Optimization of a Monte Carlo variance reduction method based on sensitivity derivatives

Liu, Yaning; Hussaini, M. Y.; Ökten, Giray

doi:10.1016/j.apnum.2013.06.005

Cited by 9 publications

(15 citation statements)

References 28 publications

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“…This example is very similar to the one presented in Liu et al [30]. We have chosen to replicate a similar result for two reasons; firstly, we wish to independently demonstrate the effectiveness of the sensitivity derivative estimator, and secondly we want to publish a simple and complete working example of the method that can be modified in a relatively straightforward way to the user's time-independent problem if it can be expressed in the Unified Form Language (UFL).…”

Section: Generalised Burgers Equation With Stochastic Viscositymentioning

confidence: 99%

“…[26]. Clearly the whether these requirements actually are intrusive depends on whether the user has models that can provide the required derivative information [14] or matrix-vector actions [26], respectively. In the past, easily computing derivatives of complex forward models either by the tangent linear (forward mode) or adjoint (reverse mode) of differentiation required the use of automatic differentiation tools, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…It can also be used with quasirandom low-discrepancy sequences [33], e.g. Sobol sequences, to improve the rate of convergence to O (log Z) M Z −1 [14]. An outline of this paper is as follows; in section 2 we give the general stochastic problem setting, before continuing to section 3 to specifics on the three stochastic solution methods, namely standard Monte Carlo, sensitivity derivative Monte Carlo and the multi-level polynomial chaos expansion method.…”

Section: Introductionmentioning

confidence: 99%

“…The key contributions of this paper are as follows; we present a first-order sensitivity derivative driven Monte Carlo method [13,30,15] to propagate uncertainty through two typical PDEs from the field of computational mechanics; a simple one-dimensional viscous Burgers equation with uncertain viscosity, and a complex three-dimensional Mooney-Rivlin hyperelastic equation. We concentrate on the low-dimensional case, when the stochastic parameters are real numbers, rather than random fields.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the underlying Z −1/2 convergence rate of the basic Monte Carlo estimator is not improved, but the overall error of the estimator is reduced. Methods in the wide class of variance reduction methods include multi-level techniques [8,11,12] and sensitivity derivative techniques [13,14,15]. Multi-level Monte Carlo uses a hierarchy of forward models of increasing computational complexity (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Hauseux

Hale

Bordas

2017

Computer Methods in Applied Mechanics and Engineering

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In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a onedimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney-Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.

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Section: Generalised Burgers Equation With Stochastic Viscositymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Hauseux

Hale

Bordas

2017

Computer Methods in Applied Mechanics and Engineering

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Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters

Hauseux

Hale

Cotin

et al. 2018

Applied Mathematical Modelling

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We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than the standard Monte Carlo method. The material model of interest can be altered by adjusting few lines of (FEniCS) code. The method is able to (1) provide the user with statistical confidence intervals on quantities of practical interest, such as the displacement of a tumour or target site in an organ; (2) quantify the sensitivity of the response of the organ to the associated parameters of the material model. We exercise the approach on the determination of a confidence interval on the motion of a target in the brain. We also show that for the boundary conditions under consideration five parameters of the Ogden-Holzapfel-like model have negligible influence on the displacement of the target zone compared to the three most influential parameters. The benchmark problems and all associated data are made available as supplementary material.

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Accurate construction of high dimensional model representation with applications to uncertainty quantification

Liu

Hussaini

Ökten

2016

Reliability Engineering & System Safety

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Optimization of a Monte Carlo variance reduction method based on sensitivity derivatives

Cited by 9 publications

References 28 publications

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters

Accurate construction of high dimensional model representation with applications to uncertainty quantification

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