2. Overview
Uncertainty Quantification(UQ)
The process of determining & analyzing outcomes of input uncertainties
Reducing uncertainties in both computational and real world application
How to Approach
Stochastic Polynomial Chaos Expansion(Orthogonal polynomial basis)
Galerkin method(ex: Lorenz 96)
Monte Carlo method
3. Hilbert Space
Salient point: Orthogonality
Define 𝜑𝜑𝑛𝑛 𝑥𝑥 ∈ 𝐿𝐿2
𝑀𝑀, 𝑝𝑝 as a polynomial of degree n and 𝜑𝜑𝑛𝑛 𝑥𝑥
are orthogonal with respect to p(x) if
⟨ 𝜑𝜑𝑛𝑛, ⟩𝜑𝜑𝑚𝑚 𝑝𝑝 = �
𝑀𝑀
𝜑𝜑𝑛𝑛 𝜑𝜑𝑚𝑚 𝑝𝑝(𝑥𝑥)𝑑𝑑𝑑𝑑 = 𝜑𝜑𝑛𝑛 𝑝𝑝
2
𝛿𝛿𝑛𝑛𝑛𝑛
where
𝛿𝛿𝑛𝑛𝑛𝑛 is the Kronecker delta function
4. Polynomial Chaos Expansion(PCE)
Approximating a random distribution employing a linear
combination of corresponding orthogonal polynomials
𝑓𝑓 ≈ 𝑓𝑓𝑁𝑁 𝑥𝑥 = �
𝑘𝑘=0
𝑁𝑁
�𝑓𝑓𝑘𝑘 𝜑𝜑𝑘𝑘 𝑥𝑥
where the coefficients are
�𝑓𝑓𝑘𝑘 =
𝑓𝑓, 𝜑𝜑𝑘𝑘 𝑝𝑝
𝜑𝜑𝑘𝑘, 𝜑𝜑𝑘𝑘 𝑝𝑝
8. Galerkin method
Approximating by specifying the coefficients �𝑢𝑢𝛼𝛼 𝑥𝑥, 𝑡𝑡 such that
the error of the approximation is orthogonal to the basis function
𝜙𝜙𝛼𝛼 𝑌𝑌
𝑢𝑢(𝑥𝑥, 𝑡𝑡, 𝑍𝑍) ≈ 𝑢𝑢𝑁𝑁(𝑥𝑥, 𝑡𝑡, 𝑍𝑍) = �
𝛼𝛼∈𝒥𝒥 𝑁𝑁
�𝑢𝑢𝛼𝛼(𝑥𝑥, 𝑡𝑡)𝜙𝜙𝛼𝛼 𝑌𝑌
where the multi-index is defined as
𝒥𝒥𝑁𝑁 = {𝛼𝛼 = (𝛼𝛼𝑖𝑖, 𝑖𝑖 ≥ 1|𝛼𝛼𝑖𝑖 ∈ ℤ+, 𝛼𝛼 = � 𝛼𝛼𝑖𝑖 ≤ 𝑁𝑁)}
and 𝜙𝜙𝛼𝛼 𝑌𝑌 forms an orthonormal basis of ℒ2(ℳ 𝑑𝑑, 𝑝𝑝𝑋𝑋)
9. Lorenz 96 model
Q) Approximate this model with one random parameter
̇𝑢𝑢𝑖𝑖 = 𝑢𝑢𝑖𝑖−1 𝑢𝑢𝑖𝑖+1 − 𝑢𝑢𝑖𝑖−2 − 𝑢𝑢𝑖𝑖 + 𝑍𝑍1
where 𝑢𝑢𝑖𝑖 is periodic, that is, 𝑢𝑢𝑖𝑖+𝑁𝑁 = 𝑢𝑢𝑖𝑖 and N is set to be 40, and
𝑍𝑍1~Uniform[6,16]
