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: 3170 samples with 242 evaluations
 min    118.901 ns (2 allocs: 64 bytes)
 median 120.599 ns (2 allocs: 64 bytes)
 mean   122.682 ns (2 allocs: 64 bytes)
 max    266.620 ns (2 allocs: 64 bytes)

julia> @be A_sph_nu($loc)
Benchmark: 3166 samples with 319 evaluations
 min    89.856 ns (2 allocs: 48 bytes)
 median 91.489 ns (2 allocs: 48 bytes)
 mean   92.914 ns (2 allocs: 48 bytes)
 max    205.119 ns (2 allocs: 48 bytes)

Uniform spherical interpolation

julia> @be B_sph($loc)
Benchmark: 3149 samples with 243 evaluations
 min    117.712 ns (2 allocs: 64 bytes)
 median 119.609 ns (2 allocs: 64 bytes)
 mean   121.630 ns (2 allocs: 64 bytes)
 max    217.490 ns (2 allocs: 64 bytes)

julia> @be A_sph($loc)
Benchmark: 3172 samples with 324 evaluations
 min    88.469 ns (2 allocs: 48 bytes)
 median 90.139 ns (2 allocs: 48 bytes)
 mean   91.361 ns (2 allocs: 48 bytes)
 max    173.722 ns (2 allocs: 48 bytes)

Uniform Cartesian interpolation

julia> @be B_car($loc)
Benchmark: 2403 samples with 578 evaluations
 min    48.379 ns (2 allocs: 64 bytes)
 median 50.631 ns (2 allocs: 64 bytes)
 mean   139.450 ns (2 allocs: 64 bytes, 0.04% gc time)
 max    192.468 μs (2 allocs: 64 bytes, 99.88% gc time)

julia> @be A_car($loc)
Benchmark: 3258 samples with 603 evaluations
 min    46.355 ns (2 allocs: 48 bytes)
 median 47.818 ns (2 allocs: 48 bytes)
 mean   48.476 ns (2 allocs: 48 bytes)
 max    94.506 ns (2 allocs: 48 bytes)

Non-uniform Cartesian interpolation

julia> @be B_car_nu($loc)
Benchmark: 3218 samples with 564 evaluations
 min    50.041 ns (2 allocs: 64 bytes)
 median 51.622 ns (2 allocs: 64 bytes)
 mean   52.688 ns (2 allocs: 64 bytes)
 max    105.215 ns (2 allocs: 64 bytes)

julia> @be A_car_nu($loc)
Benchmark: 3173 samples with 603 evaluations
 min    46.904 ns (2 allocs: 48 bytes)
 median 48.449 ns (2 allocs: 48 bytes)
 mean   49.134 ns (2 allocs: 48 bytes)
 max    143.852 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: 1463 samples with 616 evaluations
 min    45.490 ns (2 allocs: 64 bytes)
 median 67.937 ns (2 allocs: 64 bytes)
 mean   174.256 ns (2 allocs: 64 bytes, 0.07% gc time)
 max    165.976 μs (2 allocs: 64 bytes, 99.88% gc time)

julia> @be itp_f32($loc_f64)
Benchmark: 3235 samples with 551 evaluations
 min    50.040 ns (2 allocs: 64 bytes)
 median 51.294 ns (2 allocs: 64 bytes)
 mean   52.225 ns (2 allocs: 64 bytes)
 max    99.606 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: 931 samples with 8 evaluations
 min    3.058 μs (8 allocs: 31.750 KiB)
 median 3.418 μs (8 allocs: 31.750 KiB)
 mean   14.456 μs (8 allocs: 31.750 KiB, 0.21% gc time)
 max    6.143 ms (8 allocs: 31.750 KiB, 98.95% gc time)

julia> # Order 3: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 3)
Benchmark: 1324 samples with 7 evaluations
 min    3.474 μs (14 allocs: 32.047 KiB)
 median 3.898 μs (14 allocs: 32.047 KiB)
 mean   10.549 μs (14 allocs: 32.047 KiB, 0.15% gc time)
 max    4.500 ms (14 allocs: 32.047 KiB, 99.29% 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: 3146 samples with 83 evaluations
 min    347.880 ns (2 allocs: 64 bytes)
 median 352.831 ns (2 allocs: 64 bytes)
 mean   357.233 ns (2 allocs: 64 bytes)
 max    673.446 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: 3132 samples with 184 evaluations
 min    156.272 ns (2 allocs: 64 bytes)
 median 160.027 ns (2 allocs: 64 bytes)
 mean   162.059 ns (2 allocs: 64 bytes)
 max    310.853 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