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

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

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

Airfoil analysis and design using surrogate models

A study was performed to compare two different methods for generating surrogate models for the analysis and design of airfoils. Initial research was performed to compare the accuracy of surrogate models for predicting the lift and drag of an airfoil with data collected from highidelity simulations using a modern CFD code along with lower-order models using a panel code. This was followed by an evaluation of the Class Shape Trans- formation (CST) method for parameterizing airfoil geometries as a prelude to the use of surrogate models for airfoil design optimization and the implementation of software to use CST to modify airfoil shapes as part of the airfoil design process. Optimization routines were coupled with surrogate modeling techniques to study the accuracy and efficiency of the surrogate models to produce optimal airfoil shapes. Finally, the results of the current research are summarized, and suggestions are made for future research

Parametric Model Order Reduction Using Sparse Grids

In applications such as very large scale integration chip design models are typically huge. The same is true in mechanical engineering, especially when the models are complex finite element discretizations. To speed up simulations the large full order model is replaced by a smaller reduced order model. This is called model order reduction. The challenge is to find a reduced order model that closely resembles the full order model in order not to lose too much accuracy. Many models depend on parameters. The goal of parametric model order reduction is to preserve this dependence in the reduced order model to avoid repeatedly performing time-consuming model order reduction for every new parameter value. This is particularly interesting for parameter studies. In this thesis we develop, analyze and test a parametric model order reduction method for large symmetric linear time-invariant dynamical systems which preserves stability and is efficient in higher-dimensional parameter spaces. This method is based on sparse grid interpolation. In a pre-computation step, local reduced order models are computed at several discrete points in parameter space. Whenever we need to evaluate the model at an arbitrary point in parameter space, a reduced order model at that point is obtained by interpolating the system matrices of the local reduced order models on matrix manifolds. As a theoretical foundation we introduce appropriate norms for parametric models and state conditions for the parameter dependence such that these norms are finite. In our analysis we then derive an upper bound for the interpolation error expressed in these norms, which shows a good qualitative behavior in computational experiments. We demonstrate that interpolation on sparse grids is more efficient than interpolation on full grids and that interpolation with global polynomial basis functions is more efficient than interpolation with piece-wise linear basis functions when the parameter dependence is smooth. Furthermore, we consider a benchmark studied in a recent parametric model order reduction method comparison survey and show that parametric model order reduction methods based on matrix interpolation can be competitive to other methods when exploiting the symmetry of that system.Parametrische Modellreduktion mit Dünnen Gittern In vielen Anwendungen wie dem Entwurf von VLSI-Schaltkreisen treten sehr große Simulationsmodelle auf. Ein weiteres Beispiel sind physikalische Modelle aus dem Maschinen- oder Fahrzeugbau, die mit finiten Elementen sehr fein diskretisiert wurden. Man kann die Simulation solcher Modelle mithilfe von Modellreduktion beschleunigen. Dabei ersetzt man das große Modell durch ein kleineres, reduziertes Modell. Die Herausforderung besteht darin, ein passendes reduziertes Modell zu finden. Um möglichst wenig Genauigkeit einzubüßen, sollten sich das ursprüngliche Modell und das reduzierte Modell möglichst ähnlich verhalten. Oft sind die Modelle außerdem parameterabhängig. Bei parametrischer Modellreduktion versucht man, diese Abhängigkeit im reduzierten Modell zu erhalten, um dadurch das wiederholte Ausführen der zeitaufwändigen Modellreduktion für jeden neuen Parameterwert zu vermeiden. Dies ist besonders für Parameterstudien von wesentlicher Bedeutung. In dieser Dissertation entwickeln, analysieren und testen wir ein stabilitätserhaltendes parametrisches Modellreduktionsverfahren für große symmetrische lineare zeit-invariante dynamische Systeme, welches aufgrund seiner Effizienz auch für den Einsatz in höher-dimensionalen Parameterräumen geeignet ist. Dieses Verfahren basiert auf Dünngitterinterpolation. In einem Vorverarbeitungsschritt werden lokale reduzierte Modelle an wenigen festen Punkten im Parameterraum erzeugt. Durch Interpolation der Systemmatrizen der lokalen reduzierten Modelle auf Matrixmannigfaltigkeiten erhält man das reduzierte Modell an einem beliebigen Punkt im Parameterraum. Als theoretische Grundlage führen wir passende Normen für parametrische Modelle ein und formulieren Bedingungen an die Parameterabhängigkeit des Modells, sodass diese Normen endlich sind. In unserer Analyse leiten wir dann eine obere Schranke für den Interpolationsfehler in diesen Normen her. Diese zeigt in numerischen Experimenten ein qualitativ gutes Verhalten. Wir demonstrieren, dass Interpolation auf dünnen Gittern effizienter ist als Interpolation auf vollen Gittern und dass Interpolation mit globalen Polynomen effizienter ist als Interpolation mit stückweise linearen Basisfunktionen, vorausgesetzt die Parameterabhängigkeit ist glatt. Außerdem zeigen wir anhand eines Benchmarks, welches in einem aktuellen Vergleichsartikel über parametrische Modellreduktionsverfahren verwendet wird, dass Verfahren, die auf der Interpolation von Systemmatrizen beruhen, mit anderen parametrischen Modellreduktionsverfahren konkurrieren können, wenn man die Symmetrie des Modells ausnutzt

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

Accounting And Forms Of Accountability In Ancient Civilizations: Mesopotamia And Ancient Egypt

The aim of this paper is to identify the relevance and implications of ancient accounting practices to the contemporary theorizing of accounting. The paper provides a synthesis of the literature on ancient accounting particularly in relation to issues of human accountability, identifies its major achievements and outlines some of the key challenges facing researchers. We argue that far from being an idiosyncratic research field of marginal interest, research in ancient accounting is a rich and promising undertaking. The paper concludes by considering a number of implications of ancient accounting practices for the theorizing of accounting and identifies news avenues for future research.