Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. Developing efficient and accurate solution strategies that account for errors on the space, time and parameter domains simultaneously is highly challenging. Indeed, it is well known that standard polynomial-based approximations on the parameter domain can incur errors that grow in time. In this work, we focus on advection-diffusion problems with parameter-dependent wind fields. A novel adaptive solution strategy is proposed that allows users to combine stochastic collocation on the parameter domain with off-the-shelf adaptive timestepping algorithms with local error control. This is a non-intrusive strategy that builds a polynomial-based surrogate that is adapted sequentially in time. The algorithm is driven by a so-called hierarchical estimator for the parametric error and balances this against an estimate for the global timestepping error which is derived from a scaling argument.

翻译：含不确定输入的物理模型常被表征为参数化偏微分方程(PDEs)，即输入量以具有概率分布的参数函数形式表达的偏微分方程。同时考虑空间、时间及参数域误差的高效精确求解策略开发极具挑战性。众所周知，标准参数域多项式逼近方法会产生随时间增长的误差。本文聚焦于含参数依赖风场的对流扩散问题，提出一种新型自适应求解策略，允许用户将参数域随机配位法与具备局部误差控制的现有时步自适应算法相结合。该非侵入式策略通过构建时域序贯自适应的多项式代理模型，采用参数误差分层估计器驱动算法，并与基于尺度分析推导的全局时步误差估计进行平衡。