Inverse problem is ubiquitous in science and engineering, and Bayesian methodologies are often used to infer the underlying parameters. For high-dimensional temporal-spatial models, classical Markov chain Monte Carlo methods are often slow to converge, and it is necessary to apply Metropolis-within-Gibbs (MwG) sampling on parameter blocks. However, the computation cost of each MwG iteration is typically \(O(n^2)\), where n is the model dimension. This can be too expensive in practice. This paper introduces a new reduced computation method to bring down the computation cost to O(n), for the inverse initial value problem of a stochastic differential equation (SDE) with local interactions. The key observation is that each MwG proposal is only different from the original iterate at one parameter block, and this difference will only propagate within a local domain in the SDE computations. Therefore, we can approximate the global SDE computation with a surrogate updated only within the local domain for reduced computation cost. Both theoretically and numerically, we show that the approximation errors can be controlled by the local domain size. We discuss how to implement the local computation scheme using Euler–Maruyama and fourth-order Runge–Kutta methods. We numerically demonstrate the performance of the proposed method with the Lorenz 96 model and a linear stochastic flow model.

中文翻译：

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通过微分方程的局部计算加速大都市内吉布斯采样器

反问题在科学和工程中无处不在，并且经常使用贝叶斯方法来推断基本参数。对于高维时空模型，经典的马尔可夫链蒙特卡洛方法通常收敛缓慢，因此有必要在参数块上应用大都市内吉布斯（MwG）采样。但是，每次MwG迭代的计算成本通常为\（O（n ^ 2）\），其中n是模型维。实际上这可能太昂贵了。本文介绍了一种新的简化计算方法，将计算成本降低到O（n），对于具有局部相互作用的随机微分方程（SDE）的逆初值问题。关键的观察结果是，每个MwG提议仅与一个参数块上的原始迭代不同，并且这种差异将仅在SDE计算中的局部域内传播。因此，我们可以使用仅在本地域内更新的代理近似全局SDE计算，以降低计算成本。无论是从理论上还是从数值上，我们都表明近似误差可以由局部域大小控制。我们讨论了如何使用Euler-Maruyama和四阶Runge-Kutta方法来实现局部计算方案。我们用Lorenz 96模型和线性随机流模型数值证明了该方法的性能。