Monte Carlo Methods and deep neural network approximations for high-dimensional Markov decision processes and elliptic partial differential equations / vorgelegt von Konrad Malte Kleinberg. Wuppertal, August 2026
Inhalt
- Introduction
- Motivation
- Two High-Dimensional Problems
- A Common Approximation Strategy
- Literature
- Contributions of the Dissertation
- Organization
- Monte Carlo Methods and Deep Neural Network Approximations for High-Dimensional Markov Decision Processes
- Nonlinear Monte Carlo Methods for Bellman Equations of High-Dimensional Markov Decision Processes
- Existence and Uniqueness of Solutions of Functional Fixed-Point Equations
- Full-History Recursive Multilevel Fixed-Point (MLFP) Approximations
- Mathematical Description of MLFP Approximations
- Measurability and Distributional Properties for MLFP Approximations
- Integrability Properties for MLFP Approximations
- Recursive Error Bounds for MLFP Approximations
- Non-Recursive Error Bounds for MLFP Approximations
- Computational Complexity Analysis for MLFP Approximations
- Artificial Neural Network (ANN) Calculus
- Structured Descriptions of ANNs
- Compositions, Affine Linear Transformations, Powers, and Extensions of ANNs
- Parallelizations of ANNs
- Linear Combinations of ANNs
- Deep Neural Network Approximations for Bellman Equations of High-Dimensional Markov Decision Processes
- A Stability Result for Solutions of Certain Stochastic Fixed-Point Equations
- Auxiliary ANN Representations and Approximations
- ANN Representations and Approximations for the Maximum Function
- ANN Representations for MLFP Approximations
- ANN Approximations for Functional Fixed-Point Equations
- ANN Approximations for Functional Fixed-Point Equations with General Activation Functions
- ANN Approximations for Functional Fixed-Point Equations with Specific Activation Functions and One-Step Transitions
- ANN Approximations for Bellman Equations with Specific Activation Functions and One-Step Transitions
- Monte Carlo and Deep Neural Network Approximations of Elliptic PDEs with Drift and Killing
- Walk-on-Spheres Monte Carlo Methods for Elliptic PDEs with Drift and Killing
- A Girsanov-Based Feynman-Kac Representation
- A Modified Walk-on-Spheres Algorithm
- Error Analysis for the Modified Walk-on-Spheres Algorithm
- Deep Neural Network Approximation of Elliptic PDEs with Drift and Killing
- ANN Approximations of Modified Walk-on-Spheres Algorithms
- ANN Approximations for Solutions of Elliptic PDEs
- Conclusion and Outlook