Exported Functions

RadialBasisFunctions.PHS Method

function PHS(n::T=3; poly_deg::T=2) where {T<:Int}

Convienience contructor for polyharmonic splines.

source

RadialBasisFunctions.classify_stencil Method

classify_stencil(is_boundary, boundary_conditions, eval_idx,
                neighbors, global_to_boundary)

Classify stencil type for dispatch in kernel execution.

source

RadialBasisFunctions.directional Method

function directional(data, eval_points, v, basis, is_boundary, boundary_conditions, normals; k=autoselect_k(data, basis))

Builds a Hermite-compatible RadialBasisOperator where the operator is the directional derivative, Directional. The additional boundary information enables Hermite interpolation with proper boundary condition handling.

source

RadialBasisFunctions.directional Method

function directional(data, eval_points, v, basis; k=autoselect_k(data, basis))

Builds a RadialBasisOperator where the operator is the directional derivative, Directional.

source

RadialBasisFunctions.directional Method

function directional(data, v, basis; k=autoselect_k(data, basis))

Builds a RadialBasisOperator where the operator is the directional derivative, Directional.

source

RadialBasisFunctions.gradient Method

function gradient(data, basis; k=autoselect_k(data, basis))

Builds a RadialBasisOperator where the operator is the gradient, Gradient.

source

RadialBasisFunctions.gradient Method

function gradient(data, eval_points, basis, is_boundary, boundary_conditions, normals; k=autoselect_k(data, basis))

Builds a Hermite-compatible RadialBasisOperator where the operator is the gradient, Gradient. The additional boundary information enables Hermite interpolation with proper boundary condition handling.

source

RadialBasisFunctions.gradient Method

function gradient(data, eval_points, basis; k=autoselect_k(data, basis))

Builds a RadialBasisOperator where the operator is the gradient, Gradient. The resulting operator will only evaluate at eval_points.

source

RadialBasisFunctions.laplacian Method

function laplacian(data, eval_points, basis, is_boundary, boundary_conditions, normals; k=autoselect_k(data, basis))

Builds a Hermite-compatible RadialBasisOperator where the operator is the Laplacian, Laplacian. The additional boundary information enables Hermite interpolation with proper boundary condition handling.

source

RadialBasisFunctions.partial Method

function partial(data, eval_points, order, dim, basis, is_boundary, boundary_conditions, normals; k=autoselect_k(data, basis))

Builds a Hermite-compatible RadialBasisOperator where the operator is the partial derivative, Partial. The additional boundary information enables Hermite interpolation with proper boundary condition handling.

source

RadialBasisFunctions.partial Method

function partial(data, eval_points, order, dim, basis; k=autoselect_k(data, basis))

Builds a RadialBasisOperator where the operator is the partial derivative, Partial. The resulting operator will only evaluate at eval_points.

source

RadialBasisFunctions.partial Method

function partial(data, order, dim, basis; k=autoselect_k(data, basis))

Builds a RadialBasisOperator where the operator is the partial derivative, Partial, of order with respect to dim.

source

RadialBasisFunctions.regrid Method

function regrid(data, eval_points, basis=PHS(3; poly_deg=2); k=autoselect_k(data, basis))

Builds a RadialBasisOperator where the operator is an regrid from one set of points to another, data -> eval_points.

source

RadialBasisFunctions.update_hermite_stencil_data! Method

update_hermite_stencil_data!(hermite_data, global_data, neighbors,
                             is_boundary, boundary_conditions, normals,
                             global_to_boundary)

Populate local Hermite stencil data from global arrays. Used within kernels to extract boundary info for specific neighborhoods.

source

RadialBasisFunctions.∂virtual Method

function ∂virtual(data, eval_points, dim, Δ, basis; k=autoselect_k(data, basis))

Builds a virtual RadialBasisOperator whichi will be evaluated at eval_points where the operator is the partial derivative with respect to dim. Virtual operators interpolate the data to structured points at a distance Δ for which standard finite difference formulas can be applied.

source

RadialBasisFunctions.∂virtual Method

function ∂virtual(data, dim, Δ, basis; k=autoselect_k(data, basis))

