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Existence, uniqueness, and numerical approximations for stochastic Burgers equationsJan 26 2019In this paper we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. ... More

A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientsSep 19 2018In recent years deep artificial neural networks (DNNs) have very successfully been employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, ... More

Existence and uniqueness properties for solutions of a class of Banach space valued evolution equationsDec 17 2018In this note we provide a self-contained proof of an existence and uniqueness result for a class of Banach space valued evolution equations with an additive forcing term. The framework of our abstract result includes, for example, finite dimensional ordinary ... More

Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equationsApr 07 2016May 17 2016This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly ... More

Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equationsOct 19 2017In this paper we propose and analyze explicit space-time discrete numerical approximations for additive space-time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers ... More

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearitiesMar 14 2019The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing ... More