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: 3128 samples with 225 evaluations
 min    129.129 ns (2 allocs: 64 bytes)
 median 130.560 ns (2 allocs: 64 bytes)
 mean   132.571 ns (2 allocs: 64 bytes)
 max    365.751 ns (2 allocs: 64 bytes)

julia> @be A_sph_nu($loc)
Benchmark: 3170 samples with 289 evaluations
 min    99.737 ns (2 allocs: 48 bytes)
 median 101.453 ns (2 allocs: 48 bytes)
 mean   102.894 ns (2 allocs: 48 bytes)
 max    193.512 ns (2 allocs: 48 bytes)

Uniform spherical interpolation

julia> @be B_sph($loc)
Benchmark: 3121 samples with 227 evaluations
 min    127.991 ns (2 allocs: 64 bytes)
 median 129.802 ns (2 allocs: 64 bytes)
 mean   131.694 ns (2 allocs: 64 bytes)
 max    262.432 ns (2 allocs: 64 bytes)

julia> @be A_sph($loc)
Benchmark: 3158 samples with 283 evaluations
 min    101.286 ns (2 allocs: 48 bytes)
 median 103.516 ns (2 allocs: 48 bytes)
 mean   104.880 ns (2 allocs: 48 bytes)
 max    218.431 ns (2 allocs: 48 bytes)

Uniform Cartesian interpolation

julia> @be B_car($loc)
Benchmark: 2469 samples with 481 evaluations
 min    58.466 ns (2 allocs: 64 bytes)
 median 80.547 ns (2 allocs: 64 bytes)
 mean   174.102 ns (2 allocs: 64 bytes, 0.04% gc time)
 max    246.612 μs (2 allocs: 64 bytes, 99.85% gc time)

julia> @be A_car($loc)
Benchmark: 3199 samples with 498 evaluations
 min    56.532 ns (2 allocs: 48 bytes)
 median 57.920 ns (2 allocs: 48 bytes)
 mean   58.886 ns (2 allocs: 48 bytes)
 max    165.048 ns (2 allocs: 48 bytes)

Non-uniform Cartesian interpolation

julia> @be B_car_nu($loc)
Benchmark: 3157 samples with 503 evaluations
 min    56.109 ns (2 allocs: 64 bytes)
 median 57.724 ns (2 allocs: 64 bytes)
 mean   58.995 ns (2 allocs: 64 bytes)
 max    129.149 ns (2 allocs: 64 bytes)

julia> @be A_car_nu($loc)
Benchmark: 3263 samples with 520 evaluations
 min    53.619 ns (2 allocs: 48 bytes)
 median 55.085 ns (2 allocs: 48 bytes)
 mean   55.932 ns (2 allocs: 48 bytes)
 max    101.015 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: 3111 samples with 559 evaluations
 min    50.758 ns (2 allocs: 64 bytes)
 median 52.388 ns (2 allocs: 64 bytes)
 mean   53.791 ns (2 allocs: 64 bytes)
 max    112.932 ns (2 allocs: 64 bytes)

julia> @be itp_f32($loc_f64)
Benchmark: 1224 samples with 333 evaluations
 min    82.078 ns (2 allocs: 64 bytes)
 median 85.294 ns (2 allocs: 64 bytes)
 mean   321.077 ns (2 allocs: 64 bytes, 0.08% gc time)
 max    283.993 μs (2 allocs: 64 bytes, 99.88% gc time)

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: 1284 samples with 7 evaluations
 min    2.961 μs (8 allocs: 31.781 KiB)
 median 3.608 μs (8 allocs: 31.781 KiB)
 mean   10.605 μs (8 allocs: 31.781 KiB, 0.23% gc time)
 max    4.391 ms (8 allocs: 31.781 KiB, 98.24% gc time)

julia> # Order 3: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 3)
Benchmark: 1188 samples with 7 evaluations
 min    3.514 μs (14 allocs: 32.109 KiB)
 median 4.079 μs (14 allocs: 32.109 KiB)
 mean   12.115 μs (14 allocs: 32.109 KiB, 0.24% gc time)
 max    4.689 ms (14 allocs: 32.109 KiB, 98.39% 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: 3359 samples with 43 evaluations
 min    653.558 ns (2 allocs: 64 bytes)
 median 663.791 ns (2 allocs: 64 bytes)
 mean   673.445 ns (2 allocs: 64 bytes)
 max    1.500 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: 3138 samples with 162 evaluations
 min    179.407 ns (2 allocs: 64 bytes)
 median 182.444 ns (2 allocs: 64 bytes)
 mean   184.755 ns (2 allocs: 64 bytes)
 max    377.994 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