Builds a virtual RadialBasisOperator whichi will be evaluated at the input points (data) where the operator is the partial derivative with respect to dim. Virtual operators interpolate the data to structured points at a distance Δ for which standard finite difference formulas can be applied.

source

RadialBasisFunctions.AbstractPHS Type

abstract type AbstractPHS <: AbstractRadialBasis

Supertype of all Polyharmonic Splines.

source

RadialBasisFunctions.AbstractRadialBasis Type

abstract type AbstractRadialBasis <: AbstractBasis end

source

RadialBasisFunctions.BoundaryCondition Type

BoundaryCondition{T}

Unified boundary condition representation: Bu = α_u + β_∂ₙu

Special cases:

• Dirichlet: α=1, β=0

• Neumann: α=0, β=1

• Robin: α≠0, β≠0

• Internal: α=0, β=0 (sentinel for interior points)

source

RadialBasisFunctions.Custom Type

Custom <: ScalarValuedOperator

Builds an operator for a first order partial derivative.

source

RadialBasisFunctions.Directional Type

Directional{Dim,T} <: ScalarValuedOperator

Operator for the directional derivative, or the inner product of the gradient and a direction vector.

source

RadialBasisFunctions.Gaussian Type

struct Gaussian{T,D<:Int} <: AbstractRadialBasis

Gaussian radial basis function:$\varphi (r)=e^{-(\epsilon r{)}^2}$

source

RadialBasisFunctions.Gradient Type

Gradient{Dim} <: VectorValuedOperator

Builds an operator for the gradient of a function.

source

RadialBasisFunctions.HermiteStencilData Type

HermiteStencilData{T}

Local stencil data for Hermite interpolation with boundary conditions.

Fields:

• data: Coordinates of k stencil points

• is_boundary: Boolean flags for each point

• boundary_conditions: BC for each point (use Internal() for interior)

• normals: Normal vectors (zero for interior points)

Note: For interior points (is_boundary[i] == false), boundary_conditions[i] and normals[i] contain sentinel values and should not be accessed.

source

RadialBasisFunctions.HermiteStencilData Method

Pre-allocation constructor for HermiteStencilData

source

RadialBasisFunctions.Interpolator Type

struct Interpolator

Construct a radial basis interpolation.

source

RadialBasisFunctions.Interpolator Method

function Interpolator(x, y, basis::B=PHS())

Construct a radial basis interpolator.

source

RadialBasisFunctions.Laplacian Type

Laplacian <: ScalarValuedOperator

Operator for the sum of the second derivatives w.r.t. each independent variable.

source

RadialBasisFunctions.MonomialBasis Type

struct MonomialBasis{Dim,Deg} <: AbstractBasis

Dim dimensional monomial basis of order Deg.

source

RadialBasisFunctions.PHS1 Type

struct PHS1{T<:Int} <: AbstractPHS

Polyharmonic spline radial basis function:$\varphi (r)=r$

source

RadialBasisFunctions.PHS3 Type

struct PHS3{T<:Int} <: AbstractPHS

Polyharmonic spline radial basis function:$\varphi (r)=r^3$

source

RadialBasisFunctions.PHS5 Type

struct PHS5{T<:Int} <: AbstractPHS

Polyharmonic spline radial basis function:$\varphi (r)=r^5$

source

RadialBasisFunctions.PHS7 Type

struct PHS7{T<:Int} <: AbstractPHS

Polyharmonic spline radial basis function:$\varphi (r)=r^7$

source

RadialBasisFunctions.Partial Type

Partial <: ScalarValuedOperator

Builds an operator for a first order partial derivative.

source

RadialBasisFunctions.RadialBasisOperator Type

struct RadialBasisOperator

Operator of data using a radial basis with potential monomial augmentation.

source

RadialBasisFunctions.RadialBasisOperator Method

function (op::RadialBasisOperator)(y, x)

Evaluate the operator at x in-place and store the result in y.

source

RadialBasisFunctions.RadialBasisOperator Method

function (op::RadialBasisOperator)(x)

Evaluate the operator at x.

source

RadialBasisFunctions.Regrid Type

Regrid

Builds an operator for interpolating from one set of points to another.

source

Private

RadialBasisFunctions._build_collocation_matrix! Method

_build_collocation_matrix!(A, data, basis, mon, k)

Build RBF collocation matrix for interior stencil (no boundary conditions).

