大规模微分方程反问题的算法设计与应用

Inverse problem is the most rapidly developing and growing fields of applied mathematics in recent years, which holds a broad application background. Moreover, the popularity of big data sets an even higher standard for the designing of algorithms to tackle all types of inverse problems. In this project, we will mainly concentrate on solving several types of large-scale inverse problem on differential equations. We would analyze deeply on the theoretical properties of the problems, and fathom the characteristics, to figure out the proper reformulation of the model, and materialize dimension reduction on the problem, to finally design efficient computational algorithms for solving extremely large-scale inverse problems on differential equation. In the meantime, in this project, we would research on the theoretical properties of the problem, which contains: 1) Analyzing complexity of the algorithms, and propose the proof of global convergence and the conditions of converging to the global optimal solution. 2) Analyzing the model, and discuss the conditions under which the optimal solution of the model coincides with the real solution. Finally, inasmuch as the problems we analyze in this project originate from applications, the algorithms above should be applied to the applications to be tested for their performance. We would apply the newly designed algorithms to problems in fields like life sciences and financial mathematics, on problems which are hardly conquered by traditional methods, and try to answer several core questions highly concerning researchers, to stimulate the development of interdisciplinary research. We look forward to establishment of the project, to carry more favorable theoretic research and practical applications.

反问题是近年来应用数学领域中发展和成长最快的领域之一，具有广泛的应用背景。而近年来大数据的流行，更是为反问题的算法设计提出了更高的要求。本项目将主要围绕几类大规模微分方程反问题展开研究。我们将深入分析问题的理论性质，抓住问题的特殊性，寻找对模型的合理变换，实现对问题的降维，并由此设计出可以求解大规模微分方程反问题的高效数值算法。同时，本项目还将对问题的理论性质展开研究，这主要包含两个方面：一是分析所设计算法的复杂度，证明算法的全局收敛性及给出收敛到全局最优解的条件；二是分析建模本身的合理性，讨论在何种条件下，模型的最优解与真实解吻合。最后，本项目的问题来源于实际，最终也需要回到实际中去检验算法的实际表现。我们将把这些新算法应用在生命科学与金融数学等领域中的一些传统算法难以解决的问题上，回答这些领域中所关心的几个核心问题，促进交叉学科的发展。我们期待该课题的立项，以便开展理论研究和实际应用。

反问题是近年来应用数学领域中发展和成长最快的领域之一，具有广泛的应用背景。而近年来大数据的流行，更是为反问题的算法设计提出了更高的要求。本项目主要围绕两类大规模微分方程反问题展开了研究，即参数型微分方程反问题以及欠定型稀疏微分方程反问题。我们深入分析了问题的理论性质，抓住问题的特殊性，寻找对模型的合理变换，成功实现了对问题的降维，并由此设计出了可以求解大规模微分方程反问题的高效数值算法，并在理论上分析了问题与算法的理论性质。目前，我们的工作已经在顶级期刊上发表，并将代码在Fortran，Matlab，R等多个平台实现，目前已在业界广泛使用。而我们的后续工作包括把这个方法在稀疏型微分方程反问题上使用，目前已经进得了不错的进展，文章已经整理完成并投稿。同时，我们也将类似的思想用到优化领域的其他问题上，目前已经有一篇文章发表，另有一篇被接收。