We demonstrate that the recently developed Optimal Uncertainty Quantification (OUQ) theory, combined with recent software enabling fast global solutions of constrained non-convex optimization problems, provides a methodology for rigorous model certification, validation, and optimal design under uncertainty. In particular, we show the utility of the OUQ approach to understanding the behavior of a system that is governed by a partial differential equation -- Burgers' equation. We solve the problem of predicting shock location when we only know bounds on viscosity and on the initial conditions. Through this example, we demonstrate the potential to apply OUQ to complex physical systems, such as systems governed by coupled partial differential equations. We compare our results to those obtained using a standard Monte Carlo approach, and show that OUQ provides more accurate bounds at a lower computational cost. We discuss briefly about how to extend this approach to more complex systems, and how to integrate our approach into a more ambitious program of optimal experimental design.

中文翻译：

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模型验证、验证和实验设计中非线性偏微分方程的最优边界

我们证明了最近开发的最优不确定性量化 (OUQ) 理论，结合最近能够快速解决约束非凸优化问题的全局解决方案的软件，提供了一种在不确定性下进行严格的模型验证、验证和优化设计的方法。特别是，我们展示了 OUQ 方法在理解由偏微分方程（Burgers 方程）控制的系统的行为方面的效用。当我们只知道粘度和初始条件的界限时，我们解决了预测冲击位置的问题。通过这个例子，我们展示了将 OUQ 应用于复杂物理系统的潜力，例如由耦合偏微分方程控制的系统。我们将我们的结果与使用标准蒙特卡罗方法获得的结果进行比较，并表明 OUQ 以较低的计算成本提供了更准确的边界。我们简要讨论了如何将这种方法扩展到更复杂的系统，以及如何将我们的方法整合到一个更雄心勃勃的优化实验设计计划中。