Thanks! This is very helpful. I would have had no idea how to construct a basis and use it.
If I compare the performance with EvalMultiPoly.jl using only orders up to 3, EvalMultiPoly.jl is significantly faster (81µs for order 3, compared to 2.8 ms for SmolyakParameters(2, 2)) but then this is not a fair comparison. However, with order 4 or more EvalMultiPoly.jl starts to allocate and then the performance get really bad.
If you can add a TensorBasis that would allow to create simple mutivariate polynomials similar to valMultiPoly.jl, this may be a nice addition!
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… another thing is that linear_combination leads to a different way that the order of the multivariate polynomial is limited than one would typically limit the order in a multivariate expansion (e.g. yx^2 not being part of the second order multivariate polynomial, but x^2 being part of it). Any idea how to handle this?