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: 3117 samples with 226 evaluations
 min    127.673 ns (2 allocs: 64 bytes)
 median 129.885 ns (2 allocs: 64 bytes)
 mean   131.544 ns (2 allocs: 64 bytes)
 max    449.040 ns (2 allocs: 64 bytes)

julia> @be A_sph_nu($loc)
Benchmark: 3104 samples with 294 evaluations
 min    98.075 ns (2 allocs: 48 bytes)
 median 100.524 ns (2 allocs: 48 bytes)
 mean   101.967 ns (2 allocs: 48 bytes)
 max    208.340 ns (2 allocs: 48 bytes)

Uniform spherical interpolation

julia> @be B_sph($loc)
Benchmark: 3167 samples with 228 evaluations
 min    125.890 ns (2 allocs: 64 bytes)
 median 127.956 ns (2 allocs: 64 bytes)
 mean   129.463 ns (2 allocs: 64 bytes)
 max    197.627 ns (2 allocs: 64 bytes)

julia> @be A_sph($loc)
Benchmark: 884 samples with 293 evaluations
 min    98.205 ns (2 allocs: 48 bytes)
 median 99.913 ns (2 allocs: 48 bytes)
 mean   629.363 ns (2 allocs: 48 bytes, 0.11% gc time)
 max    467.091 μs (2 allocs: 48 bytes, 99.86% gc time)

Uniform Cartesian interpolation

julia> @be B_car($loc)
Benchmark: 3208 samples with 530 evaluations
 min    52.438 ns (2 allocs: 64 bytes)
 median 54.064 ns (2 allocs: 64 bytes)
 mean   54.796 ns (2 allocs: 64 bytes)
 max    97.979 ns (2 allocs: 64 bytes)

julia> @be A_car($loc)
Benchmark: 3203 samples with 564 evaluations
 min    49.613 ns (2 allocs: 48 bytes)
 median 51.408 ns (2 allocs: 48 bytes)
 mean   52.213 ns (2 allocs: 48 bytes)
 max    101.004 ns (2 allocs: 48 bytes)

Non-uniform Cartesian interpolation

julia> @be B_car_nu($loc)
Benchmark: 3193 samples with 527 evaluations
 min    53.457 ns (2 allocs: 64 bytes)
 median 55.169 ns (2 allocs: 64 bytes)
 mean   55.868 ns (2 allocs: 64 bytes)
 max    88.977 ns (2 allocs: 64 bytes)

julia> @be A_car_nu($loc)
Benchmark: 3172 samples with 567 evaluations
 min    48.875 ns (2 allocs: 48 bytes)
 median 51.224 ns (2 allocs: 48 bytes)
 mean   52.591 ns (2 allocs: 48 bytes)
 max    131.875 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: 3209 samples with 552 evaluations
 min    51.002 ns (2 allocs: 64 bytes)
 median 52.234 ns (2 allocs: 64 bytes)
 mean   53.049 ns (2 allocs: 64 bytes)
 max    106.411 ns (2 allocs: 64 bytes)

julia> @be itp_f32($loc_f64)
Benchmark: 3131 samples with 519 evaluations
 min    54.185 ns (2 allocs: 64 bytes)
 median 55.961 ns (2 allocs: 64 bytes)
 mean   57.229 ns (2 allocs: 64 bytes)
 max    142.644 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: 720 samples with 7 evaluations
 min    3.082 μs (8 allocs: 31.750 KiB)
 median 3.625 μs (8 allocs: 31.750 KiB)
 mean   25.297 μs (8 allocs: 31.750 KiB, 0.27% gc time)
 max    10.962 ms (8 allocs: 31.750 KiB, 98.41% gc time)

julia> # Order 3: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 3)
Benchmark: 1326 samples with 6 evaluations
 min    3.580 μs (14 allocs: 32.047 KiB)
 median 4.294 μs (14 allocs: 32.047 KiB)
 mean   12.305 μs (14 allocs: 32.047 KiB, 0.15% gc time)
 max    5.560 ms (14 allocs: 32.047 KiB, 99.69% 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: 3132 samples with 90 evaluations
 min    320.267 ns (2 allocs: 64 bytes)
 median 325.822 ns (2 allocs: 64 bytes)
 mean   330.010 ns (2 allocs: 64 bytes)
 max    570.189 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: 3118 samples with 173 evaluations
 min    165.566 ns (2 allocs: 64 bytes)
 median 170.376 ns (2 allocs: 64 bytes)
 mean   172.541 ns (2 allocs: 64 bytes)
 max    333.913 ns (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