Events

SSD - Helzel Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Dr.Christiane Helzel - A Third Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Differential Equations

Institute of Mathematics, Heinrich-Heine-Universität Düsseldorf

Abstract

The wave propagation algorithm of LeVeque and its implementation in the software package Clawpack are widely used for the approximation of hyperbolic problems. The method belongs to the class of truly multidimensional, high-resolution finite volume methods. Furthermore, it can be characterised as a one-step Lax-Wendroff type method, i.e. the PDE is solved simultaneously in space and time. Approximations obtained with this method are second order accurate for smooth solutions and avoid unphysical oscillations near discontinuities or steep gradients. 
 
Second order accurate methods are often a good choice in terms of balance between computational cost and desired resolution, especially for solutions dominated by shock waves or contact discontinuities and relatively simple structures between these discontinuities. However, for problems containing complicated smooth solution structures, where the accurate resolution of small scales is require, schemes with a higher order of accuracy are more efficient and computationally affordable.
 
I will present my recent work towards the construction of a third order accurate wave propagation algorithm for hyperbolic pdes. The resulting method shares main properties with the original method, i.e. it is based on a wave decomposition at grid cell interfaces, it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of
a wave limiter. Furthermore, I will compare this new method with other recently proposed third order accurate finite volume methods. 

 

SSD - Benigni Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Dr. Andrea Benigni - Energy Systems Simulation: Challenges and Solutions

Institute of Energy and Climate Research, Forschungszentrum Jülich

Abstract

One can define an energy system as a system that converts one or more energy fluxes into other energy fluxes of a different kind. This definition may describe a relatively small system – for instance a power plant, a chemical plant, the heating and cooling system of a single-family house – as well as one covering larger energy needs – as for instance the needs of a city, of a country, or even of a continent. As energy systems are developed through the centuries, the way we structure these systems goes through changes affected by contextual conditions. Recently, concerns about the availability of traditional fossil energy sources and their environmental effects are revolutionizing the way energy systems are planned, designed, and operated.

Modern energy systems are expected to be multi-modal and incorporate electrical, gas, and heat networks – to achieve maximum usage of every form of energy available – and to include storage capacity. The distributed nature of new resources (generation and storage) and the partici- pation of loads in energy management require fast, reactive control, and protection. In this context the monitoring and control of modern energy systems are expected to be characterized by distribution of functions. At the same time – to ensure optimal coordination – a large use of communication media is envisioned. Interactions between continuous dynamics and dis- crete events are becoming more relevant due to the increasing number of controllable devices (e.g., power electronic converters in the electrical grids) and the use of networked control schemes. Energy systems, furthermore, are increasingly driven by market competition. Because of these characteristics and because of human involvement, modern energy systems can therefore be classified as complex and concerns about emerging behaviors might be raised.

The complexity of such systems poses significant challenges on how these systems are planned, designed, and operated and numerical simulation it is a fundamental tool to tackle those challenges.

In the seminar we will review recent development in simulation methods for energy systems.

SSD - Hoel Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Dr. Hakon Hoel - Multilevel Monte Carlo Methods for Data Assimilation

Department of Mathematics, RWTH Aachen University

Abstract

The ensemble Kalman filter (EnKF) is a sequential filtering method
based on an ensemble of particle paths and sample moment (Monte Carlo)
approximations of true moments required in the filter update
step. EnKF is often both robust and efficient, but its performance may
suffer in settings where the computational cost of accurate
simulations of particles is high. The multilevel Monte Carlo method
(MLMC) is an extension of the classical Monte Carlo method which by
sampling stochastic realizations on a hierarchy of resolutions may
reduce the computational cost of moment approximations by orders of
magnitude.  In this talk I will present recent results on combining MLMC and EnKF 
to construct the multilevel ensemble Kalman filter (MLEnKF).
Theoretical results and numerical evidence of the performance gain of
MLEnKF over EnKF will be presented.

SSD - Zunino Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Paolo Zunino, Ph.D. - Mixed-dimensional PDEs, Derivation, Analysis, Approximation and Applications

Department of Mathematics - MOX, Politecnico di Milano, Italy

Abstract

The coupling of three-dimensional (3D) continua with embedded (1D) networks is not well investigated yet from the standpoint of mathematical analysis and numerical approximation, although it arises in applications of paramount importance such as microcirculation, flow through perforated media and the study of reinforced materials, just to make a few examples.

