On the use of gradient information in Gaussian process quadratures

Prüher, Jakub; Särkkä, Simo

doi:10.1109/mlsp.2016.7738903

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…It seems that the beginning of renewed interest in Bayesian cubature can be dated to 2012, a year that witnessed the publication of a number of articles (Huszár and Duvenaud, 2012;Osborne et al, 2012b,a). The outpouring of work that soon followed contained contributions, to name a few, on selection of the integration points by numerical optimisation (Briol et al, 2015), convergence results in the misspecified setting when f ∉ H K (Kanagawa et al, 2016), selection of the sampling distribution in Bayesian Monte Carlo (Briol et al, 2017), relationship between Bayesian cubature and random feature expansions (Bach, 2017), and generalisations where also the measure μ is considered unknown (Oates et al, 2017b), derivative evaluations are employed (Prüher and Särkkä, 2016;Wu et al, 2018), 4 and the integrand is allowed to be vector-valued (Xi et al, 2018). Other recent works are Hamrick and Griffiths (2013); Ma et al (2014) A topic that has seen a fair amount of activity is modelling of positivity of the integrand (Osborne et al, 2012a;Gunter et al, 2014;Chai and Garnett, 2018;Wagstaff et al, 2018).…”

Section: Literature Reviewmentioning

confidence: 99%

Classical quadrature rules via Gaussian processes

Karvonen

Särkkä

2017

2017 IEEE 27th International Workshop on Machine Learning for Signal Processing (MLSP)

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all of whom have been kind enough to host me at their home institutions, some of them several times, during the past four years. The probabilistic numerics research community is still compact enough for one to meet almost everybody during the short span of a few years. The various conferences, workshops, and visits have been made much more enjoyable and productive by the presence, often recurring, of in particular Dr.

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Section: Literature Reviewmentioning

confidence: 99%

Classical quadrature rules via Gaussian processes

Karvonen

Särkkä

2017

2017 IEEE 27th International Workshop on Machine Learning for Signal Processing (MLSP)

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“…The fibers are modeled as linear elastic with properties E = 74 000 MPa and ν = 0.2. For the matrix we employ the pressure-dependent elastoplastic model proposed by Melro et al [43], with E = 3130 MPa, ν = 0.37, ν p = 0.32 (plastic Poisson's ratio) and yield stresses given by: σ t = 64.80 − 33.6e −ε p eq /0.003407 − 10.21e −ε p eq /0.06493 (39) σ c = 81.00 − 42.0e −ε p eq /0.003407 − 12.77e −ε p eq /0.06493 (40) where ε p eq is the equivalent plastic strain. The model is solved for 100 load steps, at which point the global macroscopic response is almost perfectly plastic and the strain localizes around the center of the specimen.…”

Section: Fe 2 Demonstrationmentioning

confidence: 99%

On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

Rocha¹,

Kerfriden²,

Meer³

2020

Preprint

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Concurrent multiscale finite element analysis (FE 2 ) is a powerful approach for high-fidelity modeling of materials for which a suitable macroscopic constitutive model is not available. However, the extreme computational effort associated with computing a nested micromodel at every macroscopic integration point makes FE 2 prohibitive for most practical applications. Constructing surrogate models able to efficiently compute the microscopic constitutive response is therefore a promising approach in enabling concurrent multiscale modeling. This work presents a reduction framework for adaptively constructing surrogate models for FE 2 based on statistical learning. The nested micromodels are replaced by a machine learning surrogate model based on Gaussian Processes (GP). The need for offline data collection is bypassed by training the GP models online based on data coming from a small set of fully-solved anchor micromodels that undergo the same strain history as their associated macroscopic integration points. The Bayesian formalism inherent to GP models provides a natural tool for online uncertainty estimation through which new observations or inclusion of new anchor micromodels are triggered. The surrogate constitutive manifold is constructed with as few micromechanical evaluations as possible by enhancing the GP models with gradient information and the solution scheme is made robust through a greedy data selection approach embedded within the conventional finite element solution loop for nonlinear analysis. The sensitivity to model parameters is studied with a tapered bar example with plasticity, while the applicability of the model to more complex cases is demonstrated with the elastoplastic analysis of a plate with multiple cutouts and a crack growth example for mixed-mode bending. The framework is found to be a promising approach in reducing the computational cost of FE 2 , with significant efficiency gains being obtained without resorting to offline training.

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“…Podgórski et al (2015) used a Slepian model to describe vehicle movements on a non-Gaussian road. Gradient information in Gaussian processes regression is still another example, (Prüher and Särkkä, 2016).…”

Section: Slepian Modelsmentioning

confidence: 99%

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

Lindgren¹

2019

Statist. Sci.

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We describe and compare how methods based on the classical Rice's formula for the expected number, and higher moments, of level crossings by a Gaussian process stand up to contemporary numerical methods to accurately compute the statistical distribution of the maximum over a xed interval, the length of excursions above a level, and the geometry of oscillations, and other characteristics of the sample paths.We illustrate the relative merits in accuracy and computing time of the Rice moment methods and the exact numerical method, developed since the late 1990s, on three groups of distribution problems, the maximum over a nite interval and the waiting time to rst crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.We also treat the notoriously dicult problem of dependence between successive zero crossing distances. The exact solution has been known since at least 2000, but it has remained largely unnoticed outside the ocean science community.Extensive simulation studies illustrate the accuracy of the numerical methods. As a historical introduction an attempt is made to illustrate the relation between Rice's original formulation and arguments and the exact numerical methods.MSC 2010 subject classications: Primary 60G17; secondary 60-08, 60G10, 60G70, 65C50.

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On the use of gradient information in Gaussian process quadratures

Cited by 7 publications

References 12 publications

Classical quadrature rules via Gaussian processes

Classical quadrature rules via Gaussian processes

On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

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