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Finite orthonormal polynomial tensor basis
Definition
Let be the Gram–Schmidt polynomials from Positive moment Gram matrices produce orthonormal polynomial families, and let be a finite subset of the admissible degree box in Finite product-law polynomial-chaos input data. For , define Here uses the full-mass restriction to specified in Finite product-law polynomial-chaos input data, which the product rectangle formula identifies with the product of the restricted marginal laws. The finite polynomial-chaos space is inside . Whenever (in particular, for the stated total-degree and hyperbolic-cross sets), the family contains the constant mode . No density of in is part of this definition.
Depends on
Used by
- Christoffel-weighted least-squares polynomial collocation Algorithm
- Smolyak mixed-regularity bounds control chaos coefficient error Corollary
- Tensor Gaussian rules recover polynomial-chaos coefficients exactly Corollary
- Finite polynomial-chaos orthogonal projection Definition
- Quadrature estimates of polynomial-chaos coefficients Definition
- A two-state uncertain reaction rate has an exact degree-one chaos expansion Example
- Polynomial-chaos coefficients group by variable support Example
- Finite chaos projection, Parseval identity, mean, and variance Theorem
- Independent polynomial-chaos tensor products are orthonormal Theorem
- Stochastic Galerkin coercivity and Cea quasi-optimality Theorem
- Tensor products are dense in Bochner L2 under scalar and spatial density Theorem
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Dongbin Xiu and George Em Karniadakis, The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations, §4 (standard reference, not scraped)
- Bruno Sudret, Polynomial Chaos Expansions and Stochastic Finite Element Methods, §§3.2.1–3.2.2 (standard reference, not scraped)