arXiv subjects

Alexander Vogler

Publications and source records attributed to Alexander Vogler.

5 recordsLinked to original sources

arXiv2024

The Lions Derivative in Infinite Dimensions -- Application to Higher Order Expansion of Mean-Field SPDEs

In this paper we present a new interpretation of the Lions derivative as the Radon-Nikodym derivative of a vector measure, which provides a canonical extension of the Lions derivative for functions taking values in infinite dimensional Banach spaces. This is of particular relevance for the analysis of Hilbert space valued Mean-Field equations. As an illustration we derive a mild Ito-formula for Mean-Field stochastic partial differential equations (SPDEs), which provides the basis for a higher order Taylor expansion and higher order numerical schemes.

math.PR↗

arXiv2023

Approximation of Optimal Feedback Controls for Stochastic Reaction-Diffusion Equations

In this paper we present a method to approximate optimal feedback controls for stochastic reaction-diffusion equations. We derive two approximation results providing the theoretical foundation of our approach and allowing for explicit error estimates. The approximation of optimal feedback controls by neural networks is discussed as an explicit application of our method. Finally we provide a numerical examples to illustrate our findings.

math.OC↗

arXiv2023

Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations

We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computational complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can also be applied to stochastic control problems for high dimensional stochastic differential equations and more general stochastic partial differential equations.

math.OC↗

arXiv2022

An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations

We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovičs, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature approximation on Wiener space introduced by Lyons and Victoir. Our analysis allows to derive stability estimates and explicit weak convergence rates. As a particular example, a cubature approximation for stochastic differential equations driven by continuous Gaussian martingales is given.

math.PR↗

arXiv2020

Optimal control of mean field equations with monotone coefficients and applications in neuroscience

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^α$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),α_t)dt+σ(t,X_t,\mathcal L(X_t),α_t)dW_t,$ $X_0\sim μ$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $α_t$ is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipshitz condition, and that the dynamics is subject to a (convex) level set constraint of the form $π(X_t)\leq0$. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipshitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle and then numerically investigate a gradient algorithm for the approximation of the optimal control.

math.PR↗