Sparse-grid integration

Scientific question¶

How can we avoid resolving every high-order interaction in a tensor product? Sparse grids retain selected hierarchical tensor increments and are effective when the integrand has mixed smoothness or low effective dimension.

Geometric picture¶

Imagine each axis carrying a refinement level. A multi-index selects one combination of levels. A downward-closed set fills the lower-left portion of this index lattice before admitting more expensive interactions.

Derivation¶

Define nested one-dimensional differences (\Delta_\ell=Q_\ell-Q_{\ell-1}), with (Q_0=0). For a downward-closed set (\mathcal{I}),

A_{\mathcal{I}}[f] = \sum_{\boldsymbol{\ell}\in\mathcal{I}} \left(\Delta_{\ell_1}\otimes\cdots\otimes \Delta_{\ell_d}\right)[f].

(1)

Dimension-adaptive selection uses a profit such as

P_{\boldsymbol{\ell}} = \frac{\|\Delta_{\boldsymbol{\ell}}[f]\|} {\max(1,\Delta N_{\boldsymbol{\ell}})},

(2)

and reports the active-frontier sum

E_{\mathcal{F}} = \sum_{\boldsymbol{\ell}\in\mathcal{F}} \|\Delta_{\boldsymbol{\ell}}[f]\|.

(3)

Computational cost¶

Cost is the number of unique nested nodes, not the sum of all tensor sizes. Jaxstro coalesces nodes by exact dyadic-angle identity. Capacities independently bound accepted indices, frontier rows, nodes, and evaluations.

What the estimator means¶

ErrorKind.SPARSE_GRID_SURPLUS measures the executed hierarchy’s frontier. It can be conservative, as in the frozen level-5 exponential case, or miss a feature not represented by the current index set.

JAX and differentiation¶

First-order replay materializes the accepted sparse formula and differentiates its coalesced weighted sum. Index admission and surplus diagnostics are not differentiated.

Quantities and units¶

Heterogeneous axes work through CoordinatePoint. ProductMeasure can attach one Lebesgue or weighted component to each axis while sharing explicit args.

Worked astrophysical example¶

A population moment can combine heterogeneous coordinates:

I=\int M^2(1+Z)tD^2\,dM\,dZ\,dt\,dD.

(4)

When high-order interactions are weak, sparse increments can require far fewer unique evaluations than a uniform tensor rule.

Failure modes¶

Audit recipe¶

Save the level or initial level, anisotropy, all four capacities, unique-node count, accepted-index count, frontier evidence, status, and an independent truth or convergence sequence.

Warranted claim¶

Jaxstro provides fixed and dimension-adaptive Smolyak formulas with exact node reuse and honest surplus evidence. The surplus is not a universal error bound.