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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.
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@@ -12,31 +12,26 @@ of the stochastic Galerkin (SG) methods implemented in this repository.
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## Where to find documentation/implementations of the key blocks
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1. Parametric model / KL representation of the random coefficient.
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- See also:
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-[Stochastic coefficients](coefficients.md) — available random coefficients
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- and polynomial chaos expansions.
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- See:
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-[Stochastic coefficients](coefficients.md) — available random coefficients and polynomial chaos expansions.
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-[Model problems](modelproblems.md) — available model problems that involve random coefficients. See also source: `src/modelproblems/`.
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2. Choose stochastic basis, orthogonal polynomials, suitable for the parameter space
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- See also:
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- See:
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-[Orthogonal polynomials and recurrence relations](orthogonal_polynomials.md) — Legendre / Hermite polynomials and recurrence coefficients.
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-[ONBasis (one-dimensional orthonormal basis)](onbasis.md) — construction and utilities for evaluating 1D orthonormal polynomial bases (norms, quadrature, evaluations).
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-[Tensorized / multivariate basis (TensorizedBasis)](tonbasis.md) — assembling multivariate bases from ONBasis instances and precomputing triple products.
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3. Build spatial FE spaces and blocks of system matrix.
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1. Build spatial FE spaces and blocks of system matrix.
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- Use FESpace types (H1, HDIV, ...) from the ExtendableFEM ecosystem.
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- See: [ExtendableFEMBase.jl](https://github.com/WIAS-PDELib/ExtendableFEMBase.jl) — basis finite-element spaces, basis evaluations and low-level FE utilities like standard interpolations.
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- See: [ExtendableFEM.jl](https://github.com/WIAS-PDELib/ExtendableFEM.jl) — high-level deterministic operator assembly and helpers used throughout the codebase.
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4. Assemble and solve the full SG system
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2. Assemble and solve the full SG system
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- Galerkin projection yields a coupled deterministic block system. Briefly:
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- Expand u(y,x)=∑_μ u_μ(x) H_μ(y), test with H_ν ⇒ block matrix with entries a_{μ,ν} A(·).
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- Solver choices:
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- direct assembly + dense solver (for debugging / small problems)
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