Numerical integration using Birkhoff theorem

Question

There is a method for numerical integration that uses Brikhoff ergodic theorem?

For example if we have a irrational number $\alpha$ we know that for every continuous function $f \colon [0,1] \to \mathbb{R}$ with $f(0)=f(1)$ we have

$$\frac{1}{n} \sum_{i=0}^{n-1} f(i\cdot \alpha \mod 1) \to \int_0^1f(x)dx $$

because the irrational rotation in $\mathbb{R} / \mathbb{Z}$ is uniquely ergodic.

The problem here is the rate of convergence and that computers don't have irrational numbers :(

Any help or reference will be appriciated

Answer 1

If you have only a fixed precision available, then there cannot be a numerical integration method of this type, because you cannot represent irrational numbers in fixed finite precision, as you pointed out. If your precision depended on $n$, then you could attempt to control the precision and $n$ simultaneously to get an error control. I think this would require some estimate of the modulus of continuity of $f$ along with careful estimates of the error in actually computing $f$ as well. This is all very inconvenient on the hardware we have available to us, because performance is much higher in the floating point arithmetic that we have natively available, which on most hardware maxes out at double precision. Put differently, the ratio of the speed of double precision to the speed of single precision is smaller than the ratio of the speed of quadruple precision to the speed of double precision, and the difference is quite large.

That said, I would not be surprised if this general idea were a common technique to accelerate convergence of Monte Carlo methods.

Answer 2

If $f$ is continuous and bounded, and $a_k$ is uniformly distributed, then by Weyl's Theorem$$\lim_{n\to\infty}{1\over n}\sum_1^nf(a_k)=\int_0^1f$$ The sequence where $a_k$ is the fractional part of $k\alpha$, $\alpha$ irrational, is known to be uniformly distributed. Furthermore, there are estimates of how quickly the sum converges to the integral, e.g., Koksma's inequality says $$\Biggl|{1\over n}\sum_1^nf(a_k)-\int_0^1f\,\Biggr|\le D(a_1,\dots,a_n)V(f)$$ where $D$ is the discrepancy and $V$ is the variation.

For details, see the book by Kuipers and Niederreiter, or search for Weyl Criterion, Koksma's Inequality, and other phrases I've used.