论文标题
通用多项式膨胀的后验验证
A posteriori validation of generalized polynomial chaos expansions
论文作者
论文摘要
广义多项式混乱扩展是研究具有随机系数的微分方程的强大工具，尤其可以有效地近似与此类方程相关的随机不变集。在这项工作中，我们使用经过验证的数字中的想法来获得严格的后验错误估计以及存在有关随机不变集的GPC扩展的结果。这种方法还提供了一个新的框架，用于进行验证的延续，即，在参数依赖性系统中严格计算解决方案的孤立分支，该系统以直接的方式概括到多参数延续。我们通过严格计算Lorenz系统中的随机不变周期轨道以及Swift-Hohenberg方程的稳态的分支和二维流形来说明所提出的方法。
Generalized polynomial chaos expansions are a powerful tool to study differential equations with random coefficients, allowing in particular to efficiently approximate random invariant sets associated to such equations. In this work, we use ideas from validated numerics in order to obtain rigorous a posteriori error estimates together with existence results about gPC expansions of random invariant sets. This approach also provides a new framework for conducting validated continuation, i.e. for rigorously computing isolated branches of solutions in parameter-dependent systems, which generalizes in a straightforward way to multi-parameter continuation. We illustrate the proposed methodology by rigorously computing random invariant periodic orbits in the Lorenz system, as well as branches and 2-dimensional manifolds of steady states of the Swift-Hohenberg equation.
