Adaptivity and error control

Residual-based a posteriori error estimation

Spatial error estimation refers to classical residual-based error estimation for the zero-th multi index that refers to the mean value. Stochastic error control has to estimate which stochastic mode needs to be refined in the sense that either the polynomial degree is increased or neighbouring stochastic modes are activated. Both is represented by the multi-indices. Then unified error control allows to perform residual-based error estimation for the subresiduals that are associated to each multi-index and depend on the model problem (see references on the main page for details).

ExtendableASGFEM.estimate — Method

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

computes the residual-based a posteriori error estimator for the current solution vector sol for the given model problem and stochastic coefficient C and returns:

eta4modes = array with total error estimators for each multi-index (corresponding to the enriched set of multi-indices with the current active modes coming first)
eta4cell = array with total error estimators for each cell in grid (for spatial refinement)
multi_indices_extended = enriched set of multi-indices used for the computations

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Multi index management

Depending on the model problem and stochastic coefficient the amount of multi indices that should be added to the error estimation varies. Here are some methods that help with enriching the set of multi-indices.

ExtendableASGFEM.add_boundary_modes — Method

add_boundary_modes(
    multi_indices;
    p_extension,
    tail_extension
) -> Any

adds new stochastic modes:

for each existing mode all neighbouring modes are added (= all possible copies of that mode where one dimension is increased by 1)
p_extension additionally ensures that the polynomial degree of the first dimension is increased by this amount
tail_extension activates tail_expansion[1] many new stochastic modes with order 1, for each existing mode also the next tail_expansion[2] many higher stochastic modes are activated or increased

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ExtendableASGFEM.classify_modes — Function

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

classifies all multi indices into these categories:

active_int = mode and all its direct neighbours (+1 in all active and the next dimensions) are in active_modes
active_bnd = mode but not all its direct neighbours are in active_modes
inactive_bnd = inactive mode that has an active neighbour
inactive_bnd2 = inactive mode that has an neighbour in inactive_bnd
inactive_else = all other

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ExtendableASGFEM.generate_multiindices — Method

generate_multiindices(M, deg) -> Any

generates multi-indices for M dimensions up to degree deg

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ExtendableASGFEM.prepare_multi_indices! — Method

prepare_multi_indices!(multi_indices; minimal_length)

ensures that all multi-indices have the same length

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Monte carlo sampling estimator

There is also a hierarchical Monte carlo error estimator available that compares the solution with a higher order discrete solution for sampled deterministic problems. This is merely intended as a way to compute the reference error to assess the efficiency of the residual-based error estimator.

ExtendableASGFEM.calculate_sampling_error — Method

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 model problem problem by Monte carlo sampling: for each sample a discrete finite element solution of the deterministic model problem with fixed (sampled) coefficient with polynomial order order is computed and compared to the given stochastic Galerkin solution SolutionSGFEM. Return values (each arrays of length M+1 where M is the length of the multi-indices) are:

totalerrorL2stress_weighted : mean L2 error of the stress with samples weighted by the distribution
totalerrorL2u_weighted : mean L2 error with samples weighted by the distribution
totalerrorL2stress_uniform : mean L2 error of the stress with samples weighted uniformly
totalerrorL2u_uniform : mean L2 error with samples weighted uniformly

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

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