Field Interpolation

A robust field interpolation is the prerequisite for pushing particles. This example demonstrates the construction of scalar/vector field interpolators for Cartesian/Spherical grids. If the field is analytic, you can directly pass the generated function to prepare.

using TestParticle
using Meshes
using StaticArrays
using Chairmarks

function setup_spherical_field(ns = 16)
   r = logrange(0.1, 10.0, length = ns)
   r_uniform = range(0.1, 10.0, length = ns)
   θ = range(0, π, length = ns)
   ϕ = range(0, 2π, length = ns)

   B₀ = 1e-8 # [nT]
   B = zeros(3, length(r), length(θ), length(ϕ)) # vector
   A = zeros(length(r), length(θ), length(ϕ)) # scalar

   for (iθ, θ_val) in enumerate(θ)
      sinθ, cosθ = sincos(θ_val)
      B[1, :, iθ, :] .= B₀ * cosθ
      B[2, :, iθ, :] .= -B₀ * sinθ
      A[:, iθ, :] .= B₀ * sinθ
   end

   B_field_nu = build_interpolator(StructuredGrid, B, r, θ, ϕ)
   A_field_nu = build_interpolator(StructuredGrid, A, r, θ, ϕ)
   B_field = build_interpolator(StructuredGrid, B, r_uniform, θ, ϕ)
   A_field = build_interpolator(StructuredGrid, A, r_uniform, θ, ϕ)

   return B_field_nu, A_field_nu, B_field, A_field
end

function setup_cartesian_field(ns = 16)
   x = range(-10, 10, length = ns)
   y = range(-10, 10, length = ns)
   z = range(-10, 10, length = ns)
   B = zeros(3, length(x), length(y), length(z)) # vector
   B[3, :, :, :] .= 10e-9
   A = zeros(length(x), length(y), length(z)) # scalar
   A[:, :, :] .= 10e-9

   B_field = build_interpolator(B, x, y, z)
   A_field = build_interpolator(A, x, y, z)

   return B_field, A_field
end

function setup_cartesian_nonuniform_field()
   x = logrange(0.1, 10.0, length = 16)
   y = range(-10, 10, length = 16)
   z = range(-10, 10, length = 16)
   B = zeros(3, length(x), length(y), length(z)) # vector
   B[3, :, :, :] .= 10e-9
   A = zeros(length(x), length(y), length(z)) # scalar
   A[:, :, :] .= 10e-9

   B_field = build_interpolator(RectilinearGrid, B, x, y, z)
   A_field = build_interpolator(RectilinearGrid, A, x, y, z)

   return B_field, A_field
end

function setup_time_dependent_field(ns = 16)
   x = range(-10, 10, length = ns)
   y = range(-10, 10, length = ns)
   z = range(-10, 10, length = ns)

   # Create two time snapshots
   B0 = zeros(3, length(x), length(y), length(z))
   B0[3, :, :, :] .= 1.0 # Bz = 1 at t=0

   B1 = zeros(3, length(x), length(y), length(z))
   B1[3, :, :, :] .= 2.0 # Bz = 2 at t=1

   times = [0.0, 1.0]

   function loader(i)
       if i == 1
           # For demonstration, we assume we load from disk here
           return build_interpolator(CartesianGrid, B0, x, y, z)
       elseif i == 2
           return build_interpolator(CartesianGrid, B1, x, y, z)
       else
           error("Index out of bounds")
       end
   end

   # B_field_t(x, t)
   B_field_t = LazyTimeInterpolator(times, loader)

   return B_field_t
end

function setup_mixed_precision_field(ns = 11, order = 1, bc = FillExtrap(NaN); coeffs = OnTheFly())
   x = range(0.0f0, 10.0f0, length = ns)
   y = range(0.0f0, 10.0f0, length = ns)
   z = range(0.0f0, 10.0f0, length = ns)
   B = fill(0.0f0, 3, ns, ns, ns)
   B[3, :, :, :] .= 1.0f-8

   itp = build_interpolator(B, x, y, z, order, bc; coeffs)
   return itp
end

B_sph_nu, A_sph_nu, B_sph, A_sph = setup_spherical_field();
B_car, A_car = setup_cartesian_field();
B_car_nu, A_car_nu = setup_cartesian_nonuniform_field();
B_td = setup_time_dependent_field();
itp_f32 = setup_mixed_precision_field();

loc = SA[1.0, 1.0, 1.0];
loc_f32 = SA[1.0f0, 1.0f0, 1.0f0];
loc_f64 = SA[1.0, 1.0, 1.0];

