Descripción del proyecto
Las nuevas aproximaciones permiten aplicar potentes solucionadores convencionales
Las ecuaciones diferenciales parciales (EDP) son fundamentales para describir el mundo que nos rodea, ya que explican matemáticamente cómo cambia un determinado fenómeno con respecto a los cambios de otros factores. Pueden modelar la dinámica relacionada con el intercambio de gases en el sistema circulatorio, los cambios en el mercado de valores y la propagación de los rayos X en los materiales. Las EDP totalmente no lineales son un caso muy complicado para el que puede que no existan soluciones convencionales o, incluso, que no estén definidas. El proyecto DAFNE, financiado con fondos europeos, sentará las bases para aplicar enfoques basados en soluciones a través de potentes métodos de elementos finitos con impacto en áreas que van desde la física y la geometría hasta los fenómenos de transporte y las finanzas.
Objetivo
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.
Ámbito científico
- natural sciencescomputer and information sciencescomputational science
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesmathematicsapplied mathematicsnumerical analysis
Palabras clave
Programa(s)
Régimen de financiación
ERC-STG - Starting GrantInstitución de acogida
07743 JENA
Alemania