## Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter StudiesScientists and engineers use computer simulations to study relationships between a model's input parameters and its outputs. However, thorough parameter studies are challenging, if not impossible, when the simulation is expensive and the model has several inputs. To enable studies in these instances, the engineer may attempt to reduce the dimension of the model's input parameter space. Active subspaces are an emerging set of dimension reduction tools that identify important directions in the parameter space. This book describes techniques for discovering a model's active subspace and proposes methods for exploiting the reduced dimension to enable otherwise infeasible parameter studies. Readers will find new ideas for dimension reduction, easy-to-implement algorithms, and several examples of active subspaces in action. |

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active subspace active variable Algorithm 1.3 analysis approximate gradients assume average bootstrap intervals bootstrap replicates bound Chapter choose coefficients components computed conditional density conditional expectation construct Corollary define the active density function derived dimension reduction Dimension Subspace Dimension directions domain drag eigenvalue decomposition eigenvectors error estimates estimated subspace evaluations exit pressure exploit the active Figure finite difference global linear model gradient gradient samples Hellinger distance high dimensions high-dimensional hypercube HyShot inactive variables Index Index input parameters input space integration inverse kriging kriging surface Lemma lift linear combinations linear model local linear Markov chain matrix methods one-dimensional active subspace optimization orthogonal parameter studies parameterized Pmax Poincaré inequality polynomial prediction quadratic quantity of interest random sampling range reduce the dimension regression surface response surface scramjet sensitivity simulation’s subset Subspace Dimension Subspace sufficient summary plot Theorem 4.3 tion zero zonotope