Multiscale Stochastic Representation in High-Dimensional Data Using Gaussian Processes with Implicit Diffusion Metrics

Thimmisetty, Charanraj; Khodabakhshnejad, Arman; Jabbari, Nima; Aminzadeh, Fred; Ghanem, Roger; Rose, Kelly; Bauer, Jennifer; Disenhof, Corinne

doi:10.1007/978-3-319-25138-7_15

Cited by 12 publications

(9 citation statements)

References 10 publications

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“…From Equation 16, it can be deduced that pW(w 0 ) = ∫ R pW ,R (w 0 ,r) dr can be estimated, for sim sufficiently large, by…”

Section: Nonparametric Statistical Estimation Of the Conditional Mathmentioning

confidence: 99%

“…Recent research in the field of uncertainty quantification [13][14][15][16][17] has underscored the need for optimization algorithms with underlying stochastic operators and constraints. In these situations, referred to as optimization under uncertainty (OUU), the challenge is magnified because for each design point along the optimization path, a sufficiently large statistical sample of function outputs must be computed to evaluate the required expectations.…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic nonconvex constrained optimization with fixed number of function evaluations

Ghanem

Soize

2017

Numerical Meth Engineering

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Summary A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.

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“…From Equation 16, it can be deduced that pW(w 0 ) = ∫ R pW ,R (w 0 ,r) dr can be estimated, for sim sufficiently large, by…”

Section: Nonparametric Statistical Estimation Of the Conditional Mathmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

Ghanem

Soize

2017

Numerical Meth Engineering

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“…Although Gaussian process models are most commonly used in this context [10,11], more robust alternatives based on Bayesian optimization [8,12] have also proven useful. Recent research in the field of uncertainty quantification [13,14,15,16,17] has underscored the need for optimization algorithms with underlying stochastic operators and constraints. In these situations, that we have previously referred to as OUU, the challenge is magnified since for each design point along the optimization path, a sufficiently large statistical sample of function outputs must be computed to evaluate the required expectations.…”

Section: A Few Words About Optimization Under Uncertaintiesmentioning

confidence: 99%

Probabilistic Learning on Manifold for Optimization Under Uncertainties

Soize¹,

Ghanem²

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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“…This method relies upon targeting a covariance function through numerical experiments, rather than an explicit a priori definition. In [27], a diffusion maps metric was used to parametrise the covariance function of Gaussian processes, in order to account for intrinsic correlation structures occurring in large time-dependent datasets, which are insufficiently captured using a Euclidean metric. Also of note is the investigation in [28], in which it was found that the mathematical structure of a random field can be significantly altered through large geometrical transformations of the spatial domain, such as those induced during manufacturing processes.…”

Section: Introductionmentioning

confidence: 99%

Random field simulation over curved surfaces: Applications to computational structural mechanics

Scarth

Adhikari

Cabral

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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It is important to account for inherent variability in the material properties in the design and analysis of engineering structures. These properties are typically not homogeneous, but vary across the spatial coordinates within a structure, as well as from specimen to specimen. This form of uncertainty is commonly modelled using random fields within the Stochastic Finite Element Method. Simulation within this framework can be complicated by the dependence of a random field's correlation function upon the geometry of the domain over which it is defined. In this paper, a new method is proposed for simulating random fields over a general two-dimension curved surface, represented as a finite element mesh. The covariance function is parametrised using the geodesic distance, evaluated using the solution to the 'discrete geodesic problem,' and a point discretisation approach is subsequently applied in order to sample the random field at the nodes of the model. The major contribution of the present work is the development of a methodology for simulating random fields over curved surfaces of arbitrary geometry, with a focus upon non-intrusive application to industrial finite element models using 'off the shelf' commercial software. In order to demonstrate the potential impact of the the proposed approach, the algorithm is applied in an uncertainty quantification case study concerning vibration and buckling of an industrial composite aircraft wing model.

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Multiscale Stochastic Representation in High-Dimensional Data Using Gaussian Processes with Implicit Diffusion Metrics

Cited by 12 publications

References 10 publications

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

Probabilistic Learning on Manifold for Optimization Under Uncertainties

Random field simulation over curved surfaces: Applications to computational structural mechanics

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