SSD - Zunino Seminar
Prof. Paolo Zunino, Ph.D. - Mixed-dimensional PDEs, Derivation, Analysis, Approximation and Applications
Department of Mathematics - MOX, Politecnico di Milano, Italy
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
Dr. Brendan Keith - The Surrogate Matrix Methodology: Low-cost Assembly for Isogeometric Analysis
Chair for Numerical Mathematics, the Technical University of Munich
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 .
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  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.
 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.
 L. B. Da Veiga, A. Buffa, G. Sangalli, and R. Vázquez. Mathematical analysis of variational isogeometric methods. Acta Numerica, 23:157–287, 2014.
 D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: a priori error estimation. SIAM Journal on Scientific Computing (to appear), 2019.
 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.
 D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: Low-cost assembly for isogeometric analysis. arXiv preprint arXiv:1904.06971, 2019.
 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.
 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.
EU Regional School - Fritzen Seminar
Dr. Felix Fritzen - Model Order Reduction and Data-assisted Surrogates in Materials Science
In engineering applications the use of manufactured or natural composite materials is a key technology enabling leight-weight constructions and outstanding physical properties. However, the prediction of the impact of changes of the microstructural configuration on the physical properties are nontrivial to foresee. In order to better understand the microstructure-property relations, the use of model order reduction can be of great use. First, it enables parametric studies that account for deviations of the phase properties. Second, it allows for the consideration of nonlinearities at a fraction of the compute time of fully resolved models. The link between model reduction and computational homogenization in linear and nonlinear settings will be explored in this lecture. Further, the use of state of the art data-assisted techniques for the surrogation of nonlinear material models will be discussed. Here the use of reduced order models for the generation of the samples of the full order model are found to be rather useful.
EU Regional School - Geuzaine Seminar
Prof. Christoph Geuzaine, Ph.D. - Recent Developments in Gmsh
Department of Electrical Engineering and Computer Science, University of Liege, Belgium
Gmsh (http://gmsh.info) is an open source finite element mesh generator with built-in pre- and post-processing facilities. Under continuous development for the last two decades, it has become the de facto standard for open source finite element mesh generation, with a large user community in both academia and industry. In this talk I will present an overview of Gmsh, and highlight recent developments including the support for constructive solid geometry, new robust and parallel meshing algorithms, flexible solver integration and a new multi-language Application Programming Interface. Time permitting I will also present an overview of current research directions for meshing based on the solution of partial differential equations: from surface remeshing to frame-based hex-meshing.
EU Regional School - Uciński Seminar
Prof. Dariusz Uciński Ph.D. - Optimum Experimental Design for Distributed Parameter System Identification
Institute of Control and Computation Engineering, University of Zielona Góra, Poland
The impossibility of observing the states of distributed parameter systems over the entire spatial domain raises the question of where to locate measurement sensors so as to estimate the unknown system parameters as accurately as possible. Both researchers and practitioners do not doubt that making use of sensors placed in an ‘intelligent’ manner may lead to dramatic gains in the achievable accuracy of the parameter estimates, so efficient sensor location strategies are highly desirable. In turn, the complexity of the sensor location problem implies that there are few sensor placement methods which are readily applicable to practical situations. What is more, they are not well known among researchers. The aim of the minicourse is to give account of both classical and recent original work on optimal sensor placement strategies for parameter identification in dynamic distributed systems modeled by partial differential equations. The reported work constitutes an attempt to meet the needs created by practical applications, especially regarding environmental processes, through the development of new techniques and algorithms or adopting methods which have been successful in akin fields of optimal control and optimum experimental design. While planning, real-valued functions of the Fisher information matrix of parameters are primarily employed as the performance indices to be minimized with respect to the sensor positions. Particular emphasis is placed on determining the ‘best’ way to guide moving and scanning sensors, and making the solutions independent of the parameters to be identified. A couple of case studies regarding the design of air quality monitoring networks are adopted as an illustration aiming at showing the strength of the proposed approach in studying practical problems. The course will be complemented by a discussion of more advanced topics including the related problem of optimum input design and the Bayesian approach to deal with the ill-posedness of parameter estimation.