Sparse Collocation Methods for Stochastic Interpolation and Quadrature

Gunzburger, Max; Webster, Clayton G.; Zhang, Guannan

doi:10.1007/978-3-319-12385-1_29

Cited by 7 publications

(4 citation statements)

References 60 publications

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“…Recently, the SG interpolation is developed with the aim of using fewer interpolation points to achieve a comparable accuracy with the full tensor interpolation (Barthelmann et al, ; Gunzburger et al ):

{frakturT}^{L, d} ([]|, η) (|, bold-italicθ) ≔ \underset{{L + 1 \leq |ℓ|}_{1} \leq L + d}{false\sum} {{(|, - 1)}^{L - {(|||, bold-scriptℓ)}_{1}} (|, \frac{0pt}{d - 1}) scriptU}^{{N_{scriptℓ}}_{1}} \otimes \dots \otimes {scriptU}^{N_{{scriptℓ}_{d}}} ([]|, η) .

The SG interpolant

{frakturT}^{L, d} ([]|, η) (|, bold-italicθ)

in equation is a weighted sum of a series of tensor‐product interpolants

{scriptU}^{N_{{scriptℓ}_{1}}} \otimes \dots \otimes {scriptU}^{N_{{scriptℓ}_{d}}} ([]|, η)

. Each of these interpolants is similar to the one in equation , but defined on a coarse grid with different resolution levels in different dimensions, i.e .,

{scriptℓ}_{i}

and

N_{{scriptℓ}_{i}}

may differ from each other for

i = 1, \dots, d

.…”

Section: Methodsmentioning

confidence: 99%

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Calibration of the E3SM Land Model Using Surrogate‐Based Global Optimization

Ricciuto

Stoyanov

et al. 2018

J Adv Model Earth Syst

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Calibration of the Energy Exascale Earth System Model (E3SM), land model (ELMv0) is challenging because of its model complexity, strong model nonlinearity, and significant computational requirements. Therefore, only a limited number of simulations can be allowed in any attempt to find a near‐optimal solution within an affordable time. The goal of this study is to calibrate some of the ELMv0 parameters to improve model projection of carbon fluxes. We propose a computationally efficient global optimization procedure using sparse‐grid based surrogates. We first use advanced sparse grid (SG) interpolation to construct a surrogate system of the ELMv0, and then calibrate the surrogate model in the optimization process. As the surrogate model is a polynomial whose evaluation is fast, it can be efficiently evaluated a sufficiently large number of times in the optimization, which facilitates the global search. We calibrate eight parameters against five years of net ecosystem exchange, total leaf area index, and latent heat flux data from the U.S. Missouri Ozark flux tower. The calibrated model is then used for predicting the three variables in the following 4 years. The results indicate that an accurate surrogate model can be created for the ELMv0 with a relatively small number of SG points, i.e., a few ELMv0 simulations that can be fully parallel. And, the application of the optimized parameters leads to a better model performance and a higher predictive capability than the default parameter values in the ELMv0.

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{frakturT}^{L, d} ([]|, η) (|, bold-italicθ) ≔ \underset{{L + 1 \leq |ℓ|}_{1} \leq L + d}{false\sum} {{(|, - 1)}^{L - {(|||, bold-scriptℓ)}_{1}} (|, \frac{0pt}{d - 1}) scriptU}^{{N_{scriptℓ}}_{1}} \otimes \dots \otimes {scriptU}^{N_{{scriptℓ}_{d}}} ([]|, η) .

The SG interpolant

{frakturT}^{L, d} ([]|, η) (|, bold-italicθ)

in equation is a weighted sum of a series of tensor‐product interpolants

{scriptU}^{N_{{scriptℓ}_{1}}} \otimes \dots \otimes {scriptU}^{N_{{scriptℓ}_{d}}} ([]|, η)

. Each of these interpolants is similar to the one in equation , but defined on a coarse grid with different resolution levels in different dimensions, i.e .,

{scriptℓ}_{i}

and

N_{{scriptℓ}_{i}}

may differ from each other for

i = 1, \dots, d

.…”

Section: Methodsmentioning

confidence: 99%

Section: Sparse-grid Interpolation and Surrogate Modelsmentioning

confidence: 99%

Calibration of the E3SM Land Model Using Surrogate‐Based Global Optimization

Ricciuto

Stoyanov

et al. 2018

J Adv Model Earth Syst

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show abstract

“…A key feature of these works is the use of sparse tensor products either for the construction of the basis or for the representation of the solution. This idea constitutes a cornerstone of high-dimensional approximation [10,18], and their applicability ranges from sparse grid approximations [15], to polynomial chaos expansion [40,29] and uncertainty quantification [28].…”

mentioning

confidence: 99%

Polynomial Approximation of High-Dimensional Hamilton--Jacobi--Bellman Equations and Applications to Feedback Control of Semilinear Parabolic PDEs

Kalise¹,

Kunisch²

2018

SIAM J. Sci. Comput.

100

111

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A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the Successive Galerkin Approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear Generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.

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“…This approach has emerged in the last half a dozen years as an alternative to more classical approximation schemes for high-dimensional functions, with the aim being to overcome some of the limitations mentioned above. Under natural the sparsity or compressibility assumptions, it enjoys a significant improvement in sample complexity over traditional methods such as discrete least-squares, projection, and interpolation [38,39]. Our intention in this chapter is to both present an overview of existing work in this area, focusing particularly on the mitigation of the curse of dimensionality, and to highlight existing open problems and challenges.…”

mentioning

confidence: 99%

Compressed Sensing Approaches for Polynomial Approximation of High-Dimensional Functions

Adcock

Brugiapaglia

Webster

2017

Applied and Numerical Harmonic Analysis

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In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the computation of polynomial approximations of high-dimensional functions on ddimensional hypercubes. We show that smooth, multivariate functions possess expansions in orthogonal polynomial bases that are not only approximately sparse, but possess a particular type of structured sparsity defined by so-called lower sets. This structure can be exploited via the use of weighted ℓ 1 minimization techniques, and, as we demonstrate, doing so leads to sample complexity estimates that are at most logarithmically dependent on the dimension d. Hence the curse of dimensionalitythe bane of high-dimensional approximation -is mitigated to a significant extent. We also discuss several practical issues, including unknown noise (due to truncation or numerical error), and highlight a number of open problems and challenges.

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Sparse Collocation Methods for Stochastic Interpolation and Quadrature

Cited by 7 publications

References 60 publications

Calibration of the E3SM Land Model Using Surrogate‐Based Global Optimization

Calibration of the E3SM Land Model Using Surrogate‐Based Global Optimization

Polynomial Approximation of High-Dimensional Hamilton--Jacobi--Bellman Equations and Applications to Feedback Control of Semilinear Parabolic PDEs

Compressed Sensing Approaches for Polynomial Approximation of High-Dimensional Functions

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