Research Area S
Coordination: Prof. Dr. S. Le Borne
Efficient and reliable implementations of simulation processes require a thorough understanding of techniques combining adaptive discretization & approximation efforts together with advanced solution procedures on the one hand, and on the other hand, of current and future high-performance computing (HPC) aspects.
The projects in this research area develop novel adaption strategies for kernel-based approximations (S1), simulation strategies for large-scale inverse problems (S3), and develop tailored preconditioning techniques for large-scale fluid dynamic applications (S2).
An efficient use of these technologies usually requires not only a mere re-editing of software parts, but necessitates re-thinking of the employed mathematical approaches and algorithms.
S1 (Adaptive Kernels) - Adaptive kernel-based approaches for fluid flow simulations
S2 (Preconditioning) - Preconditioners for RBF-FD discretized fluid flow problems
S3 (Blood Flow) - Blood Flow modeling and estimation by magnetic particle imaging
S1: Adaptive Kernels
Principal Investigator: Armin Iske
Adaptive particle methods relying on meshfree kernels (e.g. radial basis functions) are popular tools for the numerical solution of partial differential equations [Fasshauer & McCourt (2015), Griebel & Schweitzer (2017)]. In particular, the utility of smoothed particle hydrodynamic (SPH) methods [Casulli (1990),Monaghan (2005), Vila (1999)] has been demonstrated for fluid flow simulations in the geosciences [Brecht et al. (2017),Fornberg & Flyer (2015), Iske & Randen (2005)] and other areas, including e.g. (mathematical) data analysis, and kernel-based (machine) learning. The application of adaptive kernel-based particle methods requires particular care, especially for their critical interaction with high order flux evaluations [Ben-Artzi & Falcovitz (2003),Goetz & Iske (2016),Toro (2009)]. The objective of this project is the development of novel concepts for the applicationoriented design of adaptive kernel methods to
satisfy the ever increasing demands of large-scale fluid dynamic computations [Ferrari et al. (2009)]. The innovation of this research refers to recent advances in the design of non-standard kernels providing highly flexible approximations, as they are required in contemporary applications of mathematical data science.
Principal Investigator: Sabine Le Borne
Recently, a new meshfree discretization technique for partial differential equations (PDEs) based on radial basis functions (RBF) has been introduced [Fornberg & Flyer (2015)]. Given a set of scattered nodes, RBF-interpolation is used to compute a (local) differentiation stencil (RBF-FD), leading to a sparse linear system of equations. The novelty and objective of this project is the development and analysis of rank-structured preconditioners for the linear systems of equations resulting from meshfree RBF-FD discretization of fluid flow problems
S3: Blood Flow
Principal Investigator: Christina Brandt
In cardiovascular diagnostics, magnetic particle imaging (MPI) has been developed to acquire blood flow patterns inside arteries. This new imaging modality offers benefits step and of the blood flow which requires efficient minimization algorithms for real-time computations. The problem will advance existing approaches for combined image and flow reconstruction dedicated to denoising and motion estimation to real-time image reconstruction and flow estimation for MPI.