(3) pick a new point $x_\textrm{new}$ in the middle of the interval with the largest $L$, creating two new intervals around that point,
(4) calculate $f(x_\textrm{new})$,
(5) repeat the previous steps, without redoing calculations for unchanged intervals.
In this paper, we describe a class of algorithms that rely on local criteria for sampling, such as in the former example.
Here we associate a *local loss* to each of the *candidate points* within an interval, and choose the points with the largest loss.
In the case of the integration algorithm, the loss is the error estimate.
The most significant advantage of these *local* algorithms is that they allow for easy parallelization and have a low computational overhead.
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![Comparison of homogeneous sampling (top) with adaptive sampling (bottom) for different two-dimensional functions where the number of points in each column is identical.
On the left is the function $f(x) = x + a ^ 2 / (a ^ 2 + (x - x_\textrm{offset}) ^ 2)$.
In the middle a topological phase diagram from [\onlinecite{nijholt2016orbital}], where th function can take the values -1 or 1.
In the middle a topological phase diagram from [\onlinecite{nijholt2016orbital}], where the function can take the values -1 or 1.
On the right we plot level crossings for a two level quantum system.
In all cases using Adaptive results in a higher fidelity plot.