arXiv subjects

Jinpeng Lu

Publications and source records attributed to Jinpeng Lu.

27 records · Page 2Linked to original sources

arXiv2024

Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem

We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.

math.CO↗

arXiv2023

A Localization-to-Segmentation Framework for Automatic Tumor Segmentation in Whole-Body PET/CT Images

Fluorodeoxyglucose (FDG) positron emission tomography (PET) combined with computed tomography (CT) is considered the primary solution for detecting some cancers, such as lung cancer and melanoma. Automatic segmentation of tumors in PET/CT images can help reduce doctors' workload, thereby improving diagnostic quality. However, precise tumor segmentation is challenging due to the small size of many tumors and the similarity of high-uptake normal areas to the tumor regions. To address these issues, this paper proposes a localization-to-segmentation framework (L2SNet) for precise tumor segmentation. L2SNet first localizes the possible lesions in the lesion localization phase and then uses the location cues to shape the segmentation results in the lesion segmentation phase. To further improve the segmentation performance of L2SNet, we design an adaptive threshold scheme that takes the segmentation results of the two phases into consideration. The experiments with the MICCAI 2023 Automated Lesion Segmentation in Whole-Body FDG-PET/CT challenge dataset show that our method achieved a competitive result and was ranked in the top 7 methods on the preliminary test set. Our work is available at: https://github.com/MedCAI/L2SNet.

eess.IV↗

arXiv2023

Gel'fand's inverse problem for the graph Laplacian

We study the discrete Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: $X=B\cup G$, where $B$ is called the set of the boundary vertices and $G$ is called the set of the interior vertices. We consider the case where the vertices in the set $G$ and the edges connecting them are unknown. Assume that we are given the set $B$ and the pairs $(λ_j,ϕ_j|_B)$, where $λ_j$ are the eigenvalues of the graph Laplacian and $ϕ_j|_B$ are the values of the corresponding eigenfunctions at the vertices in $B$. We show that the graph structure, namely the unknown vertices in $G$ and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset $S\subseteq G$ of cardinality $|S|\geqslant 2$ contains two extreme points. A point $x\in S$ is called an extreme point of $S$ if there exists a point $z\in B$ such that $x$ is the unique nearest point in $S$ from $z$ with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.

math.SP↗

arXiv2023

Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem

We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface.

math.AP↗

arXiv2023

Inverse problems for discrete heat equations and random walks for a class of graphs

We study the inverse problem of determining a finite weighted graph $(X,E)$ from the source-to-solution map on a vertex subset $B\subset X$ for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on $B$. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under a novel geometric condition (called the Two-Points Condition), the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$, or from the observation on $B$ of one realization of the random walk.

math.SP↗

arXiv2022

Approximations of the connection Laplacian spectra

We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph connection Laplacian. We prove that for Euclidean or Hermitian connections on closed Riemannian manifolds, the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian.

math.AP↗

arXiv2022

Inverse problems for locally perturbed lattices -- Discrete Hamiltonian and quantum graph

We consider the inverse scattering problems for two types of Schrödinger operators on locally perturbed periodic lattices. For the discrete Hamiltonian, the knowledge of the S-matrix for all energies determines the graph structure and the coefficients of the Hamiltonian. For locally perturbed equilateral metric graphs, the knowledge of the S-matrix for all energies determines the graph structure.

math-ph↗

arXiv2019

Graph approximations to the Laplacian spectra

I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension.

math.DG↗

arXiv2012

On stable hypersurfaces with constant mean curvature in Euclidean spaces

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete stable hypersurfaces with constant mean curvature of arbitrary dimension, given a finite $L^p$-norm curvature condition.

math.DG↗