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Rational far fields: equivalent charges with free poles against the multipole series.

The multipole (Taylor) far field of a cell converges like rho**p with rho the cell’s half-diagonal over the target distance, which is why a fast multipole method needs rho well below one and many accepted cell pairs. A rational approximant (AAA, Nakatsukasa-Sete-Trefethen) of the same far field, read off in partial fractions, is a set of m equivalent complex charges at optimised positions; its convergence rate is set by the distance of the poles rather than by the convergence radius of a series, so it can be accurate where the series is not. This example measures what that buys, at equal cost per target (p series terms against m = p/2 charges, two parameters each).

Source: examples/rational_far_field.py. Run it yourself with python examples/rational_far_field.py [--output examples/output] [--quick] (needs pip install "tree-d-3d[examples]").

Figures

Section titled “Figures”

field error of a 32-charge cell against the number of parameters, for the Taylor series and the AAA equivalent charges, at separation ratios 0.5, 0.7 and 0.9; the rational far field wins by orders of magnitude where the series barely converges.

the gain (Taylor error over rational error) against the number of sources in the cell; it decays, because the far field of n charges has n branch points and m free poles cannot resolve them all.

error against the evaluation radius for a rational fit made on one circle; the series keeps improving with distance, the rational fit plateaus at its fit accuracy, so a fit must be made at the closest accepted separation.

error of a cell’s representation against its level in a quadtree, for the Taylor series (exact upward translation), a flat AAA fit from all the cell’s particles, and a hierarchical AAA fit that compresses the children’s equivalent charges; the hierarchy is what keeps the small-cluster gains at the top of the tree.

the number of accepted (far-field) and near-field cell pairs of the dual-tree traversal against theta, for a fixed cloud; what a larger theta would buy.

--quick uses smaller clouds and fewer orders (about 20 s instead of a few minutes). Needs Plotly (pip install 'tree-d-3d[examples]'). The scripts behind the tables in the research notes are benchmarks/rational_far_field*.py.