research
∙
05/28/2022
Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations
We prove that deep neural networks are capable of approximating solution...
share
research
∙
04/26/2022
Stopped Brownian-increment tamed Euler method
In this article we propose a new explicit Euler-type approximation metho...
research
∙
04/18/2022
Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations
Backward stochastic differential equations (BSDEs) appear in numeruous a...
research
∙
02/05/2022
Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions
The full history recursive multilevel Picard approximation method for se...
research
∙
12/13/2021
Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions
In many numerical simulations stochastic gradient descent (SGD) type opt...
research
∙
11/08/2021
Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms
Classical approximation results for stochastic differential equations an...
research
∙
10/15/2021
Strong L^p-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations
Full-history recursive multilevel Picard (MLP) approximation schemes hav...
research
∙
08/24/2021
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations
Backward stochastic differential equations (BSDEs) belong nowadays to th...
research
∙
03/19/2021
Multilevel Picard approximations for McKean-Vlasov stochastic differential equations
In the literatur there exist approximation methods for McKean-Vlasov sto...
research
∙
03/03/2021
Full history recursive multilevel Picard approximations for ordinary differential equations with expectations
We consider ordinary differential equations (ODEs) which involve expecta...
research
∙
12/22/2020
An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to app...
research
∙
09/05/2020
Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
The recently introduced full-history recursive multilevel Picard (MLP) a...
research
∙
05/20/2020
Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
One of the most challenging issues in applied mathematics is to develop ...
research
∙
12/05/2019
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Partial differential equations (PDEs) are a fundamental tool in the mode...
research
∙
11/04/2019
Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations
The main result of this article establishes strong convergence rates on ...
research
∙
07/15/2019
