arXiv subjects

Tiexiang Li

Publications and source records attributed to Tiexiang Li.

17 recordsLinked to original sources

arXiv2026

A Novel