Read e-book online Active Subspaces: Emerging Ideas for Dimension Reduction in PDF

By Paul G. Constantine

ISBN-10: 1611973856

ISBN-13: 9781611973853

ISBN-10: 1611973864

ISBN-13: 9781611973860

Scientists and engineers use desktop simulations to check relationships among a model's enter parameters and its outputs. even though, thorough parameter stories are demanding, if now not most unlikely, whilst the simulation is pricey and the version has a number of inputs. To allow experiences in those situations, the engineer may perhaps try and decrease the size of the model's enter parameter house. lively subspaces are an rising set of measurement aid instruments that determine vital instructions within the parameter area. This e-book describes thoughts for locating a model's energetic subspace and proposes tools for exploiting the diminished size to allow another way infeasible parameter reviews. Readers will locate new rules for size relief, easy-to-implement algorithms, and several other examples of energetic subspaces in action.

Parameter reviews are in every single place in computational technological know-how. advanced engineering simulations needs to run a number of instances with various inputs to successfully research 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 reviews are more challenging while every one simulation is dear and the variety of parameters is huge. In perform, the engineer could try and restrict a examine to an important parameters, which successfully reduces the measurement of the parameter examine.

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

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Let ⊆ n be the set = y∈ n , y = W1T x, x ∈ . 6) This set is the domain of g (y). Any point y ∈ is guaranteed to have at least one x ∈ such that y = W1T x. Then we have at least one choice for g (y): given y ∈ , find x ∈ such that y = W1T x, and set g (y) = f (x). , the map from y to x is ill-posed. 1. Dimension reduction and mappings 47 a particular f (x). How does one choose to set g (y) among infinitely many f (x)’s? To overcome the ill-posedness, we must choose a regularization. The choice may not matter.

Common interpolation procedures in several variables use radial basis functions [124] or polynomials [56]. Response surfaces that do not interpolate are regressions, and they assume noise or error in the qi ’s [65]. These approximation procedures suffer from the curse of dimensionality, so they may benefit tremendously from proper dimension reduction. The straightforward way to exploit the active subspace for response surfaces is to build a regression surface on the n active variables y instead of all m variables x as in the following algorithm.

This theorem is restated below; we apply it with X j = ∇x f j ∇x f jT −C. In other words, the random matrix samples are the deviance of the j th sampled gradient outer product from the true matrix C. 6 (matrix Bernstein: bounded case [121, Thm. 1]). Consider a finite sequence {X j } of independent, random, symmetric matrices with dimension m. Assume that Xj = 0 λmax (X j ) ≤ R almost surely. and Compute the norm of the total variance, σ 2 := X 2j . 7. Assume ∇x f ≤ L for all x ∈ ˆ −C ≥ C C t ≤ σ 2 /R, t ≥ σ 2 /R.

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

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