Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel

abstract: We consider how a closed Riemannian manifold $M$ and its metric tensor $g$ can be approximately reconstructed from local distance measurements. Moreover, we consider an inverse problem of determining $(M,g)$ from limited knowledge on the heat kernel. In the part 1 of the paper, we considered the approximate construction of a smooth manifold in the case when one is given the noisy distances $\tilde{d}(x,y)=d(x,y)+\varepsilon_{x,y}$ for all points $x,y\in X$, where $X$ is a $\delta$-dense subset of $M$ and $|\varepsilon_{x,y}|<\delta$. In this part 2 of the paper, we consider a similar problem with partial data, that is, the approximate construction of the manifold $(M,g)$ when we are given $\tilde{d}(x,y)$ for $x\in X$ and $y\in U\cap X$, where $U$ is an open subset of $M$. In addition, we consider the inverse problem of determining the manifold $(M,g)$ with non-negative Ricci curvature from noisy observations of the heat kernel $G(y,z,t)$. We show that a manifold approximating $(M,g)$ can be determined in a stable way, when for some unknown source points $z_j$ in $X\setminus U$, we are given the values of the heat kernel $G(y,z_k,t)$ for $y\in X\cap U$ and $t\in (0,1)$ with a multiplicative noise. We also give a uniqueness result for the inverse problem in the case when the data does not contain noise and consider applications in manifold learning. A novel feature of the inverse problem for the heat kernel is that the set $M\setminus U$ containing the sources and the observation set $U$ are disjoint.