Tensor-product integration

Scientific question¶

When is it reasonable to resolve every coordinate direction at the same one-dimensional order? Tensor products are the direct answer for smooth, low-dimensional integrands.

Geometric picture¶

Place a one-dimensional rule on each axis, then evaluate every Cartesian combination of nodes. In two dimensions the nodes form a lattice; in higher dimensions the same construction becomes expensive quickly.

Derivation¶

Given one-dimensional rules (Q_j[g]=\sum_{i=1}^{n_j}w_{j,i}g(x_{j,i})), their product is

(Q_1\otimes\cdots\otimes Q_d)[f] = \sum_{i_1=1}^{n_1}\cdots\sum_{i_d=1}^{n_d} \left(\prod_{j=1}^{d}w_{j,i_j}\right) f(x_{1,i_1},\ldots,x_{d,i_d}).

(1)

For a common order (n), the point count is

N=n^d.

(2)

AdaptiveTensorClenshawCurtis compares nested global levels:

Computational cost¶

Fixed cost is exactly the product of per-axis node counts. Global adaptive refinement inherits exponential growth and fixed-capacity storage. Dimensions five through eight therefore require explicit memory and runtime planning.

What the estimator means¶

Fixed tensor rules return ErrorKind.UNAVAILABLE; exactness is a property of the declared rule and integrand class, not a measured error. Adaptive tensor rules return a refinement difference, which is evidence from two nested formulas rather than a universal bound.

JAX and differentiation¶

Both methods support first-order accepted-formula replay. JAX differentiates the integrand, explicit args, and smooth finite bounds while treating the chosen level and work record as fixed evidence.

Quantities and units¶

Each tensor point is exposed as a CoordinatePoint in heterogeneous quantity mode. The integrand may inspect each axis with x.axis(j). Unit conversion is performed outside the raw numerical engine.

Worked astrophysical example¶

A diagonal bounded Gaussian factorizes:

\int_{-L_1}^{L_1}\cdots\int_{-L_d}^{L_d} \prod_{j=1}^{d} \frac{\exp[-x_j^2/(2\sigma_j^2)]} {\sqrt{2\pi}\sigma_j}\,d^d x = \prod_{j=1}^{d} \operatorname{erf}\!\left(\frac{L_j}{\sqrt{2}\sigma_j}\right).

(4)

This is a strong tensor-rule validation because normalization and each diagonal second moment have closed forms.

Failure modes¶

Audit recipe¶

Report per-axis rule, order or level, dimension, exact point count, payload shape, dtype, status, and comparison to a closed-form moment or higher-order formula.

Warranted claim¶

Tensor products are exact for their declared polynomial class and effective for the frozen smooth validation cases. They are not a general solution to high-dimensional integration.