Adaptive numerical methods for PDEs

Numerical Methods for PDEs

Numerical Methods for PDEs

Adaptive numerical methods for PDEs 1 Ronald A . DeVore

Résumé. Les méthodes adaptatives sont d’usage courant pour la résolution numérique des EDP. Il n’existe cependant pas de théorie bien établie analysant leur performance et justifiant leur utilisation. L’objet de cet exposé est de présenter les premiers éléments d’une telle théorie, dont les pierres angulaires sont l’approximation non-linéaire et les théorèmes de régularité pour les EDP. Une mét...

Numerical Methods for Nonlinear PDEs in Finance

Many problems in finance can be posed in terms of an optimal stochastic control. Some well-known examples include transaction cost/uncertain volatility models [17, 2, 25], passport options [1, 26], unequal borrowing/lending costs in option pricing [9], risk control in reinsurance [23], optimal withdrawals in variable annuities[13], optimal execution of trades [20, 19], and asset allocation [28,...

Gap in Nonlinear Equivalence for Numerical Methods for PDEs

For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general nonlinear equivalence between stability and conve...