Matrix structure:

┌─────────────────┬─────────┐
│  Φ(xᵢ, xⱼ)      │ P(xᵢ)   │  k×k RBF + k×nmon polynomial
├─────────────────┼─────────┤
│  P(xⱼ)ᵀ         │   0     │  nmon×k poly + nmon×nmon zero
└─────────────────┴─────────┘

source

RadialBasisFunctions._build_collocation_matrix! Method

_build_collocation_matrix!(A, data::HermiteStencilData, basis, mon, k)

Build RBF collocation matrix for Hermite stencil (with boundary conditions).

For boundary points with Neumann/Robin conditions, basis functions are modified:

• Instead of Φ(·,xⱼ), we use B₂Φ(·,xⱼ) where B is the boundary operator

• This maintains matrix symmetry

source

RadialBasisFunctions._build_rhs! Method

_build_rhs!(b, ℒrbf::Tuple, ℒmon::Tuple, data, eval_point, basis, k)

Build RHS vector for interior stencil, multiple operators.

source

RadialBasisFunctions._build_rhs! Method

_build_rhs!(b, ℒrbf::Tuple, ℒmon::Tuple, data::HermiteStencilData, eval_point, basis, mon, k)

Build RHS vector for Hermite stencil, multiple operators.

source

RadialBasisFunctions._build_rhs! Method

_build_rhs!(b, ℒrbf, ℒmon, data, eval_point, basis, k)

Build RHS vector for interior stencil, single operator.

source

RadialBasisFunctions._build_rhs! Method

_build_rhs!(b, ℒrbf, ℒmon, data::HermiteStencilData, eval_point, basis, mon, k)

Build RHS vector for Hermite stencil, single operator. Applies boundary conditions to both stencil points and evaluation point.

source

RadialBasisFunctions._build_stencil! Method

_build_stencil!(A, b, ℒrbf, ℒmon, data, eval_point, basis, mon, k)

Assemble complete stencil: build collocation matrix, build RHS, solve for weights. Works for both interior and Hermite stencils via multiple dispatch on data type.

Returns: weights (first k rows of solution, size k×num_ops)

source

RadialBasisFunctions._build_weights Method

_build_weights(ℒ, data, eval_points, adjl, basis)

Apply operator to basis functions and route to weight computation.

source

RadialBasisFunctions._build_weights Method

_build_weights(data, eval_points, adjl, basis, ℒrbf, ℒmon, mon;
              batch_size=10, device=CPU())

Build weights for interior-only problems (no boundary conditions). Creates empty BoundaryData to indicate all interior points.

source

RadialBasisFunctions._build_weights Method

_build_weights(ℒ, data, eval_points, adjl, basis,
              is_boundary, boundary_conditions, normals)

Generic Hermite dispatcher for operators. Applies operator to basis and routes to Hermite weight computation.

This eliminates repetitive _build_weights methods across operator files. Note: Type constraint removed to avoid circular dependency with operators.jl

source

RadialBasisFunctions._build_weights Method

_build_weights(data, eval_points, adjl, basis, ℒrbf, ℒmon, mon,
              is_boundary, boundary_conditions, normals;
              batch_size=10, device=CPU())

Build weights for problems with boundary conditions using Hermite interpolation. Exact allocation: Dirichlet points get single entry, others get full stencil.

source

RadialBasisFunctions._build_weights Method

_build_weights(ℒ, op)

Entry point from operator construction. Extracts configuration from operator and routes to appropriate implementation.

source

RadialBasisFunctions._build_weights Method

_build_weights(ℒ::Directional, data, eval_points, adjl, basis, is_boundary, boundary_conditions, normals)

Hermite-compatible method for building directional derivative weights with boundary condition support.

