Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations*
Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations*
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Langevin 型模型 I：给定平稳分布及其离散化的扩散*
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
R. Tweedie
中科院分区：
文献类型：
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作者：
O. Stramer;R. Tweedie
我们描述了一种算法，用于估计一个给定的测量π已知到一个比例常数，基于一大类扩散（扩展朗之万模型），其中π是不变的。我们证明了在弱条件下，我们可以从这类中选择这样一种方式，即扩散以指数速率收敛到π，并且我们甚至可以确保收敛与算法的起点无关。当收敛小于指数时，我们证明它在可验证的速率下通常是多项式。然后考虑离散扩散的方法，并找到继承连续时间过程收敛速率的方法。这些与朴素离散或欧拉离散的行为形成对比，即使在简单的情况下，它们也会表现得很糟糕。我们的结果仅在一维中详细描述，尽管也简要描述了向高维的扩展。
We describe algorithms for estimating a given measure π known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which π is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to π, and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at verifiable rates. We then consider methods of discretizing the diffusion in time, and find methods which inherit the convergence rates of the continuous time process. These contrast with the behavior of the naive or Euler discretization, which can behave badly even in simple cases. Our results are described in detail in one dimension only, although extensions to higher dimensions are also briefly described.