Email RIS file
Bookmark
Report an Issue

Mesh-less methods for solving PDEs on unknown manifolds

Author
Yan, Qile
Additional Titles
Mesh-less methods for solving partial differential equations on unknown manifolds
Published
[University Park, Pennsylvania] : Pennsylvania State University, 2023.
Physical Description
1 electronic document
Additional Creators
Harlim, John
Access Online

Availability

I Want It
I Want It

Finding items...

Graduate Program
Restrictions on Access
Restricted (PSU Only).
Summary
In this dissertation, we propose several mesh-less methods for solving partial differential equations (PDEs) on unknown manifolds identified with randomly sampled point cloud data. The first method is designed to solve the time-dependent advection-diffusion PDE on manifolds without and with boundaries. It is an extension of the class of kernel methods, the so-called diffusion maps and ghost point diffusion maps. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. Our second method is motivated by the need for an efficient PDE solver to facilitate inverse problem algorithms for estimating parameters of PDE. The proposed PDE solver is formulated as a spectral method. Here, the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. The third mesh-less method proposed in this dissertation is a finite difference type approximation for estimating the differential operators which results in a sparse discrete operator. The efficiency shown by numerical examples suggests its possible application in solving a wider range of problems such as time-dependent vector PDEs on manifolds. When possible, we provide convergence studies of these methods with error bounds in terms of the sample size under appropriate assumptions.
Other Subject(s)
Genre(s)
Dissertation Note
Ph.D. Pennsylvania State University 2023.
Technical Details
The full text of the dissertation is available as an Adobe Acrobat .pdf file ; Adobe Acrobat Reader required to view the file.
View MARC record | catkey: 42248675