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
 1.0
 1.0
 1.0

Gridded spherical interpolation

Input Location

For spherical data, the input location is still in Cartesian coordinates!

julia> @be B_sph_nu($loc)
Benchmark: 3138 samples with 226 evaluations
 min    127.358 ns (2 allocs: 64 bytes)
 median 129.796 ns (2 allocs: 64 bytes)
 mean   131.302 ns (2 allocs: 64 bytes)
 max    250.199 ns (2 allocs: 64 bytes)

julia> @be A_sph_nu($loc)
Benchmark: 3127 samples with 275 evaluations
 min    105.102 ns (2 allocs: 48 bytes)
 median 107.218 ns (2 allocs: 48 bytes)
 mean   108.467 ns (2 allocs: 48 bytes)
 max    460.113 ns (2 allocs: 48 bytes)

Uniform spherical interpolation

julia> @be B_sph($loc)
Benchmark: 3154 samples with 230 evaluations
 min    124.665 ns (2 allocs: 64 bytes)
 median 126.583 ns (2 allocs: 64 bytes)
 mean   128.108 ns (2 allocs: 64 bytes)
 max    192.378 ns (2 allocs: 64 bytes)

julia> @be A_sph($loc)
Benchmark: 3155 samples with 292 evaluations
 min    97.442 ns (2 allocs: 48 bytes)
 median 100.459 ns (2 allocs: 48 bytes)
 mean   101.515 ns (2 allocs: 48 bytes)
 max    219.716 ns (2 allocs: 48 bytes)

Uniform Cartesian interpolation

julia> @be B_car($loc)
Benchmark: 3161 samples with 536 evaluations
 min    52.392 ns (2 allocs: 64 bytes)
 median 54.075 ns (2 allocs: 64 bytes)
 mean   55.187 ns (2 allocs: 64 bytes)
 max    102.299 ns (2 allocs: 64 bytes)

julia> @be A_car($loc)
Benchmark: 3213 samples with 579 evaluations
 min    48.606 ns (2 allocs: 48 bytes)
 median 50.093 ns (2 allocs: 48 bytes)
 mean   50.727 ns (2 allocs: 48 bytes)
 max    105.841 ns (2 allocs: 48 bytes)

Non-uniform Cartesian interpolation

julia> @be B_car_nu($loc)
Benchmark: 2281 samples with 493 evaluations
 min    55.907 ns (2 allocs: 64 bytes)
 median 57.734 ns (2 allocs: 64 bytes)
 mean   179.714 ns (2 allocs: 64 bytes, 0.04% gc time)
 max    260.050 μs (2 allocs: 64 bytes, 99.89% gc time)

julia> @be A_car_nu($loc)
Benchmark: 3160 samples with 559 evaluations
 min    50.685 ns (2 allocs: 48 bytes)
 median 52.440 ns (2 allocs: 48 bytes)
 mean   53.148 ns (2 allocs: 48 bytes)
 max    98.163 ns (2 allocs: 48 bytes)

Based on the benchmarks, for the same grid size, gridded interpolation (StructuredGrid with non-uniform ranges, RectilinearGrid) is 2x slower than uniform mesh interpolation (StructuredGrid with uniform ranges, CartesianGrid).

Mixed precision interpolation

Numerical field data from files is often stored in Float32. TestParticle supports constructing interpolators from Float32 data and ranges, which can then be queried with both Float32 and Float64 location vectors.

julia> @be itp_f32($loc_f32)
Benchmark: 3179 samples with 574 evaluations
 min    49.341 ns (2 allocs: 64 bytes)
 median 51.087 ns (2 allocs: 64 bytes)
 mean   51.823 ns (2 allocs: 64 bytes)
 max    104.949 ns (2 allocs: 64 bytes)

julia> @be itp_f32($loc_f64)
Benchmark: 3242 samples with 538 evaluations
 min    51.584 ns (2 allocs: 64 bytes)
 median 52.998 ns (2 allocs: 64 bytes)
 mean   53.851 ns (2 allocs: 64 bytes)
 max    111.004 ns (2 allocs: 64 bytes)

Memory usage analysis

Large numerical field arrays can consume significant amounts of memory. It's important that interpolators are memory-efficient during both construction and storage.

