Syllabus
Updated 1/9/23 The details listed in the syllabus are subject to change during the term.
Course: Math 273A
Title: Advanced Techniques in Computational Mathematics (Topics in meshfree and particle methods)
Credit Hours: 4
Course description: This course includes a few topics in meshfree and particle methods. You will learn the mathematical foundations of scattered data approximation and their applications to numerical PDEs. We will discuss kernel-based function interpolation techniques such as moving least squares (MLS) and radial basis functions (RBF). Related concepts include weight/window function, local polynomial reproduction, positive definite functions, conditionally positive definite functions, reproducing kernel Hilbert spaces. For applications, we will discuss a few meshfree and particle methods for numerically solving PDEs (meshfree collocation method, meshfree Galerkin method, Lagrangian particle methods for time-dependent PDEs). One of our primary goals in introducing these concepts is to provide you with the necessary foundation for reading research papers in related fields. A paper list (continuously updating) is given and you are encouraged to read them early and give one or multiple presentations of the papers. Grades will be based on class participation and presentations.
Primary Reference: H. Wendland, Scatter Data Approximation, Cambridge University Press, 2004.
Additional readings: • S. Li and W. K. Liu, Meshfree Particle Methods, Springer Science & Business Media, 2007. • PA Raviart, An analysis of particle methods, Numerical methods in fluid dynamics, Springer, Berlin, Heidelberg, 1985. 243-324. • G.E. Fasshauer, Meshfree approximation methods with MATLAB, World Scientific, 2007. • L.C. Evans, Partial differential equations, American Mathematical Society, 2010.