Modelling, Computing and Data Applications for Real World Problems

本方案包括两个项目：1.发展高频波传播问题的有效数值格式。高频波传播的计算模拟在地震波、声波、电磁波和光波的许多应用中具有重要的意义。这个问题通常用亥姆霍兹方程来模拟。关于常波数的线性Helmholtz方程的求解，已经发表了大量的工作。相比之下，含不连续面的变波数Helmholtz方程和非线性Helmohtz方程得到的结果较少。在模拟克尔介质和地球物理中的电磁波传播时，会出现非线性问题和波数变化的问题。我们的工作将集中在开发高效的计算算法来解决这些困难的问题。2.将现实世界问题的模型、计算和数据与来自商业、科学、工程、医疗保健和医学研究的各种来源的海量数据进行常规生成和记录，我们计划探索如何利用现有数据来解决复杂的问题。具有机器学习算法的数据驱动方法已成为某些应用的有力工具，但仍有许多问题尚未解决。为了解决这一困难，我们考虑了另一种方法，通过整合数学模型、科学计算和数据来解决现实世界的问题。这一新方法的解决方案将用于验证数据驱动方法的结果。为了证明这种方法的有效性，我们研究了两个实际应用中的问题：(1)毒性评估中的化合物分类；(2)气动弹性响应中的异常检测和预测。这项建议需要进行强有力的HQP培训，学术界和工业界对具有应用数学、科学计算和数据科学工作经验并将其应用于现实世界问题的学员需求很高。

The present proposal consists of two projects: 1.Developing efficient numerical schemes for high frequency wave propagation problems. Computational simulation for high frequency wave propagation is important in many applications arising from seismic, acoustic, electromagnetic and optical waves. The problem is usually modelled by the Helmholtz equations. Many work has been published on solving linear Helmholtz equation with constant wave number. In contrast, there are fewer results available for Helmholtz equation with varying wave numbers containing discontinuities and the nonlinear Helmohtz equation. Nonlinear problems and varying wave numbers arise in modelling propagation of electromagnetic waves in Kerr media and geophysics. Our work will focus on developing efficient computational algorithms to solve these difficult problems. 2.Integrating model, computing and data for real world problems With massive data are now routinely generated and recorded from a wide range of sources in business, sciences, engineering, health care and medical research, we plan to explore how the available data can be utilized to solve complex problems. Data-driven method with the machine learning algorithm has emerged as a powerful tool for certain applications, but many questions remain unsolved. To address the difficulty, we consider another approach by integrating mathematical model, scientific computing and data to solve real world problems. The solution from this new approach will be used to validate the result from the data-driven method. To demonstrate the effectiveness of this approach, we examine two problems originated from two real applications: (1) chemical compounds classifications in toxicity assessment; (2) detection and prediction for anomaly in aeroelastic response. The proposal entails a strong HQP training, and trainees with working experience in applied mathematics, scientific computing and data sciences and its application for real world problems are in high demand in academic and industry sector.