| Project Title: Low-rank approximation for tensor modeling Advisors: A. Mantzaflaris (Inria), C. Giannelli (UniFI) Mentor: Michael Gabay (Artelys) |
| Objectives: This project aims at low-rank representations of numerical solutions of high-dimensional partial differential operators, for use in modeling and optimization. In particular we aim at accelerating isogeometric discretizations that are based on tensor-product or hierarchical splines. Solving using grid-based methods suffers from the, so called, curse of dimensionality and scales badly with the number of knots in the spline representation. We will tackle this problem using tensor decomposition by means of the moment method. The problem discretization will be carried out using tensor-product spline bases, enhanced by the use of low-rank approximation of the computational domain and the numerical solution. Consequently, the computational complexity of the method will depend on the resulting tensor rank and will be linear in the grid dimensions. In a second phase the discretization will be enriched using hierarchical splines, for localizing the problem and for gaining efficiency in high dimensional instances |
| Expected Results: Tackle the curse of dimensionality of high-dimensional PDE operators. Fully exploit isogeometric analysis, as well as the tensor structure of NURBS and hierarchical spaces. Develop adaptive and scalable solution methods. |
| Planned secondment(s): C. Giannelli UniFI (M25-34) to work on efficient local refinement by means of hierarchical spline technologies; with M. Gabay (Artelys) to work on optimisation tools for low-rank tensor approximation (M23-24). |
| Joint degree: Université Côte d’Azur, University of Florence |
