# Non-monotonic convergence in fixed-point problem

## Background

I am solving a variant of the Ornstein-Zernike equation from liquid theory. Abstractly, the problem can be represented as solving the fixed point problem $A c(r)=c(r)$, where $A$ is an integro-algebraic operator and $c(r)$ is the solution function (the OZ direct correlation function). I am solving by Picard iteration, where I provide an initial trial solution $c_0(r)$ and generate new trial solutions by the scheme $$c_{j+1} = \alpha (A c_j) + (1-\alpha)c_j~,$$ where $\alpha$ is an adjustable parameter that controls the mix of $c$ and $Ac$ used in the next trial solution. For this discussion, let's assume that the value of $\alpha$ is unimportant. I repeat until the iteration converges to within a desired tolerance, $\epsilon$: $$\Delta_{j+1} \equiv \int d\vec r |c_{j+1}(r)-c_j(r)| < \epsilon~.$$ In my variant of the problem, $A$ depends on a parameter $\lambda$, and my question is about how the convergence of $Ac=c$ depends on this parameter.

For a wide range of values for $\lambda$, the iteration scheme above converges exponentially quickly. However, as I decrease $\lambda$, I eventually reach a regime in which the convergence is non-monotonic, pictured below.

## Key Questions

In iterative solutions to fixed-point problems, does non-monotonic convergence have any special significance? Does it signal that my iterative scheme is on the verge of instability? Most importantly, should non-monotone convergence make me suspicious that the "converged" solution is not a good solution to the fixed-point problem?

Suppose $x$ is the unknown independent variable in the solution of $x=f(x)$, then fixed point method will converge from a point $x^*$ provided that the Jacobian $\frac{\partial f}{\partial x}(x^*) \le \alpha$, where $\alpha$ is a constant $<1$. In general $x^*$ is not a single point, but the domain traversed by the iterative scheme.
1. Your solution is converging, albeit non-monotonically. Check your Jacobian for various values of $\lambda$ and the solution variable to see if you go from satisfying convergence criteria to not satisfying it, which might explain what you are seeing.
• The difference $|x_{j+1}-x_j|$ between two successive iterations could be compared against $|x_j| \thinspace \epsilon$ where $\epsilon$ is a relative tolerance. – NameRakes Feb 10 '17 at 0:17