Descrizione del progetto
Nuove approssimazioni permettono l’applicazione di potenti solutori convenzionali
Le equazioni differenziali alle derivate parziali (EDP) sono fondamentali per descrivere il mondo che ci circonda, in quanto spiegano matematicamente come cambia un dato fenomeno rispetto ai cambiamenti di altri fattori. Esse possono modellizzare le dinamiche relative allo scambio di gas nel sistema circolatorio, i cambiamenti nel mercato azionario e la propagazione dei raggi X nei materiali. Le EDP totalmente non lineari sono un caso particolarmente difficile per il quale le soluzioni convenzionali potrebbero non esistere o persino essere definite. Il progetto DAFNE, finanziato dall’UE, getterà le basi per mettere in atto approcci risolutivi utilizzando potenti metodi degli elementi finiti con impatto in aree che vanno dalla fisica e dalla geometria ai fenomeni di trasporto e alla finanza.
Obiettivo
Fully nonlinear partial differential equations (PDE) arise in many applications ranging from physics to economy. They are different from PDEs in mechanics, and the PDE theory relies on the generalized solution concept of so-called viscosity solutions. Monotone finite difference methods (FDM) are provably convergent for approximating viscosity solutions, but are restricted to regular meshes and low-order approximations, thus having limitations in resolving realistic geometries or dealing with local mesh refinement. As viscosity solutions are lacking smoothness properties in general, adaptive approximations are desirable. In contrast to FDM, finite element methods (FEM) offer the possibility of high-order approximations with flexibility in adaptive and automatic mesh design. However, provably convergent FEM formulations for viscosity solutions to nonvariational problems are as yet unknown.
With a background in the numerical analysis of PDEs, especially the theory of FEM and adaptive algorithms, DAFNE aims at laying the theoretical and practical foundation for the application of FEM and automatic mesh-refinement algorithms to fully nonlinear equations. The focus is on the large class of Hamilton-Jacobi-Bellman (HJB) equations. They originated from stochastic control problems, but more generally comprise many classical and relevant equations like Pucci's equation or the Monge-Ampère equation
with applications in finance, optimal transport, physics, and geometry.
The novel approach is to estimate local regularity properties through the control variable in the HJB formulation. This (a) gives rise to new regularization strategies and (b) indicates where the mesh needs to be refined. Both achievements are key to the design of a new FEM formulation.
The project is at the frontiers of PDE analysis, numerical analysis, and scientific computing. The long-term goal is to establish the first convergence proofs for adaptive FEM simulations of fully nonlinear phenomena.
Campo scientifico
- natural sciencescomputer and information sciencescomputational science
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesmathematicsapplied mathematicsnumerical analysis
Parole chiave
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-STG - Starting GrantIstituzione ospitante
07743 JENA
Germania