A Full-Density Approach to Simulating Random Iteration Equations with Applications

Abstract

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.

Figures (8)

• Figure 1: Illustration of the posterior densities for different iteration numbers of the Rosenzweig McArthur RDE model. One can see the evolution of the square uniform initial density and how it evolves due to the dynamics of the Rosenberg McArthur model with random variable parameters to the steady state density.
• Figure 2: Illustration of the posterior densities for different iteration numbers of the Rosenzweig McArthur SDE model. The evolution of the densities is presented roughly over one period of the orbit in the $x_1$-$x_2$-space starting at iteration 476 ($t=23.8$), with dominant stochastic effects due to the SDE diffusion and initial state density.
• Figure 3: Numerical impression of step sizes for the discretization of the dynamic systems by deterministic simulations in the relevant time and initial value range (two starting value examples, which are vertices of each initial density box) with ode45 (Runge-Kutta-(4,5) algorithm) as ground truth and the Euler algorithm which is basis of the simulaion strategy. Left: for system 1 utilized in the simulation in Figure \ref{['fig:Res:RosMcA_iteration_posterior_004_2']} with starting values $(0.1,0.1)$ and $(0.5,0.5)$, right: for system 2 utilized in the simulation in Figure \ref{['fig:Res:RosMcA_iteration_posteriorSDE_004']} with starting values $(0.4,0.4)$ and $(0.6,0.6)$. Sufficient numerical approximation accuracy in the context of this study can be observed considering the simulated stochasticities.
• Figure 4: Illustration of the objective function. Left: With two local minima on the $x_1$-axis. Right: Himmelblau's function of optimization for four local minima.
• Figure 5: Illustration of the posterior densities for different iteration numbers for FDGD-II for the objective function with two local minima. One can see the evolution of the square initial density being first attracted to the saddle point and second the convergence to the two local minima.