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MV-SDEs are usually approximated using a stochastic interacting P<\/jats:italic>-particle system, which is a set of P<\/jats:italic> coupled d<\/jats:italic>-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a $$P \\times d$$<\/jats:tex-math>\n \n P<\/mml:mi>\n \u00d7<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et\u00a0al. 2023), generating a d<\/jats:italic>-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of $$\\mathcal {O}\\left( \\textrm{TOL}_{\\textrm{r}}^{-4}\\right) $$<\/jats:tex-math>\n \n O<\/mml:mi>\n \n \n TOL<\/mml:mtext>\n \n r<\/mml:mtext>\n <\/mml:mrow>\n \n -<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with a significantly reduced constant to achieve a prescribed relative error tolerance $$\\textrm{TOL}_{\\textrm{r}}$$<\/jats:tex-math>\n \n TOL<\/mml:mtext>\n r<\/mml:mtext>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given $$\\textrm{TOL}_{\\textrm{r}}$$<\/jats:tex-math>\n \n TOL<\/mml:mtext>\n r<\/mml:mtext>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.<\/jats:p>","DOI":"10.1007\/s11222-024-10497-3","type":"journal-article","created":{"date-parts":[[2024,10,12]],"date-time":"2024-10-12T05:02:05Z","timestamp":1728709325000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Double-loop importance sampling for McKean\u2013Vlasov stochastic differential equation"],"prefix":"10.1007","volume":"34","author":[{"given":"Nadhir","family":"Ben Rached","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdul-Lateef","family":"Haji-Ali","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shyam Mohan","family":"Subbiah Pillai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ra\u00fal","family":"Tempone","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,10,12]]},"reference":[{"issue":"1","key":"10497_CR1","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1103\/RevModPhys.77.137","volume":"77","author":"JA Acebr\u00f3n","year":"2005","unstructured":"Acebr\u00f3n, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. 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