Stable, Efficient, Adaptive Algorithms for Approximation and Integration

计算方法允许模拟那些太昂贵、太危险或在物理情况下不可行的实验。 例如，评估核反应堆设计的安全性和效率，预测电网故障的频率，以及评估金融投资的风险和回报。 该研究项目将开发计算代理模型的新方法，用于耗时的模拟。 这些新的代理模型将更快地计算，避免灾难性的计算机错误，并更忠实地代表他们的设计模型的过程。 探索未来情景的更聪明的方法将被开发出来，以减少找到上述复杂系统的平均或最坏可能行为所需的计算时间。 从这项研究中产生的新算法将公开提供给其他研究人员和从业人员。 参与开发这些算法的本科生和研究生将为科学生涯做好更好的准备。函数逼近和积分是计算数学中的两个基本问题。本研究计画将建构演算法，使函数逼近与积分在计算上稳定，避免灾难性的舍入误差。 这些算法将是自适应的，根据功能数据确定算法参数，以满足用户指定的误差容限，并具有严格的理由。该算法也将是渐近有效的，基本上需要相同的计算工作作为最好的可能的算法。 函数近似算法将基于研究人员为核方法开发的希尔伯特-施密特SVD分解。 将开发自适应确定所需样本量的积分算法。 研究人员将继续指导学生和博士后学者，并将通过与领域专家的合作来确定正在开发的算法的有用性。

Computational methods allow the simulation of experiments that are too costly, too dangerous, or otherwise infeasible to perform in physical situations. Examples include evaluating the safety and efficiency of designs for nuclear reactors, predicting the frequency of breakdowns in power grids, and assessing the risks and rewards of financial investments. This research project will develop new ways of computing surrogate models for time-consuming simulations. These new surrogate models will be quicker to compute, avoid catastrophic computer error, and more faithfully represent the processes that they are designed to model. More clever ways of exploring future scenarios will be developed to decrease the computational time required to find the average, or the worst possible, behavior of complex systems such as those mentioned. The new algorithms arising from this research will be made publicly available for other researchers and practitioners. Undergraduate and graduate students involved in developing these algorithms will be better prepared for scientific careers.Function approximation and integration are two fundamental problems in computational mathematics. This research project will construct algorithms for function approximation and integration that are computationally stable, avoiding catastrophic round-off error. These algorithms will be adaptive, determining the algorithm parameters based on function data to meet the user-specified error tolerance with rigorous justification. The algorithms will also be asymptotically efficient, requiring essentially the same computational effort as the best possible algorithms. Function approximation algorithms will be based on the Hilbert-Schmidt SVD decomposition that the investigators have developed for kernel methods. Integration algorithms will be developed that adaptively determine the sample size required. The investigators will continue to mentor students as well as post-doctoral scholars and will establish the usefulness of the algorithms under development through collaborations with domain specialists.