source

RadialBasisFunctions._num_ops Method

Get number of operators (type-stable)

source

RadialBasisFunctions._prepare_buffer Method

Prepare RHS buffer with correct type (type-stable)

source

RadialBasisFunctions.allocate_sparse_arrays Method

allocate_sparse_arrays(TD, k, N_eval, num_ops, adjl, boundary_data)

Allocate sparse matrix arrays for COO format sparse matrix construction. Exactly counts non-zeros: interior points get k entries, Dirichlet points get 1 entry.

source

RadialBasisFunctions.autoselect_k Method

autoselect_k(data::Vector, basis<:AbstractRadialBasis)

See Bayona, 2017 - https://doi.org/10.1016/j.jcp.2016.12.008

source

RadialBasisFunctions.build_weights_kernel Method

build_weights_kernel(data, eval_points, adjl, basis, ℒrbf, ℒmon, mon,
                    boundary_data; batch_size, device)

Main orchestrator: allocate memory, launch kernel, construct sparse matrix.

source

RadialBasisFunctions.calculate_batch_range Method

Calculate batch index range for kernel execution

source

RadialBasisFunctions.compute_hermite_poly_entry! Method

compute_hermite_poly_entry!(a, i, data, mon)

Compute polynomial entries for Hermite interpolation. Applies boundary operators to polynomial basis at boundary points.

source

RadialBasisFunctions.compute_hermite_rbf_entry Method

compute_hermite_rbf_entry(i, j, data, basis)

Compute single RBF matrix entry with Hermite boundary modifications. Dispatches based on point types (Interior/Dirichlet/NeumannRobin).

source

RadialBasisFunctions.construct_global_to_boundary Method

construct_global_to_boundary(is_boundary)

Construct mapping from global indices to boundary-only indices. For boundary points: global_to_boundary[i] = boundary array index For interior points: global_to_boundary[i] = 0 (sentinel)

source

RadialBasisFunctions.count_nonzeros Method

count_nonzeros(adjl, is_boundary, boundary_conditions)

Count exact number of non-zero entries for optimized allocation. Returns (total_nnz, row_offsets) where row_offsets[i] is the starting position for row i.

source

RadialBasisFunctions.fill_dirichlet_entry! Method

Fill Dirichlet identity row for optimized allocation

source

RadialBasisFunctions.fill_entries! Method

Fill sparse matrix entries using indexed storage (row_offsets)

source

RadialBasisFunctions.hermite_mono_rhs! Method

hermite_mono_rhs!(bmono, ℒmon, mon, eval_point, is_bound, bc, normal, T)

Apply boundary conditions to monomial operator evaluation.

• Interior/Dirichlet: apply operator ℒ to monomials

• Neumann/Robin: α_ℒ(P) + β_ℒ(∂ₙP)

source

RadialBasisFunctions.hermite_rbf_rhs Method

hermite_rbf_rhs(ℒrbf, eval_point, data_point, is_bound, bc, normal)

Apply boundary conditions to RBF operator evaluation.

• Interior/Dirichlet: standard evaluation ℒΦ(x_eval, x_data)

• Neumann/Robin: α_ℒΦ + β_ℒ(∂ₙΦ)

source

RadialBasisFunctions.launch_kernel! Method

launch_kernel!(I, J, V, data, eval_points, adjl, basis, ℒrbf, ℒmon, mon,
               boundary_data, row_offsets, batch_size, N_eval, n_batches,
               k, nmon, num_ops, device)

Launch parallel kernel for weight computation. Handles Dirichlet/Interior/Hermite stencil classification via dispatch.

source

RadialBasisFunctions.operator_arity Method

Extract operator arity at compile time

source

RadialBasisFunctions.point_type Method

Determine boundary type of a single point

source

RadialBasisFunctions.AbstractBasis Type

abstract type AbstractBasis end

source

RadialBasisFunctions.BoundaryData Type

BoundaryData{T}

Wrapper for global boundary information (replaces fragile tuples).

source

RadialBasisFunctions.BoundaryPointType Type

Trait types for individual point boundary classification

source

RadialBasisFunctions.OperatorArity Type

Operator arity traits for compile-time dispatch

source

RadialBasisFunctions.StencilType Type

Trait types for stencil classification

source