A Full-Density Approach to Simulating Random Iteration Equations with Applications
This work addresses the computational bottleneck of simulating random iteration equations for researchers in applied mathematics and engineering, offering a more efficient approach, though it appears incremental based on prior static random equation work.
The study introduces a unified computational framework for simulating random iteration equations by propagating full probability densities stepwise, eliminating the need for repetitive Monte Carlo simulations. It demonstrates applications including a novel full-density gradient descent method for global optimization under uncertainty, with examples in stochastic differential equations and chaotic mappings.
The goal of this study is to introduce a unified computational framework for simulating random iteration equations (RIE), understood as iteration equations containing random variables. The novelty of this work is that full probability densities of the state vectors are propagated stepwise through the iterations avoiding the need of repetitive pathwise Monte Carlo simulations of the iteration equation. The presentation of the methodology is conceptually efficient based on recent work on static random equations and intentionally accessible. The technical requirements on the RIE are minimal based on the previous work, allowing for potential nonlinearities, discontinuities and stochasticities in the transfer function, as well as nonstandard densities and diffusion processes. As results, illustrative applications of random and stochastic differential equation simulations, a novel full-density gradient descent method (FDGD) for global optimization under uncertainty and examples of chaotic mappings are presented in order to demonstrate the breadth of the utility of this framework. In total, the character of the presentation is explorative and encourages new applications and theoretical studies.