Sampling-free linear Bayesian update of polynomial chaos representations

Rosić, Bojana; Litvinenko, Alexander; Pajonk, Oliver; Matthies, Hermann G.

doi:10.1016/j.jcp.2012.04.044

Cited by 68 publications

(86 citation statements)

References 53 publications

(85 reference statements)

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“…Focusing for a moment on the specific problem of Bayesian inference, another class of approaches related to the transport map framework are the sampling-free Bayesian updates introduced in [74,48,47,54,75]. These methods treat Bayesian inference as a projection.…”

Section: Related Workmentioning

confidence: 99%

“…In particular, they approximate the conditional expectation of any prescribed function of the parameters, where conditioning is with respect to the σ-field generated by the data. The approximation of the conditional expectation may be refined by enlarging the space of functions (typically polynomials) on which one projects; hence one can generalize linear Bayesian updates [74] to nonlinear Bayesian updates [48]. The precise goal of these approximations is different from that of the transport map framework, however.…”

Section: Related Workmentioning

confidence: 99%

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Sampling via Measure Transport: An Introduction

Marzouk¹,

Moselhy²,

Parno³

et al. 2016

Handbook of Uncertainty Quantification

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We present the fundamentals of a measure transport approach to sampling. The idea is to construct a deterministic coupling-i.e., a transport map-between a complex "target" probability measure of interest and a simpler reference measure. Given a transport map, one can generate arbitrarily many independent and unweighted samples from the target simply by pushing forward reference samples through the map. If the map is endowed with a triangular structure, one can also easily generate samples from conditionals of the target measure. We consider two different and complementary scenarios: first, when only evaluations of the unnormalized target density are available, and second, when the target distribution is known only through a finite collection of samples. We show that in both settings the desired transports can be characterized as the solutions of variational problems. We then address practical issues associated with the optimization-based construction of transports: choosing finite-dimensional parameterizations of the map, enforcing monotonicity, quantifying the error of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces. Approximate transports can also be used to "Gaussianize" complex distributions and thus precondition conventional asymptotically exact sampling schemes. We place the measure transport approach in broader context, describing connections with other optimization-based samplers, with inference and density estimation schemes using optimal transport, and with alternative transformation-based approaches to simulation. We also sketch current work aimed at the construction of transport maps in high dimensions, exploiting essential features of the target distribution (e.g., conditional independence, low-rank structure). The approaches and algorithms presented here have direct applications to Bayesian computation and to broader problems of stochastic simulation.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Sampling via Measure Transport: An Introduction

Marzouk¹,

Moselhy²,

Parno³

et al. 2016

Handbook of Uncertainty Quantification

View full text Add to dashboard Cite

show abstract

“…Namely, once a set of noisy and incomplete measurement data is provided one may alter (update) the a priori assumed probability model to a more realistic one with the help of Bayesian probabilistic models [122,180,192,191]. The process of updating requires the computation of the system response when the input parameters are uncertain, which is the topic of this work.…”

Section: Significance Of the Studymentioning

confidence: 99%

“…On the other side, theoretically a constitutive tensor can be modelled with the help of the maximum entropy approach [222,52] as one possess certain information about material characteristics a priori. Such model further can be updated (see [192,191]) using the measurement and the stochastic elastoplastic formulation as presented in this work.…”

Section: Significance Of the Studymentioning

confidence: 99%

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Variational formulations and functional approximation algorithms in stochastic plasticity of materials

Rosić¹

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A class of abstract stochastic variational inequalities of the second kind described by uncertain parameters is considered within the framework of infinitesimal and large displacement elastoplasticity theory. Particularly the focus is set on the rateindependent evolutionary problem with general hardening whose material characteristics are assumed to have positively-definite distributions. By exhibiting the structure of the evolutionary equations in a convex setting the mathematical formulation is carried over to the computationally more suitable mixed variational description for which the existence and uniqueness of the solution is studied. Time discretised as usual with backward Euler, the inequality is reduced to a minimisation problem for a convex functional on discrete tensor product subspaces whose unique minimiser is obtained via a stochastic closest point projection algorithm based on "white noise analysis". To this end a description in the language of non-dissipative and dissipative operators is used, both employing the stochastic Galerkin method in its fully intrusive or non-intrusive variant. The former method represents the direct, purely algebraic way of computing the response in each iteration of Newton-like methods. As the solution is given in a form of polynomial chaos expansion, i.e. an explicit functional relationship between the independent random variables, the subsequent evaluations of its functionals (the mean, variance, or probabilities of exceedence) are shown to be very cheap, but with limited accuracy. Due to this reason, the intrusive method is contrasted to the less efficient but more accurate non-intrusive variant which evaluates the residuum in each iteration via high-dimensional integration rules based on random or deterministic sampling -Monte Carlo and related techniques. In addition to these, the problem is also solved with the help of the stochastic collocation method via sparse grid techniques. Finally, the methods are validated on a series of test examples in plain strain conditions whose reference solution is computed via direct integration methods. 5 RezimeU okviru teorije malih i velikih plastičnih deformacija razmatrana je klasa apstraktnih stohastičkih varijacionih nejednakosti opisanih slučajnim promenljivim. Poseban fokus je stavljen na asocijativni evolucioni problem sa generalnim ojačanjem cije materijalne karakteristike imaju distribuciju odredenu zakonom maksimalne entropije. Proučavajući strukturu evolucionih jednačina uz pomoć konveksne teorije analizirani su uslovi za postojanje i jedinstvenost rešenja uz dodatnu matematičku reformulaciju problema u numerički prikladan mešoviti varijacioni opis. Dobijena nejednakost se nakon implicitne diskretizacije svodi na minimizaciju konveksnog funkcionala definisanog u tenzorskom prostoru. Rešenje tako postavljenog problema se može dobiti novouvedenom stohastičkom metodom projekcije najbliže tačke uz pomoć teorije analize belogšuma. Pomenuta metoda se sastoji od dva koraka: elastičnog i plastičnog,čiji su algoritmi ...

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Uncertainty Quantification andBayesian Inversion

Matthies

2017

Encyclopedia of Computational Mechanics Second Edition

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Uncertainty estimation arises at least implicitly in any kind of modeling of the real – or phenomenological – world, and it is desirable to actually quantify the uncertainty in probabilistic terms. Here the emphasis is on uncertain systems, typically modeled by partial differential equations, where the randomness is assumed spatial. Traditional computational approaches usually use some form of perturbation or Monte Carlo simulation. Here the emphasis is on recent methods based on stochastic functional or spectral approximations. Quantifying the uncertainty in mathematical models due to uncertainties in their parameters, combined with the traditional ability to predict the system behavior given external excitations or loadings, is called solving the stochastic forward problem . Given this capability, one has the possibility to use this to estimate or identify unknown and hence uncertain parameters in the mathematical model by observing the system response. This is called a solution to the inverse problem , and a short introduction into inverse problems in a Bayesian setting using the functional or spectral approximations is provided.

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Sampling-free linear Bayesian update of polynomial chaos representations

Cited by 68 publications

References 53 publications

Sampling via Measure Transport: An Introduction

Sampling via Measure Transport: An Introduction

Variational formulations and functional approximation algorithms in stochastic plasticity of materials

Uncertainty Quantification andBayesian Inversion

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