To solve problems in domains such as filtering, optimization, and posterior sampling, interacting-particle methods have recently received much attention. These parallelizable and often gradient-free algorithms use an ensemble of particles that evolve in time, based on a combination of well-chosen dynamics and interaction between the particles. For computationally expensive dynamics -- for example, dynamics that solve inverse problems with an expensive forward model -- the cost of attaining a high accuracy quickly becomes prohibitive. We exploit a hierarchy of approximations to this forward model and apply multilevel Monte Carlo (MLMC) techniques, improving the asymptotic cost-to-error relation. More specifically, we use MLMC at each time step to estimate the interaction term within a single, globally-coupled ensemble. This technique was proposed by Hoel et al. in the context of the ensemble Kalman filter; the goal of the present paper is to study its applicability to a general framework of interacting-particle methods. After extending the algorithm and its analysis to a broad set of methods with fixed numbers of time steps, we motivate the application of the method to the class of algorithms with an infinite time horizon, which includes popular methods such as ensemble Kalman algorithms for optimization and sampling. Numerical tests confirm the improved asymptotic scaling of the multilevel approach.

翻译：为解决滤波、优化及后验采样等领域的问题，交互粒子方法近年来受到广泛关注。这类算法通常可并行化且无需梯度计算，其核心机制是通过精心设计的动力学与粒子间相互作用相结合，使粒子系综随时间演化。对于计算成本高昂的动力学过程（例如需借助昂贵正演模型求解反问题的动力学），实现高精度所需的代价会迅速变得不可承受。我们利用该正演模型的分层近似，并应用多级蒙特卡洛（MLMC）技术，从而改善渐近成本-误差关系。具体而言，我们在每个时间步内采用MLMC估计单个全局耦合系综中的相互作用项。该技术由Hoel等人在系综卡尔曼滤波框架下提出；本文旨在研究其在一般交互粒子方法框架中的适用性。在将算法及其分析推广至固定时间步数的一类广泛方法后，我们进一步论证该方法在无限时间范围算法中的应用前景——这类算法包括流行的优化与采样系综卡尔曼算法。数值实验验证了多级方法在渐近缩放性上的提升。