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 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: 3103 samples with 228 evaluations
 min    126.596 ns (2 allocs: 64 bytes)
 median 128.886 ns (2 allocs: 64 bytes)
 mean   131.514 ns (2 allocs: 64 bytes)
 max    222.478 ns (2 allocs: 64 bytes)

julia> @be A_sph_nu($loc)
Benchmark: 3146 samples with 288 evaluations
 min    99.840 ns (2 allocs: 48 bytes)
 median 102.205 ns (2 allocs: 48 bytes)
 mean   103.684 ns (2 allocs: 48 bytes)
 max    216.514 ns (2 allocs: 48 bytes)

Uniform spherical interpolation

julia> @be B_sph($loc)
Benchmark: 3143 samples with 227 evaluations
 min    127.815 ns (2 allocs: 64 bytes)
 median 129.357 ns (2 allocs: 64 bytes)
 mean   131.292 ns (2 allocs: 64 bytes)
 max    292.665 ns (2 allocs: 64 bytes)

julia> @be A_sph($loc)
Benchmark: 2945 samples with 287 evaluations
 min    100.084 ns (2 allocs: 48 bytes)
 median 103.258 ns (2 allocs: 48 bytes)
 mean   111.153 ns (2 allocs: 48 bytes)
 max    909.889 ns (2 allocs: 48 bytes)

Uniform Cartesian interpolation

julia> @be B_car($loc)
Benchmark: 719 samples with 329 evaluations
 min    68.942 ns (2 allocs: 64 bytes)
 median 86.942 ns (2 allocs: 64 bytes)
 mean   6.070 μs (2.00 allocs: 64.003 bytes, 0.14% gc time)
 max    4.298 ms (2.03 allocs: 66.043 bytes, 99.98% gc time)

julia> @be A_car($loc)
Benchmark: 3168 samples with 462 evaluations
 min    61.110 ns (2 allocs: 48 bytes)
 median 62.758 ns (2 allocs: 48 bytes)
 mean   63.771 ns (2 allocs: 48 bytes)
 max    139.613 ns (2 allocs: 48 bytes)

Non-uniform Cartesian interpolation

julia> @be B_car_nu($loc)
Benchmark: 3154 samples with 482 evaluations
 min    58.886 ns (2 allocs: 64 bytes)
 median 60.734 ns (2 allocs: 64 bytes)
 mean   61.675 ns (2 allocs: 64 bytes)
 max    122.573 ns (2 allocs: 64 bytes)

julia> @be A_car_nu($loc)
Benchmark: 3161 samples with 435 evaluations
 min    65.731 ns (2 allocs: 48 bytes)
 median 67.575 ns (2 allocs: 48 bytes)
 mean   68.547 ns (2 allocs: 48 bytes)
 max    154.152 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: 3123 samples with 459 evaluations
 min    62.667 ns (2 allocs: 64 bytes)
 median 63.669 ns (2 allocs: 64 bytes)
 mean   64.978 ns (2 allocs: 64 bytes)
 max    185.706 ns (2 allocs: 64 bytes)

julia> @be itp_f32($loc_f64)
Benchmark: 2920 samples with 403 evaluations
 min    70.777 ns (2 allocs: 64 bytes)
 median 73.313 ns (2 allocs: 64 bytes)
 mean   79.461 ns (2 allocs: 64 bytes)
 max    215.640 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: 1218 samples with 6 evaluations
 min    2.850 μs (8 allocs: 31.781 KiB)
 median 3.957 μs (8 allocs: 31.781 KiB)
 mean   21.036 μs (8 allocs: 31.781 KiB, 0.08% gc time)
 max    15.092 ms (8 allocs: 31.781 KiB, 99.32% gc time)

julia> # Order 3: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 3)
Benchmark: 1822 samples with 6 evaluations
 min    3.343 μs (14 allocs: 32.109 KiB)
 median 4.111 μs (14 allocs: 32.109 KiB)
 mean   10.496 μs (14 allocs: 32.109 KiB, 0.11% gc time)
 max    6.924 ms (14 allocs: 32.109 KiB, 98.28% 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)
16196

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}, FastInterpolations._CachedRange{Float32}, FastInterpolations._CachedRange{Float32}}, Tuple{FastInterpolations.ScalarSpacing{Float32}, FastInterpolations.ScalarSpacing{Float32}, FastInterpolations.ScalarSpacing{Float32}}, Tuple{FastInterpolations.CardinalInterp{Float64}, FastInterpolations.CardinalInterp{Float64}, FastInterpolations.CardinalInterp{Float64}}, 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)
16220

julia> # Ratios relative to raw data
       size_itp1 / size_B
1.0099775505113495

julia> size_itp3 / size_B
1.0114741830880518

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 v0.4.8

Support for PreCompute() with local Hermite cubic interpolation is currently under development and not yet available in FastInterpolations.jl v0.4.8.

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()); # Not yet supported in v0.4.8
       
       @be itp_fly($loc_f64)
Benchmark: 3196 samples with 46 evaluations
 min    643.587 ns (2 allocs: 64 bytes)
 median 650.348 ns (2 allocs: 64 bytes)
 mean   661.949 ns (2 allocs: 64 bytes)
 max    1.484 μs (2 allocs: 64 bytes)

julia> # Compare total object size
       Base.summarysize(itp_fly)
16220

As shown, OnTheFly() preserves memory efficiency while providing higher-order accuracy. Once available, PreCompute() 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: 3155 samples with 151 evaluations
 min    190.689 ns (2 allocs: 64 bytes)
 median 193.139 ns (2 allocs: 64 bytes)
 mean   195.815 ns (2 allocs: 64 bytes)
 max    382.040 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