Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with …
Recently, several deep learning (DL) methods for approximating high-dimensional partial
differential equations (PDEs) have been proposed. The interest that these methods have …
differential equations (PDEs) have been proposed. The interest that these methods have …
Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes
have been introduced and shown to overcome the curse of dimensionality in the numerical …
have been introduced and shown to overcome the curse of dimensionality in the numerical …
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling
of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a …
of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a …
Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions
The full history recursive multilevel Picard approximation method for semilinear parabolic
partial differential equations (PDEs) is the only method which provably overcomes the curse …
partial differential equations (PDEs) is the only method which provably overcomes the curse …
Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
The recently introduced full-history recursive multilevel Picard (MLP) approximation methods
have turned out to be quite successful in the numerical approximation of solutions of high …
have turned out to be quite successful in the numerical approximation of solutions of high …
On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
The classical Feynman–Kac identity builds a bridge between stochastic analysis and partial
differential equations (PDEs) by providing stochastic representations for classical solutions …
differential equations (PDEs) by providing stochastic representations for classical solutions …
Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis
In this paper we introduce a multilevel Picard approximation algorithm for semilinear
parabolic partial integro-differential equations (PIDEs). We prove that the numerical …
parabolic partial integro-differential equations (PIDEs). We prove that the numerical …
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations
Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently
studied equations in stochastic analysis and computational stochastics. BSDEs in …
studied equations in stochastic analysis and computational stochastics. BSDEs in …
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating …
We prove that multilevel Picard approximations and deep neural networks with ReLU, leaky
ReLU, and softplus activation are capable of approximating solutions of semilinear …
ReLU, and softplus activation are capable of approximating solutions of semilinear …
[PDF][PDF] Numerical approximation methods for semilinear partial differential equations with gradient-dependent nonlinearities
K Pohl - 2024 - duepublico2.uni-due.de
In this thesis we study two numerical approximation methods for the estimation of solutions
of a class of semilinear partial differential equations (PDEs) with gradient-dependent …
of a class of semilinear partial differential equations (PDEs) with gradient-dependent …