For linear interpolation (order=1), construction is near zero-allocation because it creates a wrapper around a reinterpreted view of your existing array.

For higher-order interpolation (order = 3), FastInterpolations.jl provides two modes: OnTheFly() and PreCompute(). By default, OnTheFly() is used, which calculates interpolation coefficients during each query. This approach is memory-efficient as it does not require additional storage beyond the input data. Alternatively, PreCompute() precalculates and stores coefficients in an additional array of the same size, enabling faster $O(1)$ lookup performance at the cost of higher memory usage.

We can measure this difference using @be.

julia> # Order 1: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 1)
Benchmark: 220 samples with 7 evaluations
 min    3.203 μs (8 allocs: 31.750 KiB)
 median 3.655 μs (8 allocs: 31.750 KiB)
 mean   68.693 μs (8 allocs: 31.750 KiB, 0.45% gc time)
 max    14.293 ms (8 allocs: 31.750 KiB, 97.97% gc time)

julia> # Order 3: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 3)
Benchmark: 1082 samples with 6 evaluations
 min    3.517 μs (14 allocs: 32.047 KiB)
 median 4.482 μs (14 allocs: 32.047 KiB)
 mean   15.398 μs (14 allocs: 32.047 KiB, 0.18% gc time)
 max    6.723 ms (14 allocs: 32.047 KiB, 97.95% gc time)

Comparing the ratios relative to the original array size illustrates the overhead:

julia> B = fill(0.0f0, 3, 11, 11, 11);

julia> size_B = Base.summarysize(B) # Original 4D field array size
16036

julia> size_itp1 = Base.summarysize(itp_f32) # Total size for order=1 (Essentially the input array size)
16172

julia> itp_f32_q = setup_mixed_precision_field(11, 3) # Total size for order=3
(::TestParticle.FieldInterpolator{FastInterpolations.HeteroInterpolantND{Float32, StaticArraysCore.SVector{3, Float32}, 3, Tuple{FastInterpolations._CachedRange{Float32, Float32, FastInterpolations._Generic}, FastInterpolations._CachedRange{Float32, Float32, FastInterpolations._Generic}, FastInterpolations._CachedRange{Float32, Float32, FastInterpolations._Generic}}, Tuple{FastInterpolations.CardinalInterp{Float64, FastInterpolations.NoBC}, FastInterpolations.CardinalInterp{Float64, FastInterpolations.NoBC}, FastInterpolations.CardinalInterp{Float64, FastInterpolations.NoBC}}, Tuple{FillExtrap{StaticArraysCore.SVector{3, Float32}}, FillExtrap{StaticArraysCore.SVector{3, Float32}}, FillExtrap{StaticArraysCore.SVector{3, Float32}}}, Tuple{FastInterpolations.AutoSearch, FastInterpolations.AutoSearch, FastInterpolations.AutoSearch}, Array{StaticArraysCore.SVector{3, Float32}, 3}}}) (generic function with 2 methods)

julia> size_itp3 = Base.summarysize(itp_f32_q)
16196

julia> # Ratios relative to raw data
       size_itp1 / size_B
1.008480917934647

julia> size_itp3 / size_B
1.0099775505113495

As a rule of thumb, linear interpolation and cubic interpolation with OnTheFly() coefficients have nearly zero memory overhead (ratio ≈ 1.0), as they both operate directly on the input data. When supported, cubic interpolation with PreCompute() coefficients increases the memory footprint by $2^{\mathrm{DIM}}$ to store the extra coefficients, where $\mathrm{DIM}$ is the dimension of the field.

On-the-fly vs Precomputed coefficients

Cubic interpolation (order = 3) requires high-order coefficients. By default, TestParticle uses OnTheFly() coefficients, which are calculated at query time. This saves memory but increases evaluation time. For maximum performance, you can use PreCompute(), which stores the coefficients in an additional array.

