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

Convienience contructor for polyharmonic splines.

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

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

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

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

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

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

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.

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.

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.

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.

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.

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.

abstract type AbstractPHS <: AbstractRadialBasis

Supertype of all Polyharmonic Splines.

abstract type AbstractRadialBasis <: AbstractBasis end

Custom <: ScalarValuedOperator

Builds an operator for a first order partial derivative.

Directional{Dim,T} <: ScalarValuedOperator

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

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

Gaussian radial basis function:$ϕ(r) = e^{-(ε r)^2}$

Gradient{Dim} <: VectorValuedOperator

Builds an operator for the gradient of a function.

struct Interpolator

Construct a radial basis interpolation.

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

Construct a radial basis interpolator.

Laplacian <: ScalarValuedOperator

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

struct MonomialBasis{Dim,Deg} <: AbstractBasis

Dim dimensional monomial basis of order Deg.

struct PHS1{T<:Int} <: AbstractPHS

Polyharmonic spline radial basis function:$ϕ(r) = r$

struct PHS3{T<:Int} <: AbstractPHS

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

struct PHS5{T<:Int} <: AbstractPHS

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

struct PHS7{T<:Int} <: AbstractPHS

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

Partial <: ScalarValuedOperator

Builds an operator for a first order partial derivative.

struct RadialBasisOperator

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

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

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

function (op::RadialBasisOperator)(x)

Evaluate the operator at x.

Regrid

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

_calculate_thread_offsets(adjl, boundary_flag, nchunks)

Calculate the starting offsets for each thread when filling LHS and RHS matrices.

• lhs_offsets: Starting indices for internal-to-internal connections
• rhs_offsets: Starting indices for internal-to-boundary connections

Returns a tuple of (lhsoffsets, rhsoffsets).

_return_global_matrices(I_lhs, J_lhs, V_lhs, I_rhs, J_rhs, V_rhs, boundary_flag)

Constructs sparse matrix representation of the global linear system.

Arguments

• I_lhs, J_lhs, V_lhs: COO format components for LHS matrix
• I_rhs, J_rhs, V_rhs: COO format components for RHS matrix
• boundary_flag: Boolean array indicating boundary nodes

Returns

• For single operators: tuple of (lhsmatrix, rhsmatrix)
• For multiple operators: tuple of (lhsmatrices, rhsmatrices)

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

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

abstract type AbstractBasis end