136 citations · 451 across the 26 of their papers we have counts for
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Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration con…
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
Christian Beck, Fabian Hornung, Martin Hutzenthaler +2
One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministi…
Approximating exit times of continuous Markov processes
Thomas Kruse, Mikhail Urusov
The time at which a one-dimensional continuous strong Markov process attains a boundary point of its state space is a discontinuous path functional and it is, therefore, unclear wh…
Wasserstein convergence rates for random bit approximations of continuous Markov processes
Stefan Ankirchner, Thomas Kruse, Mikhail Urusov
We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of coin tossi…
A functional limit theorem for coin tossing Markov chains
Stefan Ankirchner, Thomas Kruse, Mikhail Urusov
We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability , respectively. The theorem e…
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +1
Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential…