Methods for Solving High-Dimensional partial Differential Equations

Specialeforsvar ved: Lyudmil Tsvetanov Stanev

 Titel: Methods for Solving High-Dimensional Partial Differential Equations

Abstract:  This thesis explores the methods used in solving high-dimensional partial differential equations (PDEs) to which the notion of "curse" of dimensionality refers. Several techniques are presented with algorithms that provide convergence of the schemes. The branching diffusion representation of semilinear PDEs and Monte Carlo approximations are investigated in relation to credit valuation adaption and non-linear Monte Carlo algorithms. The Deep Learning method is referred to in connection with the non-linear Black-Scholes equation with default risk, the Hamilton-Jacobi-Bellman equation and the Allen-Cahn equation. Branching diffusion representation of semilinear, elliptic PDEs with Dirichlet condition and linear gradient are examined. Applications and numerical examples are provided at the end of the thesis.

Vejleder: David Glavind Skovmand
Censor:   Jesper Lund