Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Jakeman, John Davis; Eldred, Michael; Geraci, Gianluca; Gorodetsky, Alex

doi:10.1002/nme.6450

Cited by 6 publications

(23 citation statements)

References 35 publications

(56 reference statements)

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“…A practical adaptive algorithm to construct Λ is presented in Sect. 2.1.4, following the discussion in, e.g., [16]. Another option is to design Λ a-priori, by a careful analysis of the PDE at hand, see, e.g., [15,17,18].…”

Section: Misc Surrogate Modelmentioning

confidence: 99%

“…As already mentioned, the effectiveness of the MISC approximation depends on the choice of the multi-index set Λ : in this work such set is built with an adaptive algorithm, see [16]. We begin by introducing the following decomposition of the quadrature error…”

Section: An Adaptive Algorithm For the Multi-index Setmentioning

confidence: 99%

“…In practice, E R , can be conveniently computed by for any Λ downward-closed set such that Λ ∪ [ , ] is also downward closed. A similar quadrature-based error contribution is considered in [16], where a convex combination of the error in the computation of the mean and of the variance of the QoI is used. Another possibility is to introduce an error decomposition based on the point-wise accuracy of the surrogate model, following the same arguments above.…”

Section: An Adaptive Algorithm For the Multi-index Setmentioning

confidence: 99%

“…In practice the implementation reported in Algorithm 1 is used: it makes use of an auxiliary multi-index set, i.e., the margin of a multi-index set Λ , Mar(Λ) , which is defined as the set of multi-indices that can be reached "within one step" from Λ (see Fig. 1b) This algorithm was first proposed in the context of sparsegrids quadrature in [43] and its MISC implementation was first proposed in [16]. It is an a-posteriori algorithm and as such it determines the error contribution E S , , E R , of [ , ] after having added [ , ] to the grid.…”

Section: An Adaptive Algorithm For the Multi-index Setmentioning

confidence: 99%

“…A different option is to resort to methods that build a polynomial approximation over the parameter space: methods such as Multi-Level Stochastic Collocation [14], Multi-Index Stochastic Collocation [15][16][17][18], Multi-Level Least-Squares polynomial approximation [19], etc. fall in this category.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Comparing multi-index stochastic collocation and multi-fidelity stochastic radial basis functions for forward uncertainty quantification of ship resistance

Piazzola

Tamellini

Pellegrini

et al. 2022

Engineering with Computers

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This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passenger ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, and kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Collocation (MISC) and an adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF). The QoI is evaluated via computational fluid dynamics simulations, which are performed with the in-house unsteady Reynolds-Averaged Navier–Stokes (RANS) multi-grid solver $$\chi$$ χ navis. The different fidelities employed by both methods are obtained by stopping the RANS solver at different grid levels of the multi-grid cycle. The performance of both methods are presented and discussed: in a nutshell, the findings suggest that, at least for the current implementation of both methods, MISC could be preferred whenever a limited computational budget is available, whereas for a larger computational budget SRBF seems to be preferable, thanks to its robustness to the numerical noise in the evaluations of the QoI.

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Section: Misc Surrogate Modelmentioning

confidence: 99%

Section: An Adaptive Algorithm For the Multi-index Setmentioning

confidence: 99%

Section: An Adaptive Algorithm For the Multi-index Setmentioning

confidence: 99%

Section: An Adaptive Algorithm For the Multi-index Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Comparing multi-index stochastic collocation and multi-fidelity stochastic radial basis functions for forward uncertainty quantification of ship resistance

Piazzola

Tamellini

Pellegrini

et al. 2022

Engineering with Computers

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Adaptive experimental design for multi‐fidelity surrogate modeling of multi‐disciplinary systems

Jakeman

Friedman

Eldred

et al. 2022

Numerical Meth Engineering

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We present an adaptive algorithm for constructing surrogate models of multi-disciplinary systems composed of a set of coupled components. With this goal we introduce "coupling" variables with a priori unknown distributions that allow surrogates of each component to be built independently. Once built, the surrogates of the components are combined to form an integrated-surrogate that can be used to predict system-level quantities of interest at a fraction of the cost of the original model. The error in the integrated-surrogate is greedily minimized using an experimental design procedure that allocates the amount of training data, used to construct each component-surrogate, based on the contribution of those surrogates to the error of the integrated-surrogate. The multi-fidelity procedure presented is a generalization of multi-index stochastic collocation that can leverage ensembles of models of varying cost and accuracy, for one or more components, to reduce the computational cost of constructing the integrated-surrogate. Extensive numerical results demonstrate that, for a fixed computational budget, our algorithm is able to produce surrogates that are orders of magnitude more accurate than methods that treat the integrated system as a black-box.

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A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential

Jahnke

Stein

2022

SIAM/ASA J. Uncertainty Quantification

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We propose and analyze a numerical method for time-dependent linear Schrödinger equations with uncertain parameters in both the potential and the initial data. The random parameters are discretized by stochastic collocation on a sparse grid, and the sample solutions in the nodes are approximated with the Strang splitting method. The computational work is reduced by a multi-level strategy, i.e. by combining information obtained from sample solutions computed on different refinement levels of the discretization. We prove new error bounds for the time discretization which take the finite regularity in the stochastic variable into account, and which are crucial to obtain convergence of the multi-level approach. The predicted cost savings of the multi-level stochastic collocation method are verified by numerical examples.

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Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Cited by 6 publications

References 35 publications

Comparing multi-index stochastic collocation and multi-fidelity stochastic radial basis functions for forward uncertainty quantification of ship resistance

Comparing multi-index stochastic collocation and multi-fidelity stochastic radial basis functions for forward uncertainty quantification of ship resistance

Adaptive experimental design for multi‐fidelity surrogate modeling of multi‐disciplinary systems

A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential

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