arXiv subjects

Tianhao Ni

Publications and source records attributed to Tianhao Ni.

6 recordsLinked to original sources

arXiv2026

DTPFI: A stable algorithm for recovering nonlinear energy potentials in phase field systems

This work proposes a Dual Time Phase Field Inversion (DTPFI) method for recovering unknown potential functions in phase field models. The reconstruction is formulated as an optimization problem that minimizes the mismatch between model predictions and observed fields at the final measurement time. We prove the differentiability of the measured field with respect to the unknown potential and establish the local convexity of the regularized objective function, thereby ensuring the existence of a local optimal solution. For numerical implementation, automatic differentiation is employed to compute gradients, avoiding the expensive evaluation of analytic gradients. Extensive numerical experiments demonstrate that DTPFI accurately reconstructs both polynomial and logarithmic potentials and remains robust under measurement noise. The framework is further extended to inverse problems involving field dependent mobility and joint parameter identification in coupled Cahn-Hilliard-Allen-Cahn systems.

math.NA↗

arXiv2025

Enhanced High-Dimensional Data Visualization through Adaptive Multi-Scale Manifold Embedding

To address the dual challenges of the curse of dimensionality and the difficulty in separating intra-cluster and inter-cluster structures in high-dimensional manifold embedding, we proposes an Adaptive Multi-Scale Manifold Embedding (AMSME) algorithm. By introducing ordinal distance to replace traditional Euclidean distances, we theoretically demonstrate that ordinal distance overcomes the constraints of the curse of dimensionality in high-dimensional spaces, effectively distinguishing heterogeneous samples. We design an adaptive neighborhood adjustment method to construct similarity graphs that simultaneously balance intra-cluster compactness and inter-cluster separability. Furthermore, we develop a two-stage embedding framework: the first stage achieves preliminary cluster separation while preserving connectivity between structurally similar clusters via the similarity graph, and the second stage enhances inter-cluster separation through a label-driven distance reweighting. Experimental results demonstrate that AMSME significantly preserves intra-cluster topological structures and improves inter-cluster separation on real-world datasets. Additionally, leveraging its multi-resolution analysis capability, AMSME discovers novel neuronal subtypes in the mouse lumbar dorsal root ganglion scRNA-seq dataset, with marker gene analysis revealing their distinct biological roles.

cs.LG↗

arXiv2024

Inverse problems for a quasilinear hyperbolic equation with multiple unknowns

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=Δ_g(u+F(x, u))+G(x, u)$ on a compact Riemannian manifold $(M, g)$ with boundary. We show that if $F(x, u)$ is monomial and $G(x, u)$ is analytic in $u$, then $F, G$ and $c$ as well as the associated initial data can be uniquely determined and reconstructed by the corresponding hyperbolic DtN (Dirichlet-to-Neumann) map. Our work leverages the construction of proper Gaussian beam solutions for quasilinear hyperbolic PDEs as well as their intriguing applications in conjunction with light-ray transforms and stationary phase techniques for related inverse problems. The results obtained are also of practical importance in assorted of applications with nonlinear waves.

math.AP↗

arXiv2024

A uniqueness theory on determining the nonlinear energy potential in phase-field system

The phase-field system is a nonlinear model that has significant applications in material sciences. In this paper, we are concerned with the uniqueness of determining the nonlinear energy potential in a phase-field system consisting of Cahn-Hilliard and Allen-Cahn equations. This system finds widespread applications in the development of alloys engineered to withstand extreme temperatures and pressures. The goal is to reconstruct the nonlinear energy potential through the measurements of concentration fields. We establish the local well-posedness of the phase-field system based on the implicit function theorem in Banach spaces. Both of the uniqueness results for recovering time-independent and time-dependent energy potential functions are provided through the higher order linearization technique.

math.AP↗

arXiv2023

Inverse problems for nonlinear progressive waves

We propose and study several inverse problems associated with the nonlinear progressive waves that arise in infrasonic inversions. The nonlinear progressive equation (NPE) is of a quasilinear form $\partial_t^2 u=Δf(x, u)$ with $f(x, u)=c_1(x) u+c_2(x) u^n$, $n\geq 2$, and can be derived from the hyperbolic system of conservation laws associated with the Euler equations. We establish unique identifiability results in determining $f(x, u)$ as well as the associated initial data by the boundary measurement. Our analysis relies on high-order linearisation and construction of proper Gaussian beam solutions for the underlying wave equations. In addition to its theoretical interest, we connect our study to applications of practical importance in infrasound waveform inversion.

math.AP↗

arXiv2022

A Modified Nonlinear Conjugate Gradient Algorithm for Functions with Non-Lipschitz Gradient

In this paper, we propose a modified nonlinear conjugate gradient (NCG) method for functions with a non-Lipschitz continuous gradient. First, we present a new formula for the conjugate coefficient β_k in NCG, conducting a search direction that provides an adequate function decrease. We can derive that our NCG algorithm guarantees strongly convergent for continuous differential functions without Lipschitz continuous gradient. Second, we present a simple interpolation approach that could automatically achieve shrinkage, generating a step length satisfying the standard Wolfe conditions in each step. Our framework considerably broadens the applicability of NCG and preserves the superior numerical performance of the PRP-type methods.

math.NA↗