Improved weak convergence for the long time simulation of Mean-field Langevin equations
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| Format: | Preprint |
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2024
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| author | Chen, Xingyuan Reis, Goncalo dos Stockinger, Wolfgang Wilde, Zac |
| author_facet | Chen, Xingyuan Reis, Goncalo dos Stockinger, Wolfgang Wilde, Zac |
| contents | We study the weak convergence behaviour of the Leimkuhler--Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean--Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption.
Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate $3/2$) than the standard Euler method (of weak order $1$). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_01346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved weak convergence for the long time simulation of Mean-field Langevin equations Chen, Xingyuan Reis, Goncalo dos Stockinger, Wolfgang Wilde, Zac Numerical Analysis Probability We study the weak convergence behaviour of the Leimkuhler--Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean--Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption. Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate $3/2$) than the standard Euler method (of weak order $1$). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests. |
| title | Improved weak convergence for the long time simulation of Mean-field Langevin equations |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2405.01346 |