1 research outputs found
h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering
Practical structural engineering problems are often characterized by
significant uncertainties. Historically, one of the prevalent methods to
account for this uncertainty has been the standard Monte Carlo (MC) method.
Recently, improved sampling methods have been proposed, based on the idea of
variance reduction by employing a hierarchy of mesh refinements. We combine an
h- and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) and
Multilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicability
of these novel combination methods on three structural engineering problems,
for which the uncertainty resides in the Young's modulus: the static response
of a cantilever beam with elastic material behavior, its static response with
elastoplastic behavior, and its dynamic response with elastic behavior. The
uncertainty is either modeled by means of one random variable sampled from a
univariate Gamma distribution or with multiple random variables sampled from a
gamma random field. This random field results from a truncated Karhunen-Lo\`eve
(KL) expansion. In this paper, we compare the computational costs of these
Monte Carlo methods. We demonstrate that MQLMC and MLMC have a significant
speedup with respect to MC, regardless of the mesh refinement hierarchy used.
We empirically demonstrate that the MLQMC cost is optimally proportional to
1/epsilon under certain conditions, where epsilon is the tolerance on the
root-mean-square error (RMSE). In addition, we show that, when the uncertainty
is modeled as a random field, the multilevel methods combined with p-refinement
have a significant lower computation cost than their counterparts based on
h-refinement. We also illustrate the effect the uncertainty models have on the
uncertainty bounds in the solutions.Comment: arXiv admin note: substantial text overlap with arXiv:1808.1068