TY - JOUR

T1 - Accelerating Monte Carlo estimation with derivatives of high-level finite element models

AU - Hauseux, Paul

AU - Hale, Jack S.

AU - Bordas, Stéphane P A

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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 one-dimensional 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. ©
2017 Published by Elsevier B.V

AB - 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 one-dimensional 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. ©
2017 Published by Elsevier B.V

KW - Monte carlo methods

KW - Parallel computing

KW - Partially intrusive methods

KW - Polynomial chaos expansion

KW - Tangent linear models

KW - Uncertainty propagation

UR - http://www.scopus.com/inward/record.url?scp=85014316338&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2017.01.041

DO - 10.1016/j.cma.2017.01.041

M3 - Article

AN - SCOPUS:85014316338

VL - 318

SP - 917

EP - 936

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

ER -