We address this mathematical problem within a unified framework, designed to formulate and approximate coupled partial differential equations (PDEs) on manifolds with heterogeneous dimensionality, arising from topological model reduction. We cast such mathematical problem in the framework of mixed-dimensional PDEs. The main difficulty consists in the ill-posedness of restriction operators (such as the trace operator) applied on manifolds with co-dimension larger than one. Partial results about the analysis and the approximation of this type of problems have appeared only recently.

We will overcome the challenges of defining and approximating PDEs on manifolds with high dimensionality gap by means of nonlocal restriction operators that combine standard traces with mean values of the solution on low dimensional manifolds. This new approach has the fundamental advantage to enable the approximation of the problem using Galerkin projections on Hilbert spaces, which can not be otherwise applied because of regularity issues. Furthermore, combining the numerical error analysis with the model reduction approach, the concurrent modeling and discretization errors in the approximation of the original fully dimensional problem can be quantified and balanced.
Our ultimate objective is to exploit topological model reduction to perform large scale simulations of significant impact on medicine and geophysics.

SSD - Keith Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Dr. Brendan Keith - The Surrogate Matrix Methodology: Low-cost Assembly for Isogeometric Analysis

Chair for Numerical Mathematics, the Technical University of Munich

Abstract

This talk will introduce a new class of methods in isogeometric analysis (IGA) based on the surrogate matrix methodology [1,3–5]. The objective is to lower the cost of matrix assembly in IGA without sacrificing accuracy. It is well known that isogeometric methods face a great computational burden at the point of matrix assembly. This is due, in large part, to the quadrature involved in directly computing B-spline or NURBS basis function interactions; see, e.g., [2,6]. Nevertheless, we will show that significant classes of B-splines and NURBS bases have an intrinsic structure which can be easily exploited to avoid most of this quadrature.

The new assembly strategy we will present involves performing quadrature for only a small fraction of the IGA basis function interactions and then approximating the rest through, for example, interpolation. Therefore, most of the quadrature involved in standard IGA assembly is not performed at all and
the predominant expense in most IGA assembly algorithms is avoided. Our strategy, which may be viewed as constructing variable-scale approximations (i.e., surrogates) for each system matrix, retains the accuracy of the standard methods because the structure of our B-spline/NURBS bases allow for a simple correspondence between matrix entries and smooth functions [5].

In this talk, we will summarize the theoretical aspects of the surrogate matrix methodology which, in turn, certify the convergence of new surrogate IGA methods for Poisson’s equation, membrane vibration, plate bending, Stokes’ flow, nonlinear elasticity, and other problems. We will also focus on the implementation of the methodology in existing IGA code, using the open-source GeoPDEs library [7] as an example. For the sake of demonstration, we will show assembly speed-ups of up to fifty times, after only a few small modifications to this software. The capacity for even further speed-ups is clearly possible and similar modifications could be made to other contemporary software libraries.

[1] S. Bauer, M. Mohr, U. Rüde, J. Weismüller, M. Wittmann, and B. Wohlmuth. A two-scale approach for efficient on-the-fly operator assembly in massively parallel high performance multigrid codes. Applied Numerical Mathematics, 122:14–38, 2017.
[2] L. B. Da Veiga, A. Buffa, G. Sangalli, and R. Vázquez. Mathematical analysis of variational isogeometric methods. Acta Numerica, 23:157–287, 2014.
[3] D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: a priori error estimation. SIAM Journal on Scientific Computing (to appear), 2019.
[4] D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: A reference implementation for low-cost assembly in isogeometric analysis. arXiv preprint arXiv:1909.04029, 2019.
[5] D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: Low-cost assembly for isogeometric analysis. arXiv preprint arXiv:1904.06971, 2019.
[6] T. J. Hughes, A. Reali, and G. Sangalli. Efficient quadrature for NURBS-based isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 199(5-8):301–313, 2010.
[7] R. Vázquez. A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0. Computers & Mathematics with Applications, 72(3):523–554, 2016.