Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces

Constantine, Paul G.; Dow, Eric; Wang, Qiqi

doi:10.1137/130916138

Cited by 464 publications

(452 citation statements)

References 34 publications

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“…The problem is discretized using a finite element method on a triangulation mesh; then f and ∇ x f can be computed as a forward and adjoint problem; see [6] for more details. We choose m = 100 and examine two cases of the correlation lengths β = 1 and β = 0.01.…”

Section: Studymentioning

confidence: 99%

“…When a simulator is able to produce derivatives of the output with respect to the input parameters, such information can help predict the responses at new design points [28]. More recently, Constantine, Dow, and Wang [6] proposed finding rotations of the input space with the strongest variability in the gradients of the simulators and constructed a response surface on such a low-dimensional active subspace. This active subspace (AS) approach has been demonstrated to be effective theoretically and numerically.…”

mentioning

confidence: 99%

“…For example, it is impossible for now to produce the gradients of interest in the tsunami code VOLNA, a nonlinear shallow water equation solver using the finite volume method [12] which we consider in Study 2 (see section 5.2). The formalism for finite volumes is more difficult in terms of writing down the solution explicitly than the finite element method that is used for the PDE problem in [6]. Besides, such nonlinear hyperbolic systems are much more difficult to solve than linear hyperbolic systems and elliptic systems; and the computation of derivatives is problematic in nonlinear hyperbolic systems due to nondifferentiability issues in numerical fluxes with limiters, as well as the presence of shocks [34].…”

mentioning

confidence: 99%

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Dimension Reduction for Gaussian Process Emulation: An Application to the Influence of Bathymetry on Tsunami Heights

Liu¹,

Guillas²

2017

SIAM/ASA J. Uncertainty Quantification

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Abstract. High accuracy complex computer models, also called simulators, require large resources in time and memory to produce realistic results. Statistical emulators are computationally cheap approximations of such simulators. They can be built to replace simulators for various purposes, such as the propagation of uncertainties from inputs to outputs or the calibration of some internal parameters against observations. However, when the input space is of high dimension, the construction of an emulator can become prohibitively expensive. In this paper, we introduce a joint framework merging emulation with dimension reduction in order to overcome this hurdle. The gradient-based kernel dimension reduction technique is chosen due to its ability to drastically decrease dimensionality with little loss in information. The Gaussian process emulation technique is combined with this dimension reduction approach. Theoretical properties of the approximation are explored. Our proposed approach provides an answer to the dimension reduction issue in emulation for a wide range of simulation problems that cannot be tackled using existing methods. The efficiency and accuracy of the proposed framework is demonstrated theoretically and compared with other methods on an elliptic partial differential equation (PDE) problem. We finally present a realistic application to tsunami modeling. The uncertainties in the bathymetry (seafloor elevation) are modeled as high-dimensional realizations of a spatial process using a geostatistical approach. Our dimension-reduced emulation enables us to compute the impact of these uncertainties on resulting possible tsunami wave heights near-shore and on-shore. Considering an uncertain earthquake source, we observe a significant increase in the spread of uncertainties in the tsunami heights due to the contribution of the bathymetry uncertainties to the overall uncertainty budget. These results highlight the need to include the effect of uncertainties in the bathymetry in tsunami early warnings and risk assessments.

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Section: Studymentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Dimension Reduction for Gaussian Process Emulation: An Application to the Influence of Bathymetry on Tsunami Heights

Liu¹,

Guillas²

2017

SIAM/ASA J. Uncertainty Quantification

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“…Perturbing y changes f more, on average, than perturbing z; see Constantine et al 6 If the eigenvalues λ n+1 , . .…”

Section: Active Subspacesmentioning

confidence: 99%

Active Subspaces of Airfoil Shape Parameterizations

Grey

Constantine

2017

58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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Design and optimization benefit from understanding the dependence of a quantity of interest (e.g., a design objective or constraint function) on the design variables. A low-dimensional active subspace, when present, identifies important directions in the space of design variables; perturbing a design along the active subspace associated with a particular quantity of interest changes that quantity more, on average, than perturbing the design orthogonally to the active subspace. This low-dimensional structure provides insights that characterize the dependence of quantities of interest on design variables. Airfoil design in a transonic flow field with a parameterized geometry is a popular test problem for design methodologies. We examine two particular airfoil shape parameterizations, PARSEC and CST, and study the active subspaces present in two common design quantities of interest, transonic lift and drag coefficients, under each shape parameterization. We mathematically relate the two parameterizations with a common polynomial series. The active subspaces enable low-dimensional approximations of lift and drag that relate to physical airfoil properties.In particular, we obtain and interpret a two-dimensional approximation of both transonic lift and drag, and we show how these approximation inform a multi-objective design problem.

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“…The proofs of Theorems 3.2 and 3.3 in Constantine, Dow, and Wang [1] contain minor errors that affect the statements of the theorems. These errors propagate to the statements of Theorems 3.6 and 3.7.…”

mentioning

confidence: 99%

Erratum: Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces

Constantine¹,

Dow²,

Wang³

2014

SIAM J. Sci. Comput.

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Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces

Cited by 464 publications

References 34 publications

Dimension Reduction for Gaussian Process Emulation: An Application to the Influence of Bathymetry on Tsunami Heights

Dimension Reduction for Gaussian Process Emulation: An Application to the Influence of Bathymetry on Tsunami Heights

Active Subspaces of Airfoil Shape Parameterizations

Erratum: Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces

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