In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the righthand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine scale random fields in a hierarchical fashion from coarse scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in $L^2$ which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility.
翻译:在这项工作中,我们制定了一个新的等级多层次方法,以可有逻辑的可伸缩方式生成高斯随机字段实现,这种方法非常适合将Monte Carlo(MCMCMC)算法纳入多层次的Markov连锁 Monte Carlo(MCMCMC)算法。这个方法基于其他部分差异方程(PDE)方法,以产生高斯随机字段实现;特别是,可以通过解决反应扩散PDE(反应扩散)和空间白色噪声源功能作为右侧形成单一的场实现。虽然这些方法已经探索以加速未来的不确定性量化任务,例如多层次的蒙特卡洛,但先前的构造并不直接适用于多层次的MCMC框架,这些框架从粗略的随机字段以等级方式构建精细的随机字段。我们新的等级多层次方法依赖于以$L%2美元对白噪音源函数进行等级分解,从而使我们能够形成高斯随机字段实现多层次分解的方法,从而适应多层次的MC算法框架。在介绍我们的主要理论结果和数字缩放结果以展示其新的等级水平的可行性工具之后,在高斯马克斯四级的实地演算法方法产生了高斯。