p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

Blondeel, Philippe; Robbe, Pieterjan; hoorickx, Cédric Van; François, Stijn; Lombaert, Geert; Vandewalle, Stefan

doi:10.3390/a13050110

Cited by 13 publications

(11 citation statements)

References 44 publications

(66 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, for sufficiently smooth integrands it is possible to construct QMC rules with error bounds not depending on the number of stochastic variables while attaining faster convergence rates compared to Monte Carlo methods. For these reasons QMC methods have been very successful in applications to PDEs with random coefficients (see, e.g., [2,9,14,16,17,21,22,23,30,31,32,36,39,40]) and especially in PDE-constrained optimization under uncertainty, see [19,20]. In [29] the authors derive regularity results for the saddle point operator, which fall within the same framework as the QMC approximation of affine parametric operator equation setting considered in [40].…”

Section: Introductionmentioning

confidence: 99%

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

A.¹,

Kaarnioja²,

Kuo³

et al. 2022

Preprint

View full text Add to dashboard Cite

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem -and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

show abstract

Section: Introductionmentioning

confidence: 99%

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

A.¹,

Kaarnioja²,

Kuo³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…However, classic ML(Q)MC remains costly due to an almost geometrical increase in the number of degrees of freedom for each refined mesh. Therefore, we have developped a novel multilevel method, called p-refined Multilevel Quasi-Monte Carlo (p-MLQMC), see [6]. This multilevel method combines a mesh hierarchy based on a p-refinement scheme with a QMC sampling rule.…”

Section: Introductionmentioning

confidence: 99%

Benchmarking the P-MLQMC method on a geotechnical engineering problem

Blondeel¹,

Robbe²,

François³

et al. 2021

14th WCCM-ECCOMAS Congress

View full text Add to dashboard Cite

Problems in civil engineering are often characterized by significant uncertainty in their material parameters. Sampling methods are a straightforward manner to account for this uncertainty, which is typically modeled as a random field. A novel method developed by the authors called p-refined Multilevel Quasi-Monte Carlo (p-MLQMC), achieves a significant computational cost reduction with respect to classic Multilevel Monte Carlo (h-MLMC). p-MLQMC uses a hierarchy of p-refined Finite Element meshes, combined with a deterministic Quasi-Monte Carlo (QMC) sampling rule. A non-negligible part of modeling the stochastics in non-deterministic engineering problems consists in adequately incorporating the uncertainty in the Finite Element model. This is typically done by evaluating the random field at certain carefully chosen evaluation points, and assigning the resulting scalars to the different finite elements. For the h-MLMC method, these evaluation points consist of the centroids of the elements. For the p-MLQMC method, we distinguish two different approaches to select the evaluation points, the Non-Nested Approach (NNA) and the Local Nested Approach (LNA). We investigate how these approaches affect the variance reduction over the levels and the total computational cost. We benchmark the p-ML(Q)MC-LNA, p-ML(Q)MC-NNA and h-MLMC method on a slope stability problem where the uncertainty is located in the soil's cohesion. We show that the p-MLQMC-LNA method outperforms all other considered methods in terms of computational cost.

show abstract

“…In previous work, see [5], we obtained an order of convergence close to O(N −1 ) when combining the Multilevel Monte Carlo method with a quasi-Monte Carlo sampling method. In [6,7], we combined the p-refinement of the Finite Element method (FEM) discretization with the Multilevel quasi-Monte Carlo sampling method, applied to a slope stability problem. There, the multilevel mesh hierarchy is constructed following a p-refinement approach, i.e., the order of the elements in the subsequent meshes is increased.…”

Section: Introductionmentioning

confidence: 99%

Improving the Rate of Convergence of the Quasi-Monte Carlo Method in Estimating Expectations on a Geotechnical Slope Stability Problem

Blondeel¹,

Robbe²,

Nuyens³

et al. 2021

4th International Conference on Uncertainty Quantification in Computational Sciences and Engineering

Self Cite

View full text Add to dashboard Cite

The propagation of parameter uncertainty through engineering models is a key task in uncertainty quantification. In many cases, taking into account this uncertainty involves the estimation of expected values by means of the Monte Carlo method. While the performance of the classical Monte Carlo method is independent of the number of uncertainties, its main drawback is the slow convergence rate of the root mean square error, i.e., O(N −1/2 ) where N is the number of model evaluations. Under appropriate conditions, the quasi-Monte Carlo method improves the order of convergence to O(N −1 ) by using deterministic sample points instead of random sample points. Two examples of such point sets are rank-1 lattice sequences and Sobol' sequences. However, it is possible to further improve the order of convergence by applying the so-called "tent transformation" to a rank-1 lattice sequence, and by "interlacing" a Sobol' sequence. In this work, we benchmark these two techniques on a slope stability problem from geotechnical engineering, where the uncertainty is located in the cohesion of the soil. The soil cohesion is modeled as a lognormal random field of which realizations are computed by means of the Karhunen-Loève (KL) expansion. The quasi-Monte Carlo points are mapped to the normal distribution required in the KL expansion using a novel truncation strategy. We observe an order of convergence of O(N −1.5 ) in our numerical experiments.

show abstract

p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

Cited by 13 publications

References 44 publications

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

Benchmarking the P-MLQMC method on a geotechnical engineering problem

Improving the Rate of Convergence of the Quasi-Monte Carlo Method in Estimating Expectations on a Geotechnical Slope Stability Problem

Contact Info

Product

Resources

About