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首页> 外文期刊>SIAM/ASA Journal on Uncertainty Quantification >Multilevel Monte Carlo Approximation of Functions
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Multilevel Monte Carlo Approximation of Functions

机译：多级蒙特卡罗近似函数

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摘要

Many applications across sciences and technologies require a careful quantification of nondeterministic effects to a system output, for example, when evaluating the system's reliability or when gearing it towards more robust operation conditions. At the heart of these considerations lies an accurate characterization of uncertain system outputs. In this work we introduce and analyze novel multilevel Monte Carlo techniques for an efficient characterization of an uncertain system output's distribution. These techniques rely on accurately approximating general parametric expectations, i.e., expectations that depend on a parameter, uniformly on an interval. Applications of interest include, for example, the approximation of the characteristic function and of the cumulative distribution function of an uncertain system output. A further important consequence of the introduced approximation techniques for parametric expectations (i.e., for functions) is that they allow us to construct multilevel Monte Carlo estimators for various robustness indicators, such as for a quantile (also known as valueat-risk) and for the conditional value-at-risk. These robustness indicators cannot be expressed as moments and are thus not usually easily accessible. In fact, here we provide a framework that allows us to simultaneously estimate a cumulative distribution function, a quantile, and the associated conditional value-at-risk of an uncertain system output at the cost of a single multilevel Monte Carlo simulation, while each estimated quantity satisfies a prescribed tolerance goal.
机译：许多应用程序在科学和技术需要仔细量化不确定性系统输出的影响,例如,当评估系统的可靠性或者当它朝着更加健壮的操作条件。是一个精确的描述不确定系统输出。分析小说多层次蒙特卡罗技术对于一个有效的一个不确定的特征系统输出的分布。依赖于精确的近似参数的预期,即预期依赖于一个参数,统一在一个时间间隔。感兴趣的应用程序包括,例如,特征函数的近似和的累积分布函数不确定的系统输出。介绍了近似的结果技术参数(例如,预期函数)是他们允许我们构造多级蒙特卡罗估计不同鲁棒性指标,如分位数(也称为valueat-risk)和条件风险价值。指标并不能表示为时刻因此通常不容易。我们提供一个框架,允许我们同时估计一个累积分布函数,分位数,和相关的条件风险价值的不确定系统输出在一个多级蒙特的成本卡洛模拟,而每一个估计量满足规定的公差的目标。

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