Integrals over hyperrectangles

Scientific question¶

How do we turn a scientific integral with several bounded coordinates into a problem that numerical methods can share? Jaxstro’s Phase B methods integrate over a finite Cartesian product

\Omega=\prod_{j=1}^{d}[a_j,b_j].

(1)

The coordinates may represent different physical ideas. A stellar-population integral, for example, can combine mass, metallicity, age, and distance.

Geometric picture¶

Each axis contributes an interval. Their Cartesian product is a rectangle in two dimensions, a box in three, and a hyperrectangle in higher dimensions. Reversing one axis reverses the orientation; reversing two restores it.

Derivation¶

Map the unit cube coordinate (u_j\in[0,1]) to each physical coordinate:

x_j=a_j+(b_j-a_j)u_j, \qquad d^d x=\prod_{j=1}^{d}(b_j-a_j)\,d^d u.

(2)

Therefore

\int_{\Omega}f(\boldsymbol{x})\,d^d x = \left[\prod_{j=1}^{d}(b_j-a_j)\right] \int_{[0,1]^d} f\!\left(\boldsymbol{a}+(\boldsymbol{b}-\boldsymbol{a}) \odot\boldsymbol{u}\right)d^d u.

(3)

The signed product of widths preserves orientation.

Computational cost¶

Domain mapping costs (O(Nd)) for (N) points. The integration method, not the affine map, usually determines the dominant cost.

What the estimator means¶

The domain supplies bounds and a signed volume factor. Error evidence comes from the chosen formula: embedded rules, refinement differences, sparse surpluses, or randomized replicates.

JAX and differentiation¶

Finite bounds can be dynamic JAX values. With gradient="replay", accepted formula points are reconstructed and the first derivative includes smooth bound motion. Method choice, dimension, and capacities remain static.

Quantities and units¶

Hyperrectangle.from_axes accepts an Axis per coordinate. Each axis owns one static unit, so heterogeneous coordinates do not need to be stacked into a single artificial unit. If (f) has unit (U_f), the result has unit

U_I=U_f\prod_{j=1}^{d}U_{x_j}.

(4)

Quantity mode is alpha and opt-in; raw numeric domains remain supported.

Worked astrophysical example¶

For a projected Plummer profile with scale (a), integrate radius and angle:

\int_0^{2\pi}\int_0^R \frac{r}{\pi a^2[1+(r/a)^2]^2}\,dr\,d\phi =\frac{R^2}{R^2+a^2}.

(5)

Here the polar Jacobian (r) is explicit in the integrand.

Failure modes¶

Audit recipe¶

Record axis order, bounds, units, orientation, Jacobians, method, tolerances, capacities, status, work, and an independent truth or refinement check.

Warranted claim¶

Jaxstro supports differentiable finite hyperrectangles with homogeneous raw coordinates or heterogeneous opt-in quantity axes. Phase C geometries remain future work.