Christian Glusa

Mark Ainsworth, Christian Glusa

The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant. The purpose of this work is to provide an analysis of the behaviour of the multigrid algorithm in the presence of faults. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations and it is of considerable importance to anticipate its behaviour on an exascale machine. The analysis of resilience of algorithms is in its infancy and the current work is perhaps the first to provide a mathematical model for faults and analyse the behaviour of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method.

Matthias Mayr, Alexander Heinlein, Christian Glusa et al.

Trilinos is a community-developed, open-source software framework that facilitates building large-scale, complex, multiscale, multiphysics simulation code bases for scientific and engineering problems. Since the Trilinos framework has undergone substantial changes to support new applications and new hardware architectures, this document is an update to ``An Overview of the Trilinos project'' by Heroux et al. (ACM Transactions on Mathematical Software, 31(3):397-423, 2005). It describes the design of Trilinos, introduces its new organization in product areas, and highlights established and new features available in Trilinos. Particular focus is put on the modernized software stack based on the Kokkos ecosystem to deliver performance portability across heterogeneous hardware architectures. This paper also outlines the organization of the Trilinos community and the contribution model to help onboard interested users and contributors.

Noujoud Nader, Patrick Diehl, Marta D'Elia et al.

Local-nonlocal coupling approaches combine the computational efficiency of local models and the accuracy of nonlocal models. However, the coupling process is challenging, requiring expertise to identify the interface between local and nonlocal regions. This study introduces a machine learning-based approach to automatically detect the regions in which the local and nonlocal models should be used in a coupling approach. This identification process uses the loading functions and provides as output the selected model at the grid points. Training is based on datasets of loading functions for which reference coupling configurations are computed using accurate coupled solutions, where accuracy is measured in terms of the relative error between the solution to the coupling approach and the solution to the nonlocal model. We study two approaches that differ from one another in terms of the data structure. The first approach, referred to as the full-domain input data approach, inputs the full load vector and outputs a full label vector. In this case, the classification process is carried out globally. The second approach consists of a window-based approach, where loads are preprocessed and partitioned into windows and the problem is formulated as a node-wise classification approach in which the central point of each window is treated individually. The classification problems are solved via deep learning algorithms based on convolutional neural networks. The performance of these approaches is studied on one-dimensional numerical examples using F1-scores and accuracy metrics. In particular, it is shown that the windowing approach provides promising results, achieving an accuracy of 0.96 and an F1-score of 0.97. These results underscore the potential of the approach to automate coupling processes, leading to more accurate and computationally efficient solutions for material science applications.

Mark Ainsworth, Christian Glusa

A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $Ω$ on the extruded domain $\mathcal{C}=Ω\times[0,\infty)$ following Caffarelli and Silvestre (2007). The resulting degenerate elliptic integer order PDE is then approximated using a hybrid FEM-spectral scheme. Finite elements are used in the direction parallel to the problem domain $Ω$, and an appropriate spectral method is used in the extruded direction. The spectral part of the scheme requires that we approximate the true eigenvalues of the integer order Laplacian over $Ω$. We derive an a priori error estimate which takes account of the error arising from using an approximation in place of the true eigenvalues. We further present a strategy for choosing approximations of the eigenvalues based on Weyl's law and finite element discretizations of the eigenvalue problem. The system of linear algebraic equations arising from the hybrid FEM-spectral scheme is decomposed into blocks which can be solved effectively using standard iterative solvers such as multigrid and conjugate gradient. Numerical examples in two and three dimensions show that the approach is quasi-optimal in terms of complexity.

Mark Ainsworth, Christian Glusa

We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and, where necessary, modifying them so that they can be applied to the fractional PDE case. Improved a priori error estimates are derived for the case of quasi-uniform meshes which are seen to deliver sub-optimal rates of convergence owing to the presence of singularities. Attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure. We assume that the resulting refined meshes are locally quasi-uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods, including the multigrid method. The storage of the dense matrices along with efficient techniques for computing the dense matrix vector products needed for the iterative solution is also considered. The performance and efficiency of the resulting algorithm is illustrated for a variety of examples.

Mark Ainsworth, Christian Glusa

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples.

Mark Ainsworth, Christian Glusa

Computing at the exascale level is expected to be affected by a significantly higher rate of faults, due to increased component counts as well as power considerations. Therefore, current day numerical algorithms need to be reexamined as to determine if they are fault resilient, and which critical operations need to be safeguarded in order to obtain performance that is close to the ideal fault-free method. In a previous paper, a framework for the analysis of random stationary linear iterations was presented and applied to the two grid method. The present work is concerned with the multigrid algorithm for the solution of linear systems of equations, which is widely used on high performance computing systems. It is shown that the Fault-Prone Multigrid Method is not resilient, unless the prolongation operation is protected. Strategies for fault detection and mitigation as well as protection of the prolongation operation are presented and tested, and a guideline for an optimal choice of parameters is devised.