Bayesian uncertainty quantification in the evaluation of alloy properties with the cluster expansion method

Kristensen, Jesper; Zabaras, Nicholas

doi:10.1016/j.cpc.2014.07.013

Cited by 26 publications

(21 citation statements)

References 39 publications

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“…We refer to the framework developed in Ref. 17 for this. In summary, our task is to obtain κ and X for some data set that we will denote D, and then to solve for the ECIβ.…”

Section: The Cluster Expansionmentioning

confidence: 99%

Predicting low-thermal-conductivity Si-Ge nanowires with a modified cluster expansion method

Kristensen

Zabaras

2015

Phys. Rev. B

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We introduce the cluster expansion ghost lattice method, which extends the applicability of existing cluster expansion software, to cluster expand structures of arbitrary finite and infinite geometries in a fast, unique, and transferable way. The ghost site is introduced that zeroes the cluster function of any cluster which includes it. This enables the use of bulk clusters grouped by bulk symmetries in non-bulk systems and distinguishes the cluster expansion ghost lattice method from a regular ternary cluster expansion with an inactive vacuum atom type. Even though the method does not treat surface terms, it can be used as an efficient way of obtaining the bulk term in [Lerch, D. et al., 2009. Model Simul Mater Sc]. We use the method to learn the thermal conductivity of Si-Ge nanowires, oriented along the [111] direction on a diamond lattice, versus their configuration of Si and Ge atoms. Once learned, the ghost lattice method cluster expansion is shown to be able to predict the lowest-thermal-conductivity nanowire configuration, in agreement with the configuration recently found in [Chan, M. et al., 2010. Physical Review B].

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“…We refer to the framework developed in Ref. 17 for this. In summary, our task is to obtain κ and X for some data set that we will denote D, and then to solve for the ECIβ.…”

Section: The Cluster Expansionmentioning

confidence: 99%

Predicting low-thermal-conductivity Si-Ge nanowires with a modified cluster expansion method

Kristensen

Zabaras

2015

Phys. Rev. B

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“…The necessary values for convergence depend on the material being simulated. For SiGe, including only 2-point clusters already gives a good fit [33], but for MgLi, clusters with up to 5 points may be necessary [23,33]. This can be seen as well using the RVM, which removes clusters that are not relevant for the fit.…”

Section: Model Trainingmentioning

confidence: 95%

“…Previous works employing a Bayesian framework for uncertainty quantification in the CE used a Laplace prior for the expansion coefficients and a right-truncated Poisson prior for the number of clusters to induce sparsity in the number of clusters for the expansion [33]. Despite the generality of this approach, the posterior distribution cannot be sampled analytically and a reversible jump Markov Chain Monte Carlo (RJ-MCMC) [34] had to be used to obtain statistics on the predictions [33]. There are other works based on a Bayesian framework, but they did not exploit the resultant uncertainty.…”

Section: Introductionmentioning

confidence: 99%

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Quantifying uncertainties in first-principles alloy thermodynamics using cluster expansions

Aldegunde¹,

Zabaras²,

Kristensen³

2016

Journal of Computational Physics

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The cluster expansion is a popular surrogate model for alloy modeling to avoid costly quantum mechanical simulations. As its practical implementations require approximations, its use trades efficiency for accuracy. Furthermore, the coefficients of the model need to be determined from some known data set (training set). These two sources of error, if not quantified, decrease the confidence we can put in the results obtained from the surrogate model. This paper presents a framework for the determination of the cluster expansion coefficients using a Bayesian approach, which allows for the quantification of uncertainties in the predictions. In particular, a relevance vector machine is used to automatically select the most relevant terms of the model while retaining an analytical expression for the predictive distribution. This methodology is applied to two binary alloys, SiGe and MgLi, including the temperature dependence in their effective cluster interactions. The resulting cluster expansions are used to calculate the uncertainty in several thermodynamic quantities: ground state line, including the uncertainty in which structures are thermodynamically stable at 0 K, phase diagrams and phase transitions. The uncertainty in the ground state line is found to be of the order of meV/atom, showing that the cluster expansion is reliable to ab initio level accuracy even with limited data. We found that the uncertainty in the predicted phase transition temperature increases when including the temperature dependence of the effective cluster interactions. Also, the use of the bond stiffness versus bond length approximation to calculate temperature dependent properties from a reduced set of alloy configurations showed similar uncertainty to the approach where all training configurations are considered but at a much reduced computational cost.

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“…These statistical quantities can be found analytically for straightforward cases, but often the use of a numerical technique cannot be avoided. One such a technique is Markov chain Monte Carlo sampling (Beck and Au 2002;Marzouk et al 2007;Kristensen and Zabaras 2014). MCMC techniques are based on drawing samples from a target distribution (here the posterior) and numerically approximating the quantities of interest (e.g.…”

Section: Markov Chain Monte Carlo Methods (Mcmc)mentioning

confidence: 99%

Bayesian inference to identify parameters in viscoelasticity

Rappel

Beex

Bordas

2017

Mech Time-Depend Mater

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This contribution discusses Bayesian inference (BI) as an approach to identify parameters in viscoelasticity. The aims are (i) to show that the prior has a substantial influence for viscoelasticity, (ii) to show that this influence decreases for an increasing number of measurements and (iii) to show how different types of experiments influence the identified parameters and their uncertainties. The standard linear solid model is the material description of interest and a relaxation test, a constant strain-rate test and a creep test are the tensile experiments focused on. The experimental data are artificially created, allowing us to make a one-to-one comparison between the input parameters and the identified parameter values. Besides dealing with the aforementioned aims, we believe that this contribution forms a comprehensible start for those interested in applying BI in viscoelasticity.

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Bayesian uncertainty quantification in the evaluation of alloy properties with the cluster expansion method

Cited by 26 publications

References 39 publications

Predicting low-thermal-conductivity Si-Ge nanowires with a modified cluster expansion method

Predicting low-thermal-conductivity Si-Ge nanowires with a modified cluster expansion method

Quantifying uncertainties in first-principles alloy thermodynamics using cluster expansions

Bayesian inference to identify parameters in viscoelasticity

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