Choosing a multidimensional integration method

Scientific question¶

Which method produces evidence that matches the scientific structure, not just a number? Start with geometry, smoothness, effective dimension, uncertainty requirements, and the derivative you need.

Geometric picture¶

Tensor rules fill every coordinate combination. Cubature partitions physical space. Sparse grids navigate a level lattice. Randomized QMC spreads replicated digital nets across the whole domain.

Derivation¶

A useful first comparison is the work model:

N_{\mathrm{tensor}}\sim n^d, \qquad N_{\mathrm{RQMC}}=R2^m,

(1)

while adaptive cubature and sparse grids have problem-dependent work

N_{\mathrm{adaptive}} =N(\epsilon_{\mathrm{abs}},\epsilon_{\mathrm{rel}}, d,f,\text{capacities}).

(2)

No method removes dependence on the integrand class.

Computational cost¶

Scientific structure	First method to try	Main cost warning
Smooth, low dimension, polynomial moment	`TensorProduct`	(n^d)
Localized smooth structure	`AdaptiveCubature`	regional growth
Mixed smoothness or low effective dimension	`Smolyak`	index and node growth
Unknown anisotropy	`AdaptiveSmolyak`	frontier capacity
Moderate or high dimension, deterministic estimate	`Sobol`	no error estimate
Randomized uncertainty required	`ScrambledSobol`	replicate cost
Sequential randomized stopping	`AdaptiveScrambledSobol`	valid bounds required

What the estimator means¶

Fixed rules may have no runtime estimator. Adaptive evidence is formula disagreement, not a universal error bound. Randomized intervals describe replicate uncertainty under a declared randomization.

JAX and differentiation¶

All methods support first-order replay. Prefer fixed formulas when stable, repeatable derivatives matter more than adaptive work. Always pass live parameters through args.

Quantities and units¶

Use raw normalized coordinates when that is the clearest model. Use Hyperrectangle.from_axes when heterogeneous physical units materially improve auditability. Quantity mode is optional and alpha.

Worked astrophysical example¶

For a four-dimensional stellar population moment, start with a tensor rule if the integrand is a low-degree separable polynomial. Move to a sparse grid for smooth nonseparable corrections, or randomized QMC when selection effects add many coordinates.

I=\int p(M,Z,t,D)\,S(M,Z,t,D)\,g(M,Z,t,D)\,dM\,dZ\,dt\,dD.

(3)

Failure modes¶

Audit recipe¶

Write down why the geometry fits, what smoothness you expect, which estimator you will trust, the stopping capacities, derivative mode, units, independent truth check, and the fallback method.

Warranted claim¶

Jaxstro now spans four complementary finite-hyperrectangle strategies. Method choice remains a scientific modeling decision, and Phase C geometries are outside this guide.