Download PDF by Paul G. Constantine: Active Subspaces: Emerging Ideas for Dimension Reduction in

By Paul G. Constantine

ISBN-10: 1611973856

ISBN-13: 9781611973853

ISBN-10: 1611973864

ISBN-13: 9781611973860

Scientists and engineers use laptop simulations to check relationships among a model's enter parameters and its outputs. notwithstanding, thorough parameter experiences are not easy, if now not most unlikely, while the simulation is pricey and the version has numerous inputs. To permit reports in those cases, the engineer may possibly try and decrease the size of the model's enter parameter house. energetic subspaces are an rising set of measurement aid instruments that determine very important instructions within the parameter house. This booklet describes concepts for locating a model's energetic subspace and proposes equipment for exploiting the diminished size to let in a different way infeasible parameter reviews. Readers will locate new rules for measurement relief, easy-to-implement algorithms, and a number of other examples of energetic subspaces in action.

Parameter reviews are all over the place in computational technological know-how. advanced engineering simulations needs to run a number of occasions with diversified inputs to successfully learn the relationships among inputs and outputs. reports like optimization, uncertainty quantification, and sensitivity research produce subtle characterizations of the input/output map. yet thorough parameter reports are more challenging while each one simulation is dear and the variety of parameters is big. In perform, the engineer may possibly try and restrict a learn to crucial parameters, which successfully reduces the size of the parameter examine.

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Extra info for Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies

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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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Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies by Paul G. Constantine

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