7,255 research outputs found
Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems
In this paper, we establish that for a wide class of controlled stochastic
differential equations (SDEs) with stiff coefficients, the value functions of
corresponding zero-sum games can be represented by a deep artificial neural
network (DNN), whose complexity grows at most polynomially in both the
dimension of the state equation and the reciprocal of the required accuracy.
Such nonlinear stiff systems may arise, for example, from Galerkin
approximations of controlled stochastic partial differential equations (SPDEs),
or controlled PDEs with uncertain initial conditions and source terms. This
implies that DNNs can break the curse of dimensionality in numerical
approximations and optimal control of PDEs and SPDEs. The main ingredient of
our proof is to construct a suitable discrete-time system to effectively
approximate the evolution of the underlying stochastic dynamics. Similar ideas
can also be applied to obtain expression rates of DNNs for value functions
induced by stiff systems with regime switching coefficients and driven by
general L\'{e}vy noise.Comment: This revised version has been accepted for publication in Analysis
and Application
The physical basis for Parrondo's games
Several authors have implied that the original inspiration for Parrondo's
games was a physical system called a ``flashing Brownian ratchet''. The
relationship seems to be intuitively clear but, surprisingly, has not yet been
established with rigor. In this paper, we apply standard finite-difference
methods of numerical analysis to the Fokker-Planck equation. We derive a set of
finite difference equations and show that they have the same form as Parrondo's
games. Parrondo's games, are in effect, a particular way of sampling a
Fokker-Planck equation. Physical Brownian ratchets have been constructed and
have worked. It is hoped that the finite element method presented here will be
useful in the simulation and design of flashing Brownian ratchets.Comment: 10 pages and 2 figure
Multigrid methods for two-player zero-sum stochastic games
We present a fast numerical algorithm for large scale zero-sum stochastic
games with perfect information, which combines policy iteration and algebraic
multigrid methods. This algorithm can be applied either to a true finite state
space zero-sum two player game or to the discretization of an Isaacs equation.
We present numerical tests on discretizations of Isaacs equations or
variational inequalities. We also present a full multi-level policy iteration,
similar to FMG, which allows to improve substantially the computation time for
solving some variational inequalities.Comment: 31 page
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