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Sparse GP-FVM Solver
Sparse Cholesky approximation + Finite Volume Method discretization.
This is the main method we're advocating for in the paper.
"""
using GPFiniteVolume
using FunctionalGPs, GaussianMarkovRandomFields
using LinearAlgebra, SparseArrays
using Kronecker
using SparseConnectivityTracer, SparseMatrixColorings
import GaussianMarkovRandomFields: mean, std, precision_matrix
# Include common problem/metrics definitions
include(joinpath(@__DIR__, "..", "problem.jl"))
include(joinpath(@__DIR__, "..", "metrics.jl"))
"""
solve_sparse_fvm(instance::ProblemInstance, N::Int; kwargs...)
Solve Burgers equation using sparse GP-FVM.
# Arguments
- `instance`: Problem instance with IC and reference solution
- `N`: Number of spatial grid points
# Keyword arguments
- `n_timesteps`: Number of time steps (required for fair comparison)
- `ρ=3.0`: Sparse Cholesky threshold
- `lengthscale=nothing`: Absolute kernel lengthscale (overrides lengthscale_factor if set)
- `lengthscale_factor=3.0`: Kernel lengthscale as multiple of cell size (lengthscale = factor * Δx)
- `smoothness=2`: Matérn smoothness (2 = Matérn 5/2)
- `time_scheme=:crank_nicolson`: Time integration scheme
# Returns
`SolutionResult` with posterior mean/std and computational metrics.
"""
function solve_sparse_fvm(instance::ProblemInstance, N::Int;
n_timesteps::Int,
ρ::Float64=3.0,
lengthscale::Union{Nothing,Float64}=nothing,
lengthscale_factor::Float64=3.0,
smoothness::Int=2,
time_scheme::Symbol=:crank_nicolson,
initialization::Symbol=:ekf)
problem = instance.problem
ic = instance.ic
# Track timing
t_start = time()
# Memory tracking (approximate via GC)
GC.gc()
mem_before = Base.gc_live_bytes()
# -------------------------------------------------------------------------
# Spatial setup
# -------------------------------------------------------------------------
(; x_min, x_max, T_end, ν) = problem
L = x_max - x_min
endpoints = range(x_min, x_max, length=N)
intervals = intervals_from_endpoints(collect(endpoints))
N_int = length(intervals)
Δx = endpoints[2] - endpoints[1]
# Time discretization
n_t = n_timesteps
Δt = T_end / n_t
# -------------------------------------------------------------------------
# Build sparse spatial precision
# -------------------------------------------------------------------------
# Use absolute lengthscale if provided, otherwise scale with cell size
ls = isnothing(lengthscale) ? lengthscale_factor * Δx : lengthscale
k = HalfIntegerMaternKernel(smoothness, [ls])
approx = sparse_precision([
:f => EvaluationFunctional(endpoints),
:f_dx => EvaluationFunctional(endpoints) ∘ PartialDerivative((1,)),
:f_int => VectorizedLebesgueIntegral(intervals)
], k; ρ=ρ, ordering=:integrals_coarsest)
Q_space = approx.Q
state_layout = approx.layout
N_space = size(Q_space, 1)
# Track Cholesky nnz
cholesky_nnz = approx.info.nnz
# -------------------------------------------------------------------------
# Build initial state GMRF
# -------------------------------------------------------------------------
Q_state0 = [Q_space spzeros(size(Q_space)...);
spzeros(size(Q_space)...) Q_space]
N_state = size(Q_state0, 1)
x0 = GMRF(zeros(N_state), Q_state0)
full_state_layout = layout((
f = N,
f_dx = N,
f_int = N_int,
df_dt = N,
df_dx_dt = N,
df_int_dt = N_int
))
# -------------------------------------------------------------------------
# Apply initial conditions
# -------------------------------------------------------------------------
ys = [ic(x) for x in endpoints]
ys_dx = [evaluate_dx(ic, x) for x in endpoints]
ys_int = [evaluate_int(ic, endpoints[i], endpoints[i+1]) for i in 1:N_int]
x0_ic = prescribe_indices(x0, indices(full_state_layout, :f), ys)
x0_ic = prescribe_indices(x0_ic, indices(full_state_layout, :f_dx), ys_dx)
x0_ic = prescribe_indices(x0_ic, indices(full_state_layout, :f_int), ys_int)
# -------------------------------------------------------------------------
# FVM constraint at t=0
# -------------------------------------------------------------------------
function f_fvm_t0(x)
x_s = State(x, full_state_layout)
dint_dt = x_s.df_int_dt
u_left = x_s.f[1:end-1]
u_right = x_s.f[2:end]
ux_left = x_s.f_dx[1:end-1]
ux_right = x_s.f_dx[2:end]
F_left = 0.5 * u_left.^2 - ν * ux_left
F_right = 0.5 * u_right.^2 - ν * ux_right
return dint_dt + (F_right - F_left)
end
fvm_model_0 = NonlinearLeastSquaresModel(f_fvm_t0, length(x0_ic))
y_fvm_0 = zeros(N_int)
lik_fvm_0 = fvm_model_0(y_fvm_0; σ=0.0001)
x_fvm_0 = gaussian_approximation(x0_ic, lik_fvm_0)
# -------------------------------------------------------------------------
# Time stepping setup
# -------------------------------------------------------------------------
sde = IWPSDE(1.0)
A_t, Σ_noise_t = discretize_vanloan(sde, Δt)
Q_noise_t = inv(Σ_noise_t)
Q_noise_t = 0.5 * (Q_noise_t + Q_noise_t')
A = kronecker(A_t, sparse(I, N_space, N_space))
Q_noise = kronecker(Q_noise_t, Q_space)
# -------------------------------------------------------------------------
# Build joint spacetime GMRF
# -------------------------------------------------------------------------
ssm = ConstantLinearGaussianSSM(x_fvm_0, A, Q_noise)
x_joint = GPFiniteVolume.joint_gmrf(ssm, n_t)
timest = TimeStack(mean(x_joint), full_state_layout)
# -------------------------------------------------------------------------
# Boundary conditions
# -------------------------------------------------------------------------
bc_left = problem.u_left
bc_right = problem.u_right
ys_bc = repeat([bc_left, bc_right], n_t - 1)
bc_indices = absindices(timest, :f, [1, N], 2:n_t)
x_joint_bc = prescribe_indices(x_joint, bc_indices, ys_bc)
# -------------------------------------------------------------------------
# Causal sweep initialization (block Gauss-Seidel style)
# -------------------------------------------------------------------------
function ekf_sweep(x_t0_mean)
Q_noise_sparse = sparse(collect(Q_noise))
means = Vector{Vector{Float64}}(undef, n_t)
means[1] = x_t0_mean
for t in 2:n_t
μ_prior = collect(A * means[t-1])
x_prior = GMRF(μ_prior, Q_noise_sparse)
# Hard BC constraint via ConstrainedGMRF
A_bc = zeros(2, length(μ_prior))
A_bc[1, 1] = 1.0 # f[1] = bc_left
A_bc[2, N] = 1.0 # f[N] = bc_right
x_bc = ConstrainedGMRF(x_prior, A_bc, [bc_left, bc_right])
function f_fvm_block(x)
x_c = State(x, full_state_layout)
x_p = State(means[t-1], full_state_layout)
F_c = 0.5 * x_c.f.^2 - ν * x_c.f_dx
F_p = 0.5 * x_p.f.^2 - ν * x_p.f_dx
dF_c = F_c[2:end] - F_c[1:end-1]
dF_p = F_p[2:end] - F_p[1:end-1]
return x_c.df_int_dt + 0.5 * (dF_c + dF_p)
end
model = NonlinearLeastSquaresModel(f_fvm_block, length(x_bc))
lik = model(zeros(N_int); σ=0.0001)
x_sol = gaussian_approximation(x_bc, lik; verbose=false)
means[t] = mean(x_sol)
end
return vcat(means...)
end
# -------------------------------------------------------------------------
# FVM constraints at all time steps
# -------------------------------------------------------------------------
f_left_curr = absindices(timest, :f, 1:(N-1), 2:n_t)
f_right_curr = absindices(timest, :f, 2:N, 2:n_t)
f_dx_left_curr = absindices(timest, :f_dx, 1:(N-1), 2:n_t)
f_dx_right_curr = absindices(timest, :f_dx, 2:N, 2:n_t)
df_int_dt_curr = absindices(timest, :df_int_dt, 1:(N-1), 2:n_t)
f_left_prev = absindices(timest, :f, 1:(N-1), 1:(n_t-1))
f_right_prev = absindices(timest, :f, 2:N, 1:(n_t-1))
f_dx_left_prev = absindices(timest, :f_dx, 1:(N-1), 1:(n_t-1))
f_dx_right_prev = absindices(timest, :f_dx, 2:N, 1:(n_t-1))
function f_fvm_crank_nicolson(x)
dint_dt = x[df_int_dt_curr]
F_L_curr = 0.5 * x[f_left_curr].^2 - ν * x[f_dx_left_curr]
F_R_curr = 0.5 * x[f_right_curr].^2 - ν * x[f_dx_right_curr]
F_L_prev = 0.5 * x[f_left_prev].^2 - ν * x[f_dx_left_prev]
F_R_prev = 0.5 * x[f_right_prev].^2 - ν * x[f_dx_right_prev]
net_flux_avg = 0.5 * ((F_R_curr - F_L_curr) + (F_R_prev - F_L_prev))
return dint_dt + net_flux_avg
end
function f_fvm_euler(x)
dint_dt = x[df_int_dt_curr]
F_left = 0.5 * x[f_left_curr].^2 - ν * x[f_dx_left_curr]
F_right = 0.5 * x[f_right_curr].^2 - ν * x[f_dx_right_curr]
return dint_dt + (F_right - F_left)
end
f_fvm_fn = time_scheme == :crank_nicolson ? f_fvm_crank_nicolson : f_fvm_euler
fvm_model = NonlinearLeastSquaresModel(f_fvm_fn, length(x_joint_bc))
y_fvm = zeros(length(f_left_curr))
lik_fvm = fvm_model(y_fvm; σ=0.0001)
A_constr = zeros(1, length(x_joint_bc))
A_constr[1] = 1.0
e = [mean(x_joint_bc)[1]]
x_constr = ConstrainedGMRF(x_joint_bc, A_constr, e)
# Apply EKF-style initialization if requested
if initialization == :ekf
init_vec = ekf_sweep(mean(x_fvm_0))
Q_joint = precision_matrix(x_constr)
x_init = GMRF(init_vec, Q_joint)
x_constr = ConstrainedGMRF(x_init, A_constr, e)
end
x_solution = gaussian_approximation(x_constr, lik_fvm; verbose=false)
# -------------------------------------------------------------------------
# Extract solution statistics
# -------------------------------------------------------------------------
GC.gc()
mem_after = Base.gc_live_bytes()
peak_memory_mb = max(0.0, (mem_after - mem_before) / 1e6)
wall_time_s = time() - t_start
means_stack = TimeStack(mean(x_solution), full_state_layout)
stds_stack = TimeStack(std(x_solution), full_state_layout)
xs = collect(Float64, endpoints)
ts = [i * Δt for i in 0:(n_t-1)]
mean_matrix = hcat([means_stack[:f, t] for t in 1:n_t]...)
std_matrix = hcat([stds_stack[:f, t] for t in 1:n_t]...)
return SolutionResult(
xs, ts,
mean_matrix, std_matrix,
wall_time_s, peak_memory_mb,
cholesky_nnz, N_state * n_t,
"sparse_fvm", N
)
end
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