{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:34:16Z","timestamp":1762522456713,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,4,28]],"date-time":"2020-04-28T00:00:00Z","timestamp":1588032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen\u2013Lo\u00e8ve expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.<\/jats:p>","DOI":"10.3390\/a13050110","type":"journal-article","created":{"date-parts":[[2020,4,28]],"date-time":"2020-04-28T09:57:21Z","timestamp":1588067841000},"page":"110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2322-095X","authenticated-orcid":false,"given":"Philippe","family":"Blondeel","sequence":"first","affiliation":[{"name":"Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pieterjan","family":"Robbe","sequence":"additional","affiliation":[{"name":"Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C\u00e9dric","family":"Van hoorickx","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, KU Leuven, Kasteelpark Arenberg 40, 3001 Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3877-9748","authenticated-orcid":false,"given":"Stijn","family":"Fran\u00e7ois","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, KU Leuven, Kasteelpark Arenberg 40, 3001 Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9273-3038","authenticated-orcid":false,"given":"Geert","family":"Lombaert","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, KU Leuven, Kasteelpark Arenberg 40, 3001 Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8988-2374","authenticated-orcid":false,"given":"Stefan","family":"Vandewalle","sequence":"additional","affiliation":[{"name":"Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,28]]},"reference":[{"key":"ref_1","unstructured":"Kleiber, M., and Hien, T.D. 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