课程名称: Approximation Theory and Applications
授课人： Christoph Ortner (University of British Columbia)
授课人简介： Christoph Ortner is Professor of Mathematics at UBC (Vancouver). After his D.Phil. (Ph.D.) in Numerical Analysis in Oxford (2006) CO stayed on as an RCUK Academic Fellow in Solid Mechanics and Mathematics of Materials. In 2011 CO joined the Warwick Mathematics Institute. CO’s interdisciplinary work spans applied analysis, numerical analysis, scientific computing and atomistic (material) modelling. A substantial component of his research to date has been the development of the mathematical theory of multi-scale methods (atomistic-to-continuum, QM/MM) for materials and in particular material defects. This work was recognized by a Philip Leverhulme Prize (2012), ERC Starting Grant (2012), Whitehead Prize (2015) the Oberwolfach John Todd Award (2017) and an ERC Consolidater Grant (2020). His main research interests now are mathematical aspects of data-driven coarse-graining methods.
授课人主页： http://www.math.ubc.ca/~ortner/
授课方式：远程线上授课 （后主楼1124）
授课时间：2021年3月20日（周六） 10:00-10:45，11:00-11:45，13:00-13:45
2021年3月27日（周六） 10:00-10:45，11:00-11:45，13:00-13:45
2021年4月3日（周六） 10:00-10:45，11:00-11:45，13:00-13:45
2021年4月10日（周六） 10:00-10:45，11:00-11:45，13:00-13:45
2021年4月17日（周六） 10:00-10:45，11:00-11:45，13:00-13:45
2021年4月24日（周六） 10:00-10:45，11:00-11:45，13:00-13:45
课程简介：
Outline: This mini-course will introduce students to approximation theory, and its applications for the solution of differential equations and regression. To keep it concrete we will focus on trigonometric polynomial approximation, but will also discuss alternatives, and general concepts, in particular connections with modern numerical analysis of PDEs and machine learning to motivate further study. The emphasis is on intermixing of theory and computational experiment to provide students with an intuitive connection between theory and practise, much in the style of [3, 4]. Course material will be drawn from the list of references below.
Tentative Syllabus:
Day 1: Approximation with trigonometric polynomials: approximation rates, Cj regularity, Jackson’s theorems, analyticity, Payley-Wiener theorems; Outlook: dim 2, 3; algebraic polynomials; rational approximation
Day 2: Spectral methods: differential operators in reciprocal space, discretisation of BVPs, convergence, implementation; Outlook: spectral methods with algebraic polynomials
Day 3: Least Squares: linear least squares, fitting and approximation, iteratively reweighted least squares, inverse problems, noisy training data; Outlook: uncertainty quantification for linear and nonlinear regression
Day 4: Approximation in high dimension: curse of dimensionality, regularity in high dimension, sparse polynomials, limitations; Outlook: kernel ridge regression and artificial neural networks
Day 5: Extended research talk: machine-learning an interatomic potential using linear regression with a symmetric polynomial basis.
References:
[1] Albert Cohen, Mark A Davenport, and Dany Leviatan. On the stability and accuracy of least squares approximations. Found. Comput. Math., 13(5):819–834, 2013.
[2] M J D Powell. Approximation Theory and Methods. Cambridge University Press, 1981.
[3] Lloyd N Trefethen. Spectral Methods in MATLAB. SIAM, 2000.
[4] Lloyd N Trefethen. Approximation Theory and Approximation Practice. SIAM, 2013.
[5] M. Bachmayr, G. Csanyi, G. Dusson, S. Etter, C. van der Oord, and C. Ortner. Atomic Cluster Expansion: Completeness, Efficiency and Stability.arXiv:1911.035
