Adaptive cubature

Scientific question¶

How can an integral spend work only where a smooth multidimensional integrand needs local resolution? Adaptive cubature subdivides the most uncertain region instead of refining the entire domain.

Geometric picture¶

Begin with one hyperrectangle. Evaluate a symmetric rule, estimate local error, split the selected region along one axis, and repeat. The active leaves form a binary partition of the original domain.

Derivation¶

For a region (R), the Genz-Malik construction evaluates high- and low-degree weighted sums on shared symmetric points:

Q_R^{(h)}=\sum_{i=1}^{N_d}w_i^{(h)}f(\boldsymbol{x}_i), \qquad Q_R^{(l)}=\sum_{i=1}^{N_d}w_i^{(l)}f(\boldsymbol{x}_i).

(1)

The local evidence is

E_R=\left\|Q_R^{(h)}-Q_R^{(l)}\right\|, \qquad E=\sum_{R\in\mathcal{L}}E_R.

(2)

The controller stops when (E\leq\epsilon_{\mathrm{abs}}+ \epsilon_{\mathrm{rel}}|Q|), or when a declared capacity is reached.

Computational cost¶

Each split adds two child-rule evaluations. Cost depends on dimension, active regions, payload size, and how quickly local error concentrates. Storage is bounded by max_regions and evaluations by max_evaluations.

What the estimator means¶

ErrorKind.EMBEDDED_RULE measures disagreement between two formulas on the executed partition. It is not an exact bound. MAX_REGIONS and MAX_EVALUATIONS preserve the best accumulated value while stating why the requested tolerance was not certified.

JAX and differentiation¶

The primal controller is JIT-compatible. Replay reconstructs the accepted leaf formulas and differentiates their weighted sum to first order. Refinement decisions and diagnostic fields are stopped.

Quantities and units¶

Quantity axes and weighted measures are normalized before the raw cubature controller. Every density must declare its unit, and the result unit includes all coordinate-width factors.

Worked astrophysical example¶

For a localized likelihood contribution,

f(\boldsymbol{x})= \exp\!\left[-\alpha\|\boldsymbol{x}-\boldsymbol{\mu}\|_2^2\right],

(3)

local subdivision can focus on the neighborhood of (\boldsymbol{\mu}), unlike global tensor refinement.

Failure modes¶

Audit recipe¶

Record rule, dimension, tolerances, max_regions, max_evaluations, status, active leaves, evaluations, embedded estimate, truth error, and the exact integrand parameters.

Warranted claim¶

Adaptive Genz-Malik cubature is validated on the declared smooth and Genz cases through dimension eight. Accuracy outside those cases remains problem-dependent and capacity-bounded.