Validated Enclosure of Uncertain Nonlinear Equations Using SIVIA Monte Carlo

Abstract

The dynamical systems in various science and engineering problems are often governed by nonlinear equations (differential equations). Due to insufficiency and incompleteness of system information, the parameters in such equations may have uncertainty. Interval analysis serves as an efficient tool for handling uncertainties in terms of closed intervals. One of the major problems with interval analysis is handling “dependency problems” for computation of the tightest range of solution enclosure or exact enclosure. Such dependency problems are often observed while dealing with complex nonlinear equations. In this regard, initially, two test problems comprising interval nonlinear equations are considered. The Set Inversion via Interval Analysis (SIVIA) along with the Monte Carlo approach is used to compute the exact enclosure of the test problems. Further, the efficiency of the proposed approach has also been verified for solving nonlinear differential equation (Van der Pol oscillator) subject to interval initial conditions.

Appendix

Forward–backward contractor: The forward–backward contractor is based on constraint $f(\mathbf{x})=0$ where $\mathbf{x}\in \mathbf{[x]}$ and $\mathbf{[x]}\in \mathbb {IR}^n$ which is illustrated using an example problem.

Example A1

Perform forward–backward contractor subject to constraint $w=2u+v$ where ${[w]}=[3,20]$, ${[u]}=[-10,5]$, and ${[v]}=[0,4]$.

Here, the constraint $w=2u+v$ may be expressed in terms of function f as $f(u,v,w)=w-2u-v$. Further, the possible different forms of the constraint that may be written are

$$\begin{aligned} u=\frac{w-v}{2} \end{aligned}$$

$$\begin{aligned} v=w-2u \end{aligned}$$

$$\begin{aligned} w=2u+v \end{aligned}$$

The forward–backward steps are then followed with respect to classical interval computations mentioned in Sect. 2 as

$$\begin{aligned} {[u]}\cap \left( \frac{[w]-[v]}{2}\right) =[-10,5]\cap \left( \frac{[3,20]-[0,4]}{2}\right) =[-0.5,5] \end{aligned}$$

$$\begin{aligned} {[v]}\cap \left( {[w]-2[u]}\right) =[0,4]\cap \left( [3,20]-2[-0.5,5]\right) =[0,4] \end{aligned}$$

$$\begin{aligned} {[w]}\cap \left( 2{[u]}+{[v]}\right) =[3,20]\cap \left( 2[-0.5,5]+[0,4]\right) =[3,14] \end{aligned}$$

As such, the new interval bounds are ${[w]}=[3,14]$, $[u]=[-0.5,5]$, and ${[v]}=[0,4]$.

Fixed-point contractor: A fixed-point contraction associated with $\psi$ is implemented with respect to the constraint $f(\mathbf{x})=0$ as $\mathbf{x}=\psi (\mathbf{[x]})$, where $\mathbf{x}\in \mathbf{[x]}\in \mathbb {IR}^n$. The fixed-point contractor with respect to constraint $u^2+2u+1=0$ is performed as

$$\begin{aligned} u\in {[u]}\ and \ u=\psi (u)&\implies u\in {[u]}\ and \ u\in \psi ({[u]})\\&\implies u\in {[u]}\cap [\psi ]({[u]}) \end{aligned}$$

In case of the implementation of forward–backward contractor along with fixed-point contractor helps in the computation of the forward–backward contractor until the fixed interval is reached.