine with a fast continuous local optimization combine the best properties of both approaches, but rely on a good switchover, which requires error estimates for discrete paths relative to continuous trajectories. Based on vertex density and local complete connectivity, we derive localized and a priori bounds for the flight time of discrete paths relative to the optimal continuous trajectory, and illustrate their properties on a set of benchmark problems. It turns out that localization improves the error bound by four orders of magnitude, but still leaves ample opportunities for tighter bounds using a posteriori error estimators. Y1 - 2023 U6 - https://doi.org/10.1007/s10957-023-02264-7 VL - 198 SP - 830 EP - 856 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - Newton's Method for Global Free Flight Trajectory Optimization JF - Operations Research Forum N2 - Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton's method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case. Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-91846 VL - 4 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - A Discrete-Continuous Algorithm for Free Flight Planning JF - Algorithms N2 - We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach. KW - shortest path KW - flight planning KW - free flight KW - discrete-continuous algorithm KW - optimal control KW - discrete optimization Y1 - 2020 U6 - https://doi.org/10.3390/a14010004 SN - 1438-0064 VL - 14 IS - 1 SP - 4 PB - MDPI ER - TY - JOUR A1 - Carderera, Alejandro A1 - Pokutta, Sebastian A1 - Schütte, Christof A1 - Weiser, Martin T1 - CINDy: Conditional gradient-based Identification of Non-linear Dynamics – Noise-robust recovery JF - Journal of Computational and Applied Mathematics N2 - Governing equations are essential to the study of nonlinear dynamics, often enabling the prediction of previously unseen behaviors as well as the inclusion into control strategies. The discovery of governing equations from data thus has the potential to transform data-rich fields where well-established dynamical models remain unknown. This work contributes to the recent trend in data-driven sparse identification of nonlinear dynamics of finding the best sparse fit to observational data in a large library of potential nonlinear models. We propose an efficient first-order Conditional Gradient algorithm for solving the underlying optimization problem. In comparison to the most prominent alternative algorithms, the new algorithm shows significantly improved performance on several essential issues like sparsity-induction, structure-preservation, noise robustness, and sample efficiency. We demonstrate these advantages on several dynamics from the field of synchronization, particle dynamics, and enzyme chemistry. Y1 - 2021 ER - TY - JOUR A1 - Chegini, Fatemeh A1 - Kopanicakova, Alena A1 - Krause, Rolf A1 - Weiser, Martin T1 - Efficient Identification of Scars using Heterogeneous Model Hierarchies JF - EP Europace N2 - Aims. Detection and quantification of myocardial scars are helpful both for diagnosis of heart diseases and for building personalized simulation models. Scar tissue is generally charac­terized by a different conduction of electrical excitation. We aim at estimating conductivity-related parameters from endocardial mapping data, in particular the conductivity tensor. Solving this inverse problem requires computationally expensive monodomain simulations on fine discretizations. Therefore, we aim at accelerating the estimation using a multilevel method combining electrophysiology models of different complexity, namely the mono­domain and the eikonal model. Methods. Distributed parameter estimation is performed by minimizing the misfit between simulated and measured electrical activity on the endocardial surface, subject to the mono­domain model and regularization, leading to a constrained optimization problem. We formulate this optimization problem, including the modeling of scar tissue and different regularizations, and design an efficient iterative solver. We consider monodomain grid hierarchies and monodomain-eikonal model hierarchies in a recursive multilevel trust-region method. Results. From several numerical examples, both the efficiency of the method and the estimation quality, depending on the data, are investigated. The multilevel solver is significantly faster than a comparable single level solver. Endocardial mapping data of realistic density appears to be just sufficient to provide quantitatively reasonable estimates of location, size, and shape of scars close to the endocardial surface. Conclusion. In several situations, scar reconstruction based on eikonal and monodomain models differ significantly, suggesting the use of the more accurate but more expensive monodomain model for this purpose. Still, eikonal models can be utilized to accelerate the computations considerably, enabling the use of complex electrophysiology models for estimating myocardial scars from endocardial mapping data. Y1 - 2021 U6 - https://doi.org/10.1093/europace/euaa402 VL - 23 SP - i113 EP - i122 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Schiela, Anton A1 - Weiser, Martin T1 - Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia JF - Acta Numerica N2 - This paper surveys the required mathematics for a typical challenging problem from computational medicine, the cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem gave rise to quite a number of subtle mathematical problems, part of which had been unsolved when the common project started. Efficiency of numerical algorithms, i.e. computational speed and monitored reliability, play a decisive role for the medical treatment. Off-the-shelf software had turned out to be not sufficient to meet the requirements of medicine. Rather, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, a virtual lab, including 3D geometry processing of individual virtual patients had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analyzed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the individual patients' treatment and on the construction of an improved hyperthermia applicator. Y1 - 2012 VL - 21 SP - 307 EP - 378 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Weiser, Martin A1 - Seebass, Martin T1 - A new nonlinear elliptic multilevel FEM in clinical cancer therapy planning JF - Comput. Vis. Sci. Y1 - 2000 VL - 3 SP - 115 EP - 120 ER -