A massively parallel combination technique for the solution of high-dimensional PDEs

The solution of high-dimensional problems, especially high-dimensional partial differential equations (PDEs) that require the joint discretization of more than the usual three spatial dimensions and time, is one of the grand challenges in high performance computing (HPC). Due to the exponential growth of the number of unknowns - the so-called curse of dimensionality, it is in many cases not feasible to resolve the simulation domain as fine as required by the physical problem. Although the upcoming generation of exascale HPC systems theoretically provides the computational power to handle simulations that are out of reach today, it is expected that this is only achievable with new numerical algorithms that are able to efficiently exploit the massive parallelism of these systems. The sparse grid combination technique is a numerical scheme where the problem (e.g., a high-dimensional PDE) is solved on different coarse and anisotropic computational grids (so-called component grids), which are then combined to approximate the solution with a much higher target resolution than any of the individual component grids. This way, the total number of unknowns being computed is drastically reduced compared to the case when the problem is directly solved on a regular grid with the target resolution. Thus, the curse of dimensionality is mitigated. The combination technique is a promising approach to solve high-dimensional problems on future exascale systems. It offers two levels of parallelism: the component grids can be computed in parallel, independently and asynchronously of each other; and the computation of each component grid can be parallelized as well. This reduces the demand for global communication and synchronization, which is expected to be one of the limiting factors for classical discretization techniques to achieve scalability on exascale systems. Furthermore, the combination technique enables novel approaches to deal with the increasing fault rates expected from these systems. With the fault-tolerant combination technique it is possible to recover from failures without time-consuming checkpoint-restart mechanisms. In this work, new algorithms and data structures are presented that enable a massively parallel and fault-tolerant combination technique for time-dependent PDEs on large-scale HPC systems. The scalability of these algorithms is demonstrated on up to 180225 processor cores on the supercomputer Hazel Hen. Furthermore, the parallel combination technique is applied to gyrokinetic simulations in GENE, a software for the simulation of plasma microturbulence in fusion devices

A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations

The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability. The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov--Poisson solver show a stabilizing effect of the new bases on the simulations

A sparse-grid isogeometric solver

Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.Comment: updated version after revie

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

B-splines for sparse grids : algorithms and application to higher-dimensional optimization

In simulation technology, computationally expensive objective functions are often replaced by cheap surrogates, which can be obtained by interpolation. Full grid interpolation methods suffer from the so-called curse of dimensionality, rendering them infeasible if the parameter domain of the function is higher-dimensional (four or more parameters). Sparse grids constitute a discretization method that drastically eases the curse, while the approximation quality deteriorates only insignificantly. However, conventional basis functions such as piecewise linear functions are not smooth (continuously differentiable). Hence, these basis functions are unsuitable for applications in which gradients are required. One example for such an application is gradient-based optimization, in which the availability of gradients greatly improves the speed of convergence and the accuracy of the results. This thesis demonstrates that hierarchical B-splines on sparse grids are well-suited for obtaining smooth interpolants for higher dimensionalities. The thesis is organized in two main parts: In the first part, we derive new B-spline bases on sparse grids and study their implications on theory and algorithms. In the second part, we consider three real-world applications in optimization: topology optimization, biomechanical continuum-mechanics, and dynamic portfolio choice models in finance. The results reveal that the optimization problems of these applications can be solved accurately and efficiently with hierarchical B-splines on sparse grids.In der Simulationstechnik werden zeitaufwendige Zielfunktionen oft durch einfache Surrogate ersetzt, die durch Interpolation gewonnen werden können. Vollgitter-Interpolationsmethoden leiden unter dem sogenannten Fluch der Dimensionalität, der sie unbrauchbar macht, falls der Parameterbereich der Funktion höherdimensional ist (vier oder mehr Parameter). Dünne Gitter sind eine Diskretisierungsmethode, die den Fluch drastisch lindert und die Approximationsqualität nur leicht verschlechtert. Leider sind konventionelle Basisfunktionen wie die stückweise linearen Funktionen nicht glatt (stetig differenzierbar). Daher sind sie für Anwendungen ungeeignet, in denen Gradienten benötigt werden. Ein Beispiel für eine solche Anwendung ist gradientenbasierte Optimierung, in der die Verfügbarkeit von Gradienten die Konvergenzgeschwindigkeit und die Ergebnisgenauigkeit deutlich verbessert. Diese Dissertation demonstriert, dass hierarchische B-Splines auf dünnen Gittern hervorragend geeignet sind, um glatte Interpolierende für höhere Dimensionalitäten zu erhalten. Die Dissertation ist in zwei Hauptteile gegliedert: Der erste Teil leitet neue B-Spline-Basen auf dünnen Gittern her und untersucht ihre Implikationen bezüglich Theorie und Algorithmen. Der zweite Teil behandelt drei Realwelt-Anwendungen aus der Optimierung: Topologieoptimierung, biomechanische Kontinuumsmechanik und Modelle der dynamischen Portfolio-Wahl in der Finanzmathematik. Die Ergebnisse zeigen, dass die Optimierungsprobleme dieser Anwendungen durch hierarchische B-Splines auf dünnen Gittern genau und effizient gelöst werden können

High Performance Fault-Tolerant Solution of PDEs using the Sparse Grid Combination Technique

The data volume of Partial Differential Equation (PDE) based ultra-large-scale scientific simulations is increasing at a higher rate than that of the system’s processing power. To process the increased amount of simulation data within a reasonable amount of time, the evolution of computation is expected to reach the exascale level. One of several key challenges to overcome in these exascale systems is to handle the high rate of component failure arising due to having millions of cores working together with high power consumption and clock frequencies. Studies show that even the highly tuned widely used checkpointing technique is unable to handle the failures efficiently in exascale systems. The Sparse Grid Combination Technique (SGCT) is proved to be a cost-effective method for computing high-dimensional PDE based simulations with only small loss of accuracy, which can be easily modified to provide an Algorithm-Based Fault Tolerance (ABFT) for these applications. Additionally, the recently introduced User Level Failure Mitigation (ULFM) MPI library provides the ability to detect and identify application process failures, and reconstruct the failed processes. However, there is a gap of the research how these could be integrated together to develop fault-tolerant applications, and the range of issues that may arise in the process are yet to be revealed. My thesis is that with suitable infrastructural support an integration of ULFM MPI and a modified form of the SGCT can be used to create high performance robust PDE based applications. The key contributions of my thesis are: (1) An evaluation of the effectiveness of applying the modified version of the SGCT on three existing and complex applications (including a general advection solver) to make them highly fault-tolerant. (2) An evaluation of the capabilities of ULFM MPI to recover from a single or multiple real process/node failures for a range of complex applications computed with the modified form of the SGCT. (3) A detailed experimental evaluation of the fault-tolerant work including the time and space requirements, and parallelization on the non-SGCT dimensions. (4) An analysis of the result errors with respect to the number of failures. (5) An analysis of the ABFT and recovery overheads. (6) An in-depth comparison of the fault-tolerant SGCT based ABFT with traditional checkpointing on a non-fault-tolerant SGCT based application. (7) A detailed evaluation of the infrastructural support in terms of load balancing, pure- and hybrid-MPI, process layouts, processor affinity, and so on

Top-k aggregation queries in large-scale distributed systems

Distributed top-k query processing has recently become an essential functionality in a large number of emerging application classes like Internet traffic monitoring and Peer-to-Peer Web search. This work addresses efficient algorithms for distributed top-k queries in wide-area networks where the index lists for the attribute values (or text terms) of a query are distributed across a number of data peers. More precisely, in this thesis, we make the following distributions: We present the family of KLEE algorithms that are a fundamental building-block towards efficient top-k query processing in distributed systems. We present means to model score distributions and show how these score models can be used to reason about parameter values that play an important role in the overall performance of KLEE. We present GRASS, a family of novel algorithms based on three optimization techniques significantly increased overall performance of KLEE and related algorithms. We present probabilistic guarantees for the result quality. Moreover, we present Minerva1, a distributed search engine. Minerva offers a highly distributed (in both the data dimension and the computational dimension), scalable, and efficient solution toward the development of internet-scale search engines.Top-k Anfragen spielen eine große Rolle in einer Vielzahl von Anwendungen, insbesondere im Bereich von Informationssystemen, bei denen eine kleine, sorgfältig ausgewählte Teilmenge der Ergebnisse den Benutzern präsentiert werden soll. Beispiele hierfür sind Suchmaschinen wie Google, Yahoo oder MSN. Obwohl die Forschung in diesem Bereich in den letzten Jahren große Fortschritte gemacht hat, haben Top-k-Anfragen in verteilten Systemen, bei denen die Daten auf verschiedenen Rechnern verteilt sind, vergleichsweise wenig Aufmerksamkeit erlangt. In dieser Arbeit beschäftigen wir uns mit der effizienten Verarbeitung eben dieser Anfragen. Die Hauptbeiträge gliedern sich wie folgt. Wir präsentieren KLEE, eine Familie neuartiger Top-k-Algorithmen. Wir entwickeln Modelle mit denen Datenverteilungen beschrieben werden können. Diese Modelle sind die Grundlage für eine Schätzung diverser Parameter, die einen großen Einfluss auf die Performanz von KLEE und anderen ähnlichen Algorithmen haben. Wir präsentieren GRASS, eine Familie von Algorithmen, basierend auf drei neuartigen Optimierungstechniken, mit denen die Performanz von KLEE und ähnlichen Algorithmen verbessert wird. Wir präsentieren probabilistische Garantien für die Ergebnisgüte. Wir präsentieren Minerva, eine neuartige verteilte Peer-to-Peer-Suchmaschine

Data fusion strategies for energy efficiency in buildings: Overview, challenges and novel orientations

Recently, tremendous interest has been devoted to develop data fusion strategies for energy efficiency in buildings, where various kinds of information can be processed. However, applying the appropriate data fusion strategy to design an efficient energy efficiency system is not straightforward; it requires a priori knowledge of existing fusion strategies, their applications and their properties. To this regard, seeking to provide the energy research community with a better understanding of data fusion strategies in building energy saving systems, their principles, advantages, and potential applications, this paper proposes an extensive survey of existing data fusion mechanisms deployed to reduce excessive consumption and promote sustainability. We investigate their conceptualizations, advantages, challenges and drawbacks, as well as performing a taxonomy of existing data fusion strategies and other contributing factors. Following, a comprehensive comparison of the state-of-the-art data fusion based energy efficiency frameworks is conducted using various parameters, including data fusion level, data fusion techniques, behavioral change influencer, behavioral change incentive, recorded data, platform architecture, IoT technology and application scenario. Moreover, a novel method for electrical appliance identification is proposed based on the fusion of 2D local texture descriptors, where 1D power signals are transformed into 2D space and treated as images. The empirical evaluation, conducted on three real datasets, shows promising performance, in which up to 99.68% accuracy and 99.52% F1 score have been attained. In addition, various open research challenges and future orientations to improve data fusion based energy efficiency ecosystems are explored