Low Rank Tensor Isogeometric Analysis
The use of tensor methods in the field of numerical simulation was explored the last decade, with the aim to overcome the curse of dimensionality, ie. the exponential complexity with respect to the spatial dimension of the computational domain. With the advent of Isogeometric Analysis (IGA) during the same period of time, the very same difficulty of dimensionality has appeared, in particular in the task of matrix assembly.
Preliminary results obtained during the first period indicate that tensor methods possess significant potential for improving the overall efficiency of isogeometric simulations, which is currently one of the main bottlenecks in applications. In particular, we addressed the task of efficiently generating the required system matrices. More precisely we performed the required multi-dimensional integration via numerical quadrature applied to em univariate spline functions over real intervals.
The present subproject aims at employing the low rank tensor approximation strategy throughout the entire isogeometric simulation procedure and for all dimensions. This effectively exploits the tensor-product structure of multi-variable B-splines. A strong point of our approach is the low rank representation of intrinsic geometric quantities, which provides a Kronecker decomposition of the system matrix. By adopting this representation as the central data structure of our isogeometric simulations, we shall tackle the curse of dimensionality of isogeometric methods, which is especially important for the proposed space-time IgA in subproject 03. This subproject comprises five activities:
Activity 1 "Robust multi-dimensional numerical quadrature"
will generalize the matrix assembly technique of [A. Mantzaflaris, B. Juettler, B. Khoromskij, and U. Langer.
Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation.
In: Curves and Surfaces. Ed. by J.-D. Boissonat et al. Springer 2015, LNCS 9213, pp 321-340. Technical report available at www.gs.jku.at/pubs/NFNreport19.pdf.] to arbitrary dimension, relying on general tensor-decomposition methods instead of SVD, which was used in the bivariate case.
Activity 2 "Effective Low Rank Approximation of IgA matrices."
will be devoted to the efficient Kronecker decomposition of the system matrix for complex geometries. We will explore the behavior of the tensor rank for real CAD geometries and apply optimization techniques to reduce the complexity (more precisely, the number of terms in the resulting low rank approximations) of the parameterization of the geometry.
Activity 3 "Low rank approximations to IgA PDE solutions"
aims at exploiting the tensor-structure of the solution in 3D and 4D (space-time) IgA discretizations. Moreover, we shall make use of the Kronecker representation in iterative solving and for the computation of low rank solutions.
Activity 4 "Adaptive Methods"
explores the use of adaptivity in the low rank approximation procedure and uses these techniques in the context of adaptively refined hierarchical splines.
Finally, an efficient implementation of low rank IgA methods will be provided in Activity 5 "Efficient implementation". This activity also provides the scientific and technical coordination of the G+Smo library, which is the common frame for the software development of the entire NFN.