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: 793 samples with 212 evaluations
 min    133.547 ns (2 allocs: 64 bytes)
 median 139.075 ns (2 allocs: 64 bytes)
 mean   1.659 μs (2.00 allocs: 64.002 bytes, 0.13% gc time)
 max    1.194 ms (2.04 allocs: 65.811 bytes, 99.74% gc time)

julia> @be A_sph_nu($loc)
Benchmark: 3214 samples with 284 evaluations
 min    100.151 ns (2 allocs: 48 bytes)
 median 101.736 ns (2 allocs: 48 bytes)
 mean   103.313 ns (2 allocs: 48 bytes)
 max    1.104 μs (2 allocs: 48 bytes)

Uniform spherical interpolation

julia> @be B_sph($loc)
Benchmark: 3141 samples with 218 evaluations
 min    132.766 ns (2 allocs: 64 bytes)
 median 134.881 ns (2 allocs: 64 bytes)
 mean   136.601 ns (2 allocs: 64 bytes)
 max    289.789 ns (2 allocs: 64 bytes)

julia> @be A_sph($loc)
Benchmark: 3133 samples with 287 evaluations
 min    99.801 ns (2 allocs: 48 bytes)
 median 102.174 ns (2 allocs: 48 bytes)
 mean   103.670 ns (2 allocs: 48 bytes)
 max    191.470 ns (2 allocs: 48 bytes)

Uniform Cartesian interpolation

julia> @be B_car($loc)
Benchmark: 3203 samples with 523 evaluations
 min    54.153 ns (2 allocs: 64 bytes)
 median 55.723 ns (2 allocs: 64 bytes)
 mean   56.573 ns (2 allocs: 64 bytes)
 max    135.669 ns (2 allocs: 64 bytes)

julia> @be A_car($loc)
Benchmark: 2837 samples with 550 evaluations
 min    51.422 ns (2 allocs: 48 bytes)
 median 60.635 ns (2 allocs: 48 bytes)
 mean   60.267 ns (2 allocs: 48 bytes)
 max    150.078 ns (2 allocs: 48 bytes)

Non-uniform Cartesian interpolation

julia> @be B_car_nu($loc)
Benchmark: 3215 samples with 426 evaluations
 min    65.545 ns (2 allocs: 64 bytes)
 median 68.129 ns (2 allocs: 64 bytes)
 mean   69.125 ns (2 allocs: 64 bytes)
 max    191.505 ns (2 allocs: 64 bytes)

julia> @be A_car_nu($loc)
Benchmark: 3201 samples with 425 evaluations
 min    66.499 ns (2 allocs: 48 bytes)
 median 68.574 ns (2 allocs: 48 bytes)
 mean   69.515 ns (2 allocs: 48 bytes)
 max    118.882 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: 3161 samples with 526 evaluations
 min    53.635 ns (2 allocs: 64 bytes)
 median 55.120 ns (2 allocs: 64 bytes)
 mean   56.134 ns (2 allocs: 64 bytes)
 max    141.580 ns (2 allocs: 64 bytes)

julia> @be itp_f32($loc_f64)
Benchmark: 3142 samples with 499 evaluations
 min    57.259 ns (2 allocs: 64 bytes)
 median 58.966 ns (2 allocs: 64 bytes)
 mean   59.792 ns (2 allocs: 64 bytes)
 max    103.822 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: 131 samples with 7 evaluations
 min    3.159 μs (8 allocs: 31.750 KiB)
 median 3.598 μs (8 allocs: 31.750 KiB)
 mean   117.358 μs (8 allocs: 31.750 KiB, 0.75% gc time)
 max    14.891 ms (8 allocs: 31.750 KiB, 97.70% gc time)

julia> # Order 3: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 3)
Benchmark: 1565 samples with 5 evaluations
 min    3.475 μs (14 allocs: 32.047 KiB)
 median 4.597 μs (14 allocs: 32.047 KiB)
 mean   13.468 μs (14 allocs: 32.047 KiB, 0.13% gc time)
 max    7.687 ms (14 allocs: 32.047 KiB, 99.72% 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._CachedRange{Float32, Float32}, FastInterpolations._CachedRange{Float32, Float32}}, 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 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: 3159 samples with 60 evaluations
 min    477.883 ns (2 allocs: 64 bytes)
 median 485.217 ns (2 allocs: 64 bytes)
 mean   493.008 ns (2 allocs: 64 bytes)
 max    1.263 μs (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 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: 3133 samples with 160 evaluations
 min    178.763 ns (2 allocs: 64 bytes)
 median 183.831 ns (2 allocs: 64 bytes)
 mean   185.924 ns (2 allocs: 64 bytes)
 max    412.925 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