Status in FastInterpolations v0.4.15

PreCompute() is now available for the global natural cubic spline (CubicInterp) in ND as of FastInterpolations.jl v0.4.15. However, it is not yet supported for the local Hermite cubic spline (CardinalInterp, i.e. order = 3 in TestParticle) in ND — constructing such an interpolator with coeffs = PreCompute() raises an ArgumentError. OnTheFly() therefore remains the only option for cubic interpolation in TestParticle.

julia> # Benchmark evaluation time
       itp_fly = setup_mixed_precision_field(11, 3; coeffs = OnTheFly());

julia> # itp_pre = setup_mixed_precision_field(11, 3; coeffs = PreCompute()); # unsupported for ND cardinal cubic in v0.4.15
       
       @be itp_fly($loc_f64)
Benchmark: 3127 samples with 92 evaluations
 min    314.065 ns (2 allocs: 64 bytes)
 median 319.283 ns (2 allocs: 64 bytes)
 mean   323.505 ns (2 allocs: 64 bytes)
 max    680.924 ns (2 allocs: 64 bytes)

julia> # Compare total object size
       Base.summarysize(itp_fly)
16196

As shown, OnTheFly() preserves memory efficiency while providing higher-order accuracy. Once PreCompute() support for ND local Hermite cubic interpolation lands, it will offer a faster alternative for memory-abundant systems.

Time-dependent field interpolation

For time-dependent fields, we can use LazyTimeInterpolator. It takes a list of time points and a loader function that returns a spatial interpolator for a given time index. The interpolator will linearly interpolate between the two nearest time points.

julia> @be B_td($loc, 0.5)
Benchmark: 3109 samples with 172 evaluations
 min    165.599 ns (2 allocs: 64 bytes)
 median 171.070 ns (2 allocs: 64 bytes)
 mean   174.211 ns (2 allocs: 64 bytes)
 max    1.140 μs (2 allocs: 64 bytes)

Related API

TestParticle.build_interpolator Function

build_interpolator(gridtype, A, grids..., order::Int=1, bc=FillExtrap(NaN))
build_interpolator(A, grids..., order::Int=1, bc=FillExtrap(NaN))

Return a function for interpolating field array A on the given grids.

Arguments

• gridtype: CartesianGrid, RectilinearGrid or StructuredGrid. Usually determined by the number of grids.

• A: field array. For vector field, the first dimension should be 3 if it's not an SVector wrapper.

• order::Int=1: order of interpolation in [0,1,3].

• bc=FillExtrap(NaN): boundary condition type from FastInterpolations.jl.

  • FillExtrap(NaN): Fill with NaN (default).

  • ClampExtrap(): Clamp (flat extrapolation).

  • WrapExtrap(): Exclusive periodic wrapping ($L=N\mathrm{\Delta }x$).

• coeffs=OnTheFly(): coefficient strategy for cubic interpolation (order=3). Default is OnTheFly().

Notes

• The input array A may be modified in-place for memory optimization.

source

TestParticle.prepare Function

prepare(args...; kwargs...) -> (q2m, m, E, B, F)
prepare(E, B, F = ZeroField(); kwargs...)
prepare(grid::CartesianGrid, E, B, F = ZeroField(); kwargs...)
prepare(x, E, B, F = ZeroField(); dir = 1, kwargs...)
prepare(x, y, E, B, F = ZeroField(); kwargs...)
prepare(x, y, z, E, B, F = ZeroField(); kwargs...)
prepare(B; E = ZeroField(), F = ZeroField(), kwargs...)

Return a tuple consists of particle charge-mass ratio for a prescribed species of charge q and mass m, mass m for a prescribed species, analytic/interpolated EM field functions, and external force F.

Prescribed species are Electron and Proton; other species can be manually specified with m and q keywords or species = Ion(m̄, q̄), where m̄ and q̄ are the mass and charge numbers respectively.

Direct range input for uniform grid in 1/2/3D is supported. The grid vectors must be sorted. For 1D grid, an additional keyword dir is used for specifying the spatial direction, 1 -> x, 2 -> y, 3 -> z. For 3D grid, the default grid type is CartesianGrid. To use StructuredGrid (spherical) grid, an additional keyword gridtype is needed. For StructuredGrid (spherical) grid, dimensions of field arrays should be (Br, Bθ, Bϕ).

Keywords

• order::Int=1: order of interpolation in [0,1,3].

• bc=FillExtrap(NaN): boundary condition type from FastInterpolations.jl.

• species=Proton: particle species.

• q=nothing: particle charge.

• m=nothing: particle mass.

• gridtype: CartesianGrid, RectilinearGrid, StructuredGrid.

source