Stochastic Coefficients

Stochastic coefficients play a central role in uncertainty quantification and stochastic finite element methods. In this package, coefficients are represented using a Karhunen-Loève expansion (KLE), which allows for the efficient representation of random fields with prescribed covariance structure.

Overview

The Karhunen-Loève expansion expresses a stochastic process as a series of orthogonal functions weighted by uncorrelated random variables:

$

a(x, \omega) = a_0(x) + \sum_{n=1}^N \sqrt{\lambda_n} \phi_n(x) \xi_n(\omega)

$

where $a_0(x)$ is the mean, $\lambda_n$ and $\phi_n(x)$ are the eigenvalues and eigenfunctions of the covariance operator, and $\xi_n$ are independent standard normal random variables.

API

ExtendableASGFEM.AbstractStochasticCoefficient — Type

abstract type AbstractStochasticCoefficient{T}

A stochastic coefficient is assumed to have the Karhunen-Loeve expansion form

$a(y,x) = a_0(x) + \sum_{m=1}^\infty y_m a_m(x)$ with (centered independent) random variables $y_m$ and basis functions $a_m(x)$ that need to be specified (together with their gradients) and expectation value $a_0$ (in general they steam from a spectral analysis of the covariance operator of $a$).

ExtendableASGFEM.expa_PCE_mop — Method

expa_PCE_mop(
    TB::TensorizedBasis{T, ONBType},
    SC::AbstractStochasticCoefficient{T};
    N_truncate,
    factor
) -> Union{Tuple{ExtendableASGFEM.var"#closure#48"{_A, TensorizedBasis{T, ONBType, MIType}, ExtendableASGFEM.var"#lambda_mu#47"{T1, N_truncate, factor, SC, mu_fac_storage, multi_indices, eval_amu_storage, eval_am}, Int64} where {_A, T, ONBType, MIType, T1, N_truncate, factor, SC, mu_fac_storage, multi_indices, eval_amu_storage, eval_am}, ExtendableASGFEM.var"#lambda_mu#47"{_A, Int64, Int64, SingleStochasticCoefficient{T, k}} where {_A, T, k}}, Tuple{ExtendableASGFEM.var"#closure#48"{_A, TensorizedBasis{T, ONBType, MIType}, ExtendableASGFEM.var"#lambda_mu#47"{T1, N_truncate, factor, SC, mu_fac_storage, multi_indices, eval_amu_storage, eval_am}, Int64} where {_A, T, ONBType, MIType, T1, N_truncate, factor, SC, mu_fac_storage, multi_indices, eval_amu_storage, eval_am}, ExtendableASGFEM.var"#lambda_mu#47"{_A, Int64, Int64, StochasticCoefficientCosinus{T}} where {_A, T}}}

prepares two functions

lambda_mu!(result, input, qpinfo)
expa!(result, input, qpinfo)

that calculate λ_μ, which are orthogonal decomposition coefficient functions of exp(a) ≈ ∑ λ_μ H_μ w.r.t. to the multi-indices μ and their associated orthogonal basis functions H_μ in TB, as well as (an approximation based on this decomposition of) exp(a).

ExtendableASGFEM.get_a! — Method

get_a!(
    SC::AbstractStochasticCoefficient{T};
    factor
) -> ExtendableASGFEM.var"#closure#34"{Int64, <:AbstractStochasticCoefficient{T}} where T

prepares a function of interface

a!(result, x, y)

that evaluates the coefficient (times a factor) at space coordinates x and random variables y into result (a vector of length 1).

ExtendableASGFEM.get_am_x — Method

get_am_x(
    m,
    SC::AbstractStochasticCoefficient{T}
) -> ExtendableASGFEM.var"#closure#get_am_x##0"{_A, <:AbstractStochasticCoefficient{T}} where {_A, T}

prepares a function of interface

get_am_x!(result, input, qpinfo)

that evaluates a_m(qpinfo.x) input (used to define ExtendableFEM operator kernels).

ExtendableASGFEM.get_expa! — Method

get_expa!(
    SC::AbstractStochasticCoefficient;
    factor
) -> ExtendableASGFEM.var"#closure#38"{Int64, ExtendableASGFEM.var"#closure#34"{Int64, var"#s179", _A}} where {T, var"#s179"<:AbstractStochasticCoefficient{T}, _A}

prepares a function of interface

get_expa!(result, x, y)

that evaluates the exponential of the coefficient (times a factor) at space coordinates x and random variables y into result (a vector of length 1).

ExtendableASGFEM.get_grada! — Method

get_grada!(
    SC::AbstractStochasticCoefficient{T};
    factor
) -> ExtendableASGFEM.var"#closure#36"{Int64, <:AbstractStochasticCoefficient{T}} where T

prepares a function of interface

get_grada!(result, x, y)

that evaluates the spatial gradient of the coefficient (times a factor) at space coordinates x and random variables y into result (a vector of same length as x).

ExtendableASGFEM.get_grada_x_sigma — Method

get_grada_x_sigma(
    dim,
    SC::AbstractStochasticCoefficient{T},
    y
) -> ExtendableASGFEM.var"#closure#get_grada_x_sigma##0"{_A, <:AbstractStochasticCoefficient{T}} where {_A, T}

prepares a function of interface

get_grada_x_sigma!(result, input, qpinfo)

that evaluates ∇a(qpinfo.x) ⋅ σ(qpinfo.x) (used to define ExtendableFEM operator kernels and σ is expected to be some vector-valued quantity of same length).

ExtendableASGFEM.get_gradam_x_u — Method

get_gradam_x_u(
    m,
    SC::AbstractStochasticCoefficient{T}
) -> ExtendableASGFEM.var"#closure#get_gradam_x_u##0"{_A, <:AbstractStochasticCoefficient{T}} where {_A, T}

prepares a function of interface

get_gradam_x_u!(result, input, qpinfo)

that evaluates ∇a_m(qpinfo.x) u(qpinfo.x) (used to define ExtendableFEM operator kernels and input is expected to be some scalar quantity).

ExtendableASGFEM.plot_am — Method

plot_am(
    xgrid::ExtendableGrids.ExtendableGrid,
    m,
    SC::AbstractStochasticCoefficient;
    Plotter,
    kwargs...
)

returns a scalarplot of the m-th coefficient functiona a_m interpolated on the given grid. The Plotter backend can be changed with the Plotter argument (e.g. GLMakie, CairoMakie, PyPlot, Plots).

StochasticCoefficientCosinus

An example of a particular KLE with cosinus type basis function is given by the following subtype.

ExtendableASGFEM.StochasticCoefficientCosinus — Type

struct StochasticCoefficientCosinus{T} <: AbstractStochasticCoefficient{T}

Expansion of the form

a(x,y) = a_0(x) + ∑_m y_m a_m(x)

where the a_m are of cosinus type.

ExtendableASGFEM.StochasticCoefficientCosinus — Method

StochasticCoefficientCosinus(
;
    T,
    τ,
    start,
    decay,
    mean,
    maxm
) -> StochasticCoefficientCosinus{Float64}

constructor for StochasticCoefficientCosinus of type T (default = Float64), where decay (default = 2) steers the decay of the coefficient basis functions (the larger the faster), mean (default = 0) is the mean value of the coefficient, maxm (default = 100) is the maximal number of stochastic random variables, and τ (default = 1) is a uniform scaling factors