Anne Reinarz

Alexey Chernov, Anne Reinarz

The aim of this paper is to develop stable and accurate numerical schemes for boundary integral formulations of the heat equation with Dirichlet boundary conditions. The accuracy of Galerkin discretisations for the resulting boundary integral formulations depends mainly on the choice of discretisation space. We develop a-priori error analysis utilising a proof technique that involves norm equivalences in hierarchical wavelet subspace decompositions. We apply this to a full tensor product discretisation, showing improvements over existing results, particularly for discretisation spaces having low polynomial degrees. We then use the norm equivalences to show that an anisotropic sparse grid discretisation yields even higher convergence rates. Finally, a simple adaptive scheme is proposed to suggest an optimal shape for the sparse grid index sets.

Philipp Samfass, Tobias Weinzierl, Anne Reinarz et al.

Resilient algorithms in high-performance computing are subject to rigorous non-functional constraints. Resiliency must not increase the runtime, memory footprint or I/O demands too significantly. We propose a task-based soft error detection scheme that relies on error criteria per task outcome. They formalise how ``dubious'' an outcome is, i.e. how likely it contains an error. Our whole simulation is replicated once, forming two teams of MPI ranks that share their task results. Thus, ideally each team handles only around half of the workload. If a task yields large error criteria values, i.e.~is dubious, we compute the task redundantly and compare the outcomes. Whenever they disagree, the task result with a lower error likeliness is accepted. We obtain a self-healing, resilient algorithm which can compensate silent floating-point errors without a significant performance, I/O or memory footprint penalty. Case studies however suggest that a careful, domain-specific tailoring of the error criteria remains essential.

Jean-Matthieu Gallard, Lukas Krenz, Leonhard Rannabauer et al.

The development of a high performance PDE solver requires the combined expertise of interdisciplinary teams with respect to application domain, numerical scheme and low-level optimization. In this paper, we present how the ExaHyPE engine facilitates the collaboration of such teams by isolating three roles: application, algorithms, and optimization expert. We thus support team members in letting them focus on their own area of expertise while integrating their contributions into an HPC production code. Inspired by web application development practices, ExaHyPE relies on two custom code generation modules, the Toolkit and the Kernel Generator, which follow a Model-View-Controller architectural pattern on top of the Jinja2 template engine library. Using Jinja2's templates to abstract the critical components of the engine and generated glue code, we isolate the application development from the engine. The template language also allows us to define and use custom template macros that isolate low-level optimizations from the numerical scheme described in the templates. We present three use cases, each focusing on one of our user roles, showcasing how the design of the code generation modules allows to easily expand the solver schemes to support novel demands from applications, to add optimized algorithmic schemes (with reduced memory footprint, e.g.), or provide improved low-level SIMD vectorization support.

Anne Reinarz, Tim Dodwell, Tim Fletcher et al.

Finite element (FE) analysis has the potential to offset much of the expensive experimental testing currently required to certify aerospace laminates. However, large numbers of degrees of freedom are necessary to model entire aircraft components whilst accurately resolving micro-scale defects. The new module dune-composites, implemented within DUNE by the authors, provides a tool to efficiently solve large-scale problems using novel iterative solvers. The key innovation is a preconditioner that guarantees a constant number of iterations regardless of the problem size. Its robustness has been shown rigorously in Spillane et al. (Numer. Math. 126, 2014) for isotropic problems. For anisotropic problems in composites it is verified numerically for the first time in this paper. The parallel implementation in DUNE scales almost optimally over thousands of cores. To demonstrate this, we present an original numerical study, varying the shape of a localised wrinkle and the effect this has on the strength of a curved laminate. This requires a high-fidelity mesh containing at least four layers of quadratic elements across each ply and interface layer, underlining the need for dune-composites, which can achieve run times of just over 2 minutes on 2048 cores for realistic composites problems with 173 million degrees of freedom.