Dimension?Adaptive Tensor?Product Quadrature

Gerstner, Thomas; Griebel, Michael

doi:10.1007/s00607-003-0015-5

Cited by 493 publications

(508 citation statements)

References 39 publications

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“…The material in this section is based on previous work by many authors, including [26,28,6,3,46,48,66].…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

“…Our goal is to construct admissible index sets adaptively based on the progress of the optimization algorithm with the goal to keep the number of collocation points as small as possible while achieving a desired gradient tolerance for the optimization. One ingredient of our approach is the dimension-adaptive approach of [26], which we present next. Downloaded 01/21/16 to 128.…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

“…Adaptive selection of index sets. We will determine the index set in (3.9) adaptively using the dimension-adaptive approach presented in [26].…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

“…The algorithm proposed in [26] can be applied to vector-valued functions in f ∈ C 0 ρ (Γ; Z) if we use a function ℘ : Z → [0, ∞) to quantify the size of the contribution…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

“…(We will use the norm ℘( · ) = · Z .) The dimension-adaptive sparse-grid algorithm [26] for functions in C 0 ρ (Γ, Z) is adapted in Algorithm 3.2.…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

See 4 more Smart Citations

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

109

116

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Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

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“…The material in this section is based on previous work by many authors, including [26,28,6,3,46,48,66].…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

“…Adaptive selection of index sets. We will determine the index set in (3.9) adaptively using the dimension-adaptive approach presented in [26].…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

“…The algorithm proposed in [26] can be applied to vector-valued functions in f ∈ C 0 ρ (Γ; Z) if we use a function ℘ : Z → [0, ∞) to quantify the size of the contribution…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

“…(We will use the norm ℘( · ) = · Z .) The dimension-adaptive sparse-grid algorithm [26] for functions in C 0 ρ (Γ, Z) is adapted in Algorithm 3.2.…”

Section: Choice Of Collocation Points: Generalized Sparse Gridsmentioning

confidence: 99%

See 3 more Smart Citations

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

109

116

View full text Add to dashboard Cite

show abstract

Efficient Bayesian experimental design for contaminant source identification

et al. 2015

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In this study, an efficient full Bayesian approach is developed for the optimal sampling well location design and source parameters identification of groundwater contaminants. An information measure, i.e., the relative entropy, is employed to quantify the information gain from concentration measurements in identifying unknown parameters. In this approach, the sampling locations that give the maximum expected relative entropy are selected as the optimal design. After the sampling locations are determined, a Bayesian approach based on Markov Chain Monte Carlo (MCMC) is used to estimate unknown parameters. In both the design and estimation, the contaminant transport equation is required to be solved many times to evaluate the likelihood. To reduce the computational burden, an interpolation method based on the adaptive sparse grid is utilized to construct a surrogate for the contaminant transport equation. The approximated likelihood can be evaluated directly from the surrogate, which greatly accelerates the design and estimation process. The accuracy and efficiency of our approach are demonstrated through numerical case studies. It is shown that the methods can be used to assist in both single sampling location and monitoring network design for contaminant source identifications in groundwater.

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An overview of uncertainty quantification techniques with application to oceanic and oil‐spill simulations

Iskandarani

Wang

Srinivasan³

et al. 2016

JGR Oceans

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We give an overview of four different ensemble‐based techniques for uncertainty quantification and illustrate their application in the context of oil plume simulations. These techniques share the common paradigm of constructing a model proxy that efficiently captures the functional dependence of the model output on uncertain model inputs. This proxy is then used to explore the space of uncertain inputs using a large number of samples, so that reliable estimates of the model's output statistics can be calculated. Three of these techniques use polynomial chaos (PC) expansions to construct the model proxy, but they differ in their approach to determining the expansions' coefficients; the fourth technique uses Gaussian Process Regression (GPR). An integral plume model for simulating the Deepwater Horizon oil‐gas blowout provides examples for illustrating the different techniques. A Monte Carlo ensemble of 50,000 model simulations is used for gauging the performance of the different proxies. The examples illustrate how regression‐based techniques can outperform projection‐based techniques when the model output is noisy. They also demonstrate that robust uncertainty analysis can be performed at a fraction of the cost of the Monte Carlo calculation.

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Dimension?Adaptive Tensor?Product Quadrature

Cited by 493 publications

References 39 publications

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Efficient Bayesian experimental design for contaminant source identification

An overview of uncertainty quantification techniques with application to oceanic and oil‐spill simulations

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