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: 3149 samples with 240 evaluations
 min    119.767 ns (2 allocs: 64 bytes)
 median 121.646 ns (2 allocs: 64 bytes)
 mean   123.327 ns (2 allocs: 64 bytes)
 max    229.142 ns (2 allocs: 64 bytes)

julia> @be A_sph_nu($loc)
Benchmark: 3186 samples with 322 evaluations
 min    88.646 ns (2 allocs: 48 bytes)
 median 90.171 ns (2 allocs: 48 bytes)
 mean   91.519 ns (2 allocs: 48 bytes)
 max    198.354 ns (2 allocs: 48 bytes)

Uniform spherical interpolation

julia> @be B_sph($loc)
Benchmark: 3154 samples with 246 evaluations
 min    116.317 ns (2 allocs: 64 bytes)
 median 117.947 ns (2 allocs: 64 bytes)
 mean   120.082 ns (2 allocs: 64 bytes)
 max    250.801 ns (2 allocs: 64 bytes)

julia> @be A_sph($loc)
Benchmark: 3160 samples with 319 evaluations
 min    90.201 ns (2 allocs: 48 bytes)
 median 91.740 ns (2 allocs: 48 bytes)
 mean   92.910 ns (2 allocs: 48 bytes)
 max    189.476 ns (2 allocs: 48 bytes)

Uniform Cartesian interpolation

julia> @be B_car($loc)
Benchmark: 3207 samples with 574 evaluations
 min    49.134 ns (2 allocs: 64 bytes)
 median 50.549 ns (2 allocs: 64 bytes)
 mean   51.493 ns (2 allocs: 64 bytes)
 max    117.260 ns (2 allocs: 64 bytes)

julia> @be A_car($loc)
Benchmark: 3191 samples with 593 evaluations
 min    47.627 ns (2 allocs: 48 bytes)
 median 48.879 ns (2 allocs: 48 bytes)
 mean   49.572 ns (2 allocs: 48 bytes)
 max    99.835 ns (2 allocs: 48 bytes)

Non-uniform Cartesian interpolation

julia> @be B_car_nu($loc)
Benchmark: 3166 samples with 551 evaluations
 min    51.731 ns (2 allocs: 64 bytes)
 median 52.911 ns (2 allocs: 64 bytes)
 mean   53.794 ns (2 allocs: 64 bytes)
 max    128.753 ns (2 allocs: 64 bytes)

julia> @be A_car_nu($loc)
Benchmark: 851 samples with 574 evaluations
 min    49.293 ns (2 allocs: 48 bytes)
 median 51.247 ns (2 allocs: 48 bytes)
 mean   259.546 ns (2 allocs: 48 bytes, 0.12% gc time)
 max    173.418 μs (2 allocs: 48 bytes, 99.88% gc time)

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: 3181 samples with 597 evaluations
 min    47.342 ns (2 allocs: 64 bytes)
 median 48.583 ns (2 allocs: 64 bytes)
 mean   49.374 ns (2 allocs: 64 bytes)
 max    112.407 ns (2 allocs: 64 bytes)

julia> @be itp_f32($loc_f64)
Benchmark: 3283 samples with 548 evaluations
 min    49.343 ns (2 allocs: 64 bytes)
 median 50.807 ns (2 allocs: 64 bytes)
 mean   51.908 ns (2 allocs: 64 bytes)
 max    114.066 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: 582 samples with 8 evaluations
 min    3.135 μs (8 allocs: 31.750 KiB)
 median 3.833 μs (8 allocs: 31.750 KiB)
 mean   27.583 μs (8 allocs: 31.750 KiB, 0.34% gc time)
 max    9.575 ms (8 allocs: 31.750 KiB, 98.59% gc time)

julia> # Order 3: Minimal allocations (Uses a view)
       @be setup_mixed_precision_field(11, 3)
Benchmark: 1274 samples with 6 evaluations
 min    3.533 μs (14 allocs: 32.047 KiB)
 median 4.208 μs (14 allocs: 32.047 KiB)
 mean   12.868 μs (14 allocs: 32.047 KiB, 0.16% gc time)
 max    5.838 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._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: 3089 samples with 87 evaluations
 min    335.115 ns (2 allocs: 64 bytes)
 median 342.023 ns (2 allocs: 64 bytes)
 mean   346.747 ns (2 allocs: 64 bytes)
 max    812.230 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: 3123 samples with 189 evaluations
 min    152.561 ns (2 allocs: 64 bytes)
 median 155.323 ns (2 allocs: 64 bytes)
 mean   158.421 ns (2 allocs: 64 bytes)
 max    402.873 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