Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913865437020160 |
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| author | Fefferman, Charles Ivanov, Sergei Lassas, Matti Lu, Jinpeng Narayanan, Hariharan |
| author_facet | Fefferman, Charles Ivanov, Sergei Lassas, Matti Lu, Jinpeng Narayanan, Hariharan |
| contents | 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 $δ$-dense subset of $M$ and $|\varepsilon_{x,y}|<δ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_14528 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel Fefferman, Charles Ivanov, Sergei Lassas, Matti Lu, Jinpeng Narayanan, Hariharan Differential Geometry Analysis of PDEs 35R30, 53C20, 58J35 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 $δ$-dense subset of $M$ and $|\varepsilon_{x,y}|<δ$. 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. |
| title | Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel |
| topic | Differential Geometry Analysis of PDEs 35R30, 53C20, 58J35 |
| url | https://arxiv.org/abs/2111.14528 |