Stochastic Galerkin for ODESystem and PDESystem
#38
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…st generate a dict in the beginnign?)
…R to PolyChaos since that is really awkward
…lerkin projection
bowenszhu
changed the title
PDESystemODESystem and PDESystem
| # X => x₀ + x₁ψ₁(X) | ||
| # where x₀ and x₁ are affince PCE coefficients of X | ||
| # ψ₁(x) = x - α₀ is the first-order monic basis polynomial of X | ||
| # TODO: Use mean and std of the distribution of X to compute x₀ and x₁ | ||
| x_dict = Dict(x => op.α[1] + Ψ[i + 1] | ||
| for (i, (x, op)) in enumerate(zip(parameters, uni_basis))) |
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Create a struct for otho poly basis SemiMonomial in Symbolics.semipolynomial_form).
Evaluate tensor scalar products conveniently by defining arithmetic operations for this struct.
Use term rewriter to deal with differential operators @rule ~D(~C * ~Ψ) => ~Ψ * ~D(~C) especially for PDESystem.
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I like this idea A LOT! That will make the projection code quite a bit less cumbersome to read/write!
That said, I am not sure I like the notation/nomenclature as is. Because "inner product * inner product" is not a mathematically well-defined notion.
I think it may be a good idea to rename "Inner" to something like "BasisMonomial" or so and represent the PCE ansatz in those terms. That way we can save ourselves a lot of book-keeping and rely simply on a few arithmetic rules. Let's chat about this in more detail soon!
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Great. Thanks for the quick feedback! @FHoltorf |
supersede #27
resolve #8
resolve SciML/PolyChaos.jl#8
preview of documentation https://bowenszhu.github.io/ModelOrderReduction.jl/previews/PR1/