  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.
