Project acronym CHANGE
Project New CHallenges for (adaptive) PDE solvers: the interplay of ANalysis and GEometry
Researcher (PI) Annalisa BUFFA
Host Institution (HI) ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Call Details Advanced Grant (AdG), PE1, ERC-2015-AdG
Summary The simulation of Partial Differential Equations (PDEs) is an indispensable tool for innovation in science and technology.
Computer-based simulation of PDEs approximates unknowns defined on a geometrical entity such as the computational domain with all of its properties. Mainly due to historical reasons, geometric design and numerical methods for PDEs have been developed independently, resulting in tools that rely on different representations of the same objects.
CHANGE aims at developing innovative mathematical tools for numerically solving PDEs and for geometric modeling and processing, the final goal being the definition of a common framework where geometrical entities and simulation are coherently integrated and where adaptive methods can be used to guarantee optimal use of computer resources, from the geometric description to the simulation.
We will concentrate on two classes of methods for the discretisation of PDEs that are having growing impact:
isogeometric methods and variational methods on polyhedral partitions. They are both extensions of standard finite elements enjoying exciting features, but both lack of an ad-hoc geometric modelling counterpart.
We will extend numerical methods to ensure robustness on the most general geometric models, and we will develop geometric tools to construct, manipulate and refine such models. Based on our tools, we will design an innovative adaptive framework, that jointly exploits multilevel representation of geometric entities and PDE unknowns.
Moreover, efficient algorithms call for efficient implementation: the issue of the optimisation of our algorithms on modern computer architecture will be addressed.
Our research (and the team involved in the project) will combine competencies in computer science, numerical analysis, high performance computing, and computational mechanics. Leveraging our innovative tools, we will also tackle challenging numerical problems deriving from bio-mechanical applications.
Summary
The simulation of Partial Differential Equations (PDEs) is an indispensable tool for innovation in science and technology.
Computer-based simulation of PDEs approximates unknowns defined on a geometrical entity such as the computational domain with all of its properties. Mainly due to historical reasons, geometric design and numerical methods for PDEs have been developed independently, resulting in tools that rely on different representations of the same objects.
CHANGE aims at developing innovative mathematical tools for numerically solving PDEs and for geometric modeling and processing, the final goal being the definition of a common framework where geometrical entities and simulation are coherently integrated and where adaptive methods can be used to guarantee optimal use of computer resources, from the geometric description to the simulation.
We will concentrate on two classes of methods for the discretisation of PDEs that are having growing impact:
isogeometric methods and variational methods on polyhedral partitions. They are both extensions of standard finite elements enjoying exciting features, but both lack of an ad-hoc geometric modelling counterpart.
We will extend numerical methods to ensure robustness on the most general geometric models, and we will develop geometric tools to construct, manipulate and refine such models. Based on our tools, we will design an innovative adaptive framework, that jointly exploits multilevel representation of geometric entities and PDE unknowns.
Moreover, efficient algorithms call for efficient implementation: the issue of the optimisation of our algorithms on modern computer architecture will be addressed.
Our research (and the team involved in the project) will combine competencies in computer science, numerical analysis, high performance computing, and computational mechanics. Leveraging our innovative tools, we will also tackle challenging numerical problems deriving from bio-mechanical applications.
Max ERC Funding
2 199 219 €
Duration
Start date: 2016-10-01, End date: 2021-09-30
Project acronym NUHGD
Project Non Uniform Hyperbolicity in Global Dynamics
Researcher (PI) Sylvain CROVISIER
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Advanced Grant (AdG), PE1, ERC-2015-AdG
Summary An important part of differentiable dynamics has been developed from the uniformly hyperbolic systems. These systems have been introduced by Smale in the 60's in order to address chaotic behavior and are now deeply understood from the qualitative, symbolic and statistic viewpoints. They correspond to the structurally stable dynamics. It appeared that large classes of non-hyperbolic systems also exist. Since the 80's, different notions of relaxed hyperbolicity have been introduced: non-uniformly hyperbolic measures, partial hyperbolicity, ... They allowed to extend the previous approach to other families of systems and to handle new examples of dynamics: the fine description of the dynamics of Hénon maps for instance.
The development of local perturbative technics have brought a rebirth for the qualitative description of generic systems. It also opened the door to describe more globally the spaces of differentiable dynamics. For instance, it allowed recent progresses towards the Palis conjecture which characterizes the absence of uniform hyperbolicity by the homoclinic bifurcations — homoclinic tangencies or heterodimensional cycles. We propose in the present project to develop technics for realizing more global perturbations, yielding a breakthrough in the subject. This would settle this conjecture for C1 diffeomorphisms and imply other classification results.
These past years we have understood how qualitative dynamics of generic systems decompose into invariant pieces. We are now ready to describe more precisely the dynamics inside the pieces. We propose to combine these new geometrical ideas to the ergodic theory of non-uniformly hyperbolic systems. This will improve significantly our understanding of general smooth systems (for instance provide existence and finiteness of physical measures and measures of maximal entropy for new classes of systems beyond uniform hyperbolicity).
Summary
An important part of differentiable dynamics has been developed from the uniformly hyperbolic systems. These systems have been introduced by Smale in the 60's in order to address chaotic behavior and are now deeply understood from the qualitative, symbolic and statistic viewpoints. They correspond to the structurally stable dynamics. It appeared that large classes of non-hyperbolic systems also exist. Since the 80's, different notions of relaxed hyperbolicity have been introduced: non-uniformly hyperbolic measures, partial hyperbolicity, ... They allowed to extend the previous approach to other families of systems and to handle new examples of dynamics: the fine description of the dynamics of Hénon maps for instance.
The development of local perturbative technics have brought a rebirth for the qualitative description of generic systems. It also opened the door to describe more globally the spaces of differentiable dynamics. For instance, it allowed recent progresses towards the Palis conjecture which characterizes the absence of uniform hyperbolicity by the homoclinic bifurcations — homoclinic tangencies or heterodimensional cycles. We propose in the present project to develop technics for realizing more global perturbations, yielding a breakthrough in the subject. This would settle this conjecture for C1 diffeomorphisms and imply other classification results.
These past years we have understood how qualitative dynamics of generic systems decompose into invariant pieces. We are now ready to describe more precisely the dynamics inside the pieces. We propose to combine these new geometrical ideas to the ergodic theory of non-uniformly hyperbolic systems. This will improve significantly our understanding of general smooth systems (for instance provide existence and finiteness of physical measures and measures of maximal entropy for new classes of systems beyond uniform hyperbolicity).
Max ERC Funding
1 229 255 €
Duration
Start date: 2016-09-01, End date: 2021-08-31
