An Adaptive Multilevel Monte Carlo Method with Stochastic Bounds for Quantities of Interest with Uncertain Data
An Adaptive Multilevel Monte Carlo Method with Stochastic Bounds for Quantities of Interest with Uncertain Data
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DOI：
10.1137/15m1016448
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发表时间：
2016-10
期刊：
影响因子：
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通讯作者：
M. Eigel;C. Merdon;Johannes Neumann
中科院分区：
文献类型：
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作者：
M. Eigel;C. Merdon;Johannes Neumann
本工作的重点是引入一些可计算的后验误差控制到流行的蒙特卡罗多层抽样随机数据的PDE。我们特别感兴趣的应用程序，一些数量的兴趣应该被准确地估计。在有限元法空间离散的基础上，定义了目标泛函，并对感兴趣的数量进行编码。所设计的面向目标的后验误差估计器使人们能够确定该数量的保证路径后验误差界限。提出了一种利用计算误差估计和自适应网格控制逼近误差的自适应算法。此外，对随机误差进行了控制，使确定的边界在概率上得到保证。该方法允许多层蒙特卡罗模拟中使用的网格层次的自适应细化，用于离散化水平的问题依赖构造。数值实验证明…
The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications where some quantity of interest should be estimated accurately. Based on a spatial discretization by the finite element method, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented a posteriori error estimator enables one to determine guaranteed path-wise a posteriori error bounds for this quantity. An adaptive algorithm is proposed which employs the computed error estimates and adaptive meshes to control the approximation error. Moreover, the stochastic error is controlled such that the determined bounds are guaranteed in probability. The approach allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation which is used for a problem-dependent construction of discretization levels. Numerical experiments illustrate...