Log In Sign Up

Diffusion Maps for Embedded Manifolds with Boundary with Applications to PDEs

by   Ryan Vaughn, et al.

Given only a collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to solve elliptic partial differential equations (PDEs) supplemented with boundary conditions. Notice that the construction of triangulations on unknown manifolds can be both difficult and expensive, both in terms of computational and data requirements, our goal is to solve these problems without such constructions. Instead, we rely only on using the sample points to define quadrature formulas on the unknown manifold. Our main tool is the diffusion maps algorithm. We re-analyze this well-known method in a weak (variational) sense. The latter reduces the smoothness requirements on the underlying functions which is crucial to approximating weak solutions to PDEs. As a by-product, we also provide a rigorous justification of the well-known relationship between diffusion maps and the Neumann eigenvalue problems. We then use a recently developed method of estimating the distance to boundary function (notice that the boundary location is assumed to be unknown and must be estimated from data) in order to correct the boundary error term in the diffusion maps construction. Finally, using this estimated distance, we illustrate how to impose Dirichlet, Neumann, and mixed boundary conditions for some common PDEs based on the Laplacian. Several numerical examples confirm our theoretical findings.


page 8

page 25

page 26


Ghost Point Diffusion Maps for solving elliptic PDE's on Manifolds with Classical Boundary Conditions

In this paper, we extend the class of kernel methods, the so-called diff...

Extended Dynamic Mode Decomposition for Inhomogeneous Problems

Dynamic mode decomposition (DMD) is a powerful data-driven technique for...

Can 4th-order compact schemes exist for flux type BCs?

In this paper new innovative fourth order compact schemes for Robin and ...

A diffusion-map-based algorithm for gradient computation on manifolds and applications

We recover the gradient of a given function defined on interior points o...

Enforcing exact boundary and initial conditions in the deep mixed residual method

In theory, boundary and initial conditions are important for the wellpos...

A ghost-point smoothing strategy for geometric multigrid on curved boundaries

We present a Boundary Local Fourier Analysis (BLFA) to optimize the rela...

Probabilistic Learning on Manifolds

This paper presents mathematical results in support of the methodology o...