Adaptivity and error control

estimate(
    ::Type{ExtendableASGFEM.AbstractModelProblem},
    sol::SGFEVector,
    C::AbstractStochasticCoefficient;
    kwargs...
) -> Tuple{DataType, DataType, Vector{Vector{Int64}}}

Compute the residual-based a posteriori error estimator for a stochastic Galerkin solution vector sol.

Arguments

• ::Type{AbstractModelProblem}: The model problem type for which the estimator is called. Used for dispatching to the appropriate estimator implementation.
• sol::SGFEVector: The current stochastic Galerkin solution vector.
• C::AbstractStochasticCoefficient: The stochastic coefficient (random field or parameterization).
• rhs: (Optional) Right-hand side function for the PDE (default: nothing).
• bonus_quadorder: (Optional) Additional quadrature order for integration (default: 1).
• tail_extension: (Optional) Number of additional boundary modes to include in the multi-index set (default: 5).
• kwargs...: Additional keyword arguments passed to the estimator.

Returns

A tuple containing:

• eta4modes::Vector{Float64}: Total error estimator for each multi-index (stochastic mode), corresponding to the enriched set of multi-indices (with current active modes first).
• eta4cell::Matrix{Float64}: Error estimator for each cell in the spatial grid (for spatial refinement), for each multi-index.
• multi_indices_extended: The enriched set of multi-indices used in the computation (including boundary extensions).

Description

This function computes a residual-based a posteriori error estimator for the current SGFEM solution. It supports both spatial and stochastic adaptivity by providing error indicators for each cell and each stochastic mode. The estimator is tailored to the model problem and the stochastic coefficient, and can be extended to include additional boundary modes for improved reliability.

If no specialized estimator is available for the given model problem type, an error is raised and empty arrays are returned. ```

add_boundary_modes(
    multi_indices;
    p_extension,
    tail_extension
) -> Any

Adds new stochastic modes (multi-indices) to the current set by extending in several ways:

• For each existing mode, all neighboring modes are added (i.e., all possible copies of that mode where one dimension is increased by 1).
• p_extension ensures that the polynomial degree in the first dimension is increased by this amount (adds new modes with higher degree in the first component).
• tail_extension is a two-element vector:
  • tail_extension[1]: Adds this many new modes of order 1 (i.e., with a single nonzero entry).
  • tail_extension[2]: For each existing mode, also activates or increases the next tail_extension[2] higher stochastic modes in each dimension.

Returns the extended set of multi-indices.

Arguments

• multi_indices: The current set of multi-indices (vector of integer vectors).
• p_extension: Number of additional degrees to add in the first dimension (default: 1).
• tail_extension: Two-element vector controlling the number and type of new modes to add (default: [10, 2]).

Returns

A new vector containing the extended set of multi-indices.

classify_modes(multi_indices) -> NTuple{5, Vector{Int64}}
classify_modes(
    multi_indices,
    active_modes
) -> NTuple{5, Vector{Int64}}

Classifies all multi-indices into the following categories based on their neighborhood relations:

• active_int: Active mode where all direct neighbors (obtained by increasing any active or next dimension by 1) are also in active_modes.
• active_bnd: Active mode where not all direct neighbors are in active_modes (i.e., it lies on the active boundary).
• inactive_bnd: Inactive mode that has at least one active neighbor (first layer of inactive modes).
• inactive_bnd2: Inactive mode that has a neighbor in inactive_bnd (second layer of inactive modes).
• inactive_else: All other inactive modes.

Arguments

• multi_indices: The set of all multi-indices to classify (vector of integer vectors).
• active_modes: The set of currently active modes (default: all multi_indices).

Returns

A tuple of five vectors containing the indices of the multi-indices in the following order: (inactive_else, inactive_bnd, inactive_bnd2, active_bnd, active_int).

generate_multiindices(M, deg) -> Any

generates multi-indices for M dimensions up to degree deg

prepare_multi_indices!(multi_indices; minimal_length)

ensures that all multi-indices have the same length

calculate_sampling_error(
    SolutionSGFEM::SGFEVector,
    C::AbstractStochasticCoefficient;
    problem,
    bonus_quadorder_a,
    bonus_quadorder_f,
    order,
    rhs,
    dim,
    Msamples,
    parallel_sampling,
    dimensionwise_error,
    energy_norm,
    debug,
    nsamples
) -> NTuple{4, Vector{Float64}}

Estimates the error for the given model problem by Monte Carlo sampling. For each sample, a deterministic finite element solution is computed for a fixed (sampled) coefficient with polynomial order order and compared to the provided stochastic Galerkin solution SolutionSGFEM.

Returns

A tuple of arrays (each of length M+1, where M is the maximum order of the multi-indices):

• totalerrorL2stress_weighted: Mean L2 error of the stress, weighted by the distribution.
• totalerrorL2u_weighted: Mean L2 error of the solution, weighted by the distribution.
• totalerrorL2stress_uniform: Mean L2 error of the stress, weighted uniformly.
• totalerrorL2u_uniform: Mean L2 error of the solution, weighted uniformly.

The m-th component of these arrays gives the error when only multi-indices up to order m are included.

Arguments

• SolutionSGFEM: The stochastic Galerkin solution (SGFEVector).
• C: The stochastic coefficient.
• problem: The model problem type (default: LogTransformedPoissonProblemPrimal).
• bonus_quadorder_a: Additional quadrature order for the coefficient (default: 10).
• bonus_quadorder_f: Additional quadrature order for the right-hand side (default: 0).
• order: Polynomial order for the deterministic reference solution (default: 2).
• rhs: Right-hand side function (optional).
• dim: Spatial dimension (default: inferred from the solution).
• Msamples: Number of random variables to sample (default: maxm(C)).
• parallel_sampling: Whether to use parallel sampling (default: true).
• dimensionwise_error: If true, computes errors dimensionwise (default: false).
• energy_norm: If true, uses the energy norm for stress error (default: true).
• debug: If true, enables debug output and plotting (default: false).
• nsamples: Number of Monte Carlo samples (default: 100).

Details

This function computes deterministic reference solutions for each sample, then compares them to the SGFEM solution to estimate the error. Both weighted and unweighted (uniform) averages are returned for stress and solution errors.