GitHub

Overview

This page gives a concise overview of the main building blocks of the stochastic Galerkin (SG) methods implemented in this repository.

Key blocks

  • Represent uncertain coefficients/inputs by a finite parametric expansion (e.g. Karhunen–Loève / KL expansion).
  • Expand the solution unknowns in a tensor product of a spatial finite element basis and a stochastic polynomial basis (Galerkin ansatz).
  • Galerkin project onto the combined basis to obtain a coupled deterministic system for the expansion coefficients.
  • Solve the system with solvers that exploit the tensor/block structure.
  • Adaptivity in space and in the stochastic index set to reduce costs.

Example: Parametric Poisson Problem

To illustrate the workflow, consider a Poisson problem with parametric diffusion coefficient:

\[-\mathrm{div}(a(y,x) \nabla u(y,x)) = f(x) \quad \text{for } (y,x) \in \Gamma \times D\]

where the coefficient has the Karhunen-Loève expansion:

\[a(y,x) = a_0(x) + \sum_{m=1}^M \sqrt{\lambda_m} \phi_m(x) y_m\]

with mean field $a_0$, eigenpairs $(\lambda_m,\phi_m)$ and parameters $y_m \in [-1,1]$.

Stochastic Galerkin Formulation

  1. Expand the solution in tensorized basis functions:

    \[u(y,x) = \sum_{\mu} u_\mu(x) H_\mu(y)\]

    where $H_\mu(y) = \prod_{m=1}^M H_{\mu_m}(y_m)$ are multivariate Legendre polynomials.

  2. Galerkin projection yields the weak form:

    \[\sum_{\mu} \int_\Gamma a(y,x) \nabla u_\mu(x) \cdot \nabla v(x) H_\mu(y) H_\nu(y)\,dy = \int_\Gamma f(x)v(x) H_\nu(y)\,dy\]

    for all test functions $v(x)$ and indices $\nu$.

  3. This results in a coupled block system $\mathbf{A}\mathbf{u} = \mathbf{b}$ where:

    • Each block $\mathbf{A}_{\mu,\nu}$ involves the mean and KL terms of $a(y,x)$
    • The tensor structure allows efficient matrix-free operations

Where to find documentation/implementations of the key blocks

  1. Parametric model / KL representation of the random coefficient.
    • See:
      • Stochastic coefficients — available random coefficients and polynomial chaos expansions.
      • Model problems — available model problems that involve random coefficients. See also source: src/modelproblems/.
  2. Choose stochastic basis, orthogonal polynomials, suitable for the parameter space
  3. Build spatial FE spaces and blocks of system matrix.
    • Use FESpace types (H1, HDIV, ...) from the ExtendableFEM ecosystem.
    • See: ExtendableFEMBase.jl — basis finite-element spaces, basis evaluations and low-level FE utilities like standard interpolations.
    • See: ExtendableFEM.jl — high-level deterministic operator assembly and helpers used throughout the codebase.
  4. Assemble and solve the full SG system
    • Galerkin projection yields a coupled deterministic block system. Briefly:
      • Expand u(y,x)=∑μ uμ(x) Hμ(y), test with Hν ⇒ block matrix with entries a_{μ,ν} A(·).
    • Solver choices:
      • direct assembly + dense solver (for debugging / small problems)
      • matrix‑free iterative solvers exploiting tensor/block structure + preconditioners
    • See problem solver implementations: src/modelproblems/solvers_*.jl.
  5. Postprocessing, error estimation & adaptivity
    • Error estimators and marking criteria implemented in the script driver scripts/poisson.jl for the available Poisson model problems.
    • Error estimators are problem-dependent and can be currently found in src/estimate.jl
    • Spatial (mesh refinement) uses refinement routines from ExtendableFEMBase.jl
    • Stochastic refinement (enrich multi‑index set) uses functions from src/mopcontrol.jl
    • see documentation page on Estimators for some more details
    • Results, parameters and reproducible outputs are stored with DrWatson (see scripts/poisson.jl for naming pattern).