Interpolate

Kernel-weighted (re)construction of a field at the query positions. Kernel: coreOperations/wp_interpolate.py (computeSPHInterpolation_Kernel); JVP coreOperations/wp_interpolateJVP.py.

Equation

$$\tilde f_i \;=\; \sum_{j \in \mathcal{N}(i)} f_j\, V_j\, \widehat W_{ij}, \qquad V_j = \begin{cases} m_j / \rho_j & \text{default} \\ \mathrm{referenceVolumes}[j] & \text{if an apparent volume is supplied} \end{cases}$$

$\widehat W_{ij}$ is the kernel value sphKernel under the chosen support scheme, CRK-corrected when a CRKState is supplied:

$$\widehat W_{ij} = A_i\,\big(1 + B_i\cdot x_{ij}\big)\, W_{ij} \qquad(\text{`computeKernelCRK`}),$$

see CRK overview. Interpolate is the only value-having operator that never reads queryValues — the field is read from the reference side only — and it never reads the CRK gradient terms (gradA/gradB; if you pass a CRKState without them the backend substitutes correctly-shaped dummies, since the kernel only uses $A_i, B_i$).

Output: [N_query, ...], preserving the trailing shape of referenceValues. The query and reference counts can differ. Warp kernels support rank-1/2 field types, so fields of rank $>2$ are flattened to one vector dimension before the launch and reshaped back afterwards.

Public API

warpOperation(
    queryParticles, operationProperties, domain,
    referenceValues=f,               # required
    queryVolumes=None, referenceVolumes=None,
    crkState=None,
    adjacency=None,
    referenceParticles=None,
)

with operation == WarpOperation.Interpolate. warpOperation raises if referenceValues is missing.

Option	Default	Effect
supportMode	Gather	pairwise support of $W_{ij}$
operationMode	AllToAll	directional filtering on kinds
referenceVolumes	None	substitute $V_j$
crkState	None	CRK-corrected kernel value ($A_i, B_i$ only)
calibrateNormalization, n_h	False/None	lattice-normalization factor on $W$

gradientMode, laplacianMode, positiveDivergence and the renormalization/grad-h states are not consumed by Interpolate.

JVP / HVP

Value JVP — relaunch on tangentReferenceValues (there is no query-side value input, so warpOperationJVP only ever threads a reference tangent for Interpolate).

Geometry JVP — the volume-weighted product rule:

$$d\tilde f_i \;=\; \sum_j \Big[ f_j\big(dV_j\, W_{ij} + V_j\, dW_{ij}\big) \Big] \;+\; \underbrace{\sum_j df_j\, V_j\, W_{ij}}_{\text{value-tangent term, fused}},$$

with $f_j$ frozen at primal for the geometry part and

$$dV_j = \frac{dm_j}{\rho_j} - \frac{m_j\, d\rho_j}{\rho_j^2} \quad\text{(or the supplied apparent-volume tangent)},$$

$dW_{ij}$ from sphKernelJVP. Interpolate has no query-side density term at all and no CRK/renorm/grad-h tangent support (the geometry JVP has no CRK correction path, enforced centrally in operations.py); query-mass tangents are rejected. Omitted tangent fields default to zero.

Gated by tests/operations/test_forward_mode_geometry_jvp_interpolate.py.

No HVP exists for Interpolate.

Tests

• tests/operations/test_operations_core.py — shape/finiteness for scalar, vector and matrix fields.
• tests/operations/test_operations_consistency.py — e.g. interpolating a linear field reproduces it at interior points.
• tests/operations/test_forward_mode_value_jvp.py, tests/operations/test_forward_mode_geometry_jvp_interpolate.py, tests/operations/test_grid_modes.py.

See also

Density · CRK overview · CRK volume