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By Difillippo, F.C.; USDOE Office of Fissile Materials Disposition (MD) (US); Oak Ridge National Laboratory.; United States. Dept. of Energy. Office of Scientific and Technical Information

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Additional info for Analysis of Rod Removal Transient Experiments in VVER Reactors at Zero Power

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4) However, C is not derived from a Mercer kernel [106], so C is not a kernel matrix in the sense of kernel-based machine learning. Since C is symmetric, it has a real eigenvalue decomposition, C = W ΛW T , Λ = diag(λ1 , . . 5) where W is the m × m orthogonal matrix whose columns {w1 , . . , w m } are the normalized eigenvectors of C. The following lemma quantifies the relationship between the gradient of f and the eigendecomposition of C. 1. The mean-squared directional derivative of f with respect to the eigenvector wi is equal to the corresponding eigenvalue, 2 (∇x f )T wi ρ d x = λi , i = 1, .

This brings the interpretation that x is a random vector, where the randomness models imprecisely specified input parameters. We also 21 22 Chapter 3. Discover the Active Subspace assume that ρ is bounded to avoid integrating unbounded functions. The mathematical quantities that define the active subspace are valid for any ρ that satisfies these conditions. However, the random sampling methods we use to discover the active subspace require something much stricter, namely, that we can draw independent samples from ρ.

But these sets are relatively small. Two clarifying remarks are in order to address potential confusion. First, the active subspace is not a subset of the input space. We are not necessarily restricting our attention to input parameters in the range of the eigenvectors W1 . That would greatly limit our study of the relationship between inputs and outputs. We explore the interaction between the domain and the active subspace in Chapter 4. Second, we are not particularly interested in a low-rank approximation, C ≈ W1 Λ1 W1T .

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Analysis of Rod Removal Transient Experiments in VVER Reactors at Zero Power by Difillippo, F.C.; USDOE Office of Fissile Materials Disposition (MD) (US); Oak Ridge National Laboratory.; United States. Dept. of Energy. Office of Scientific and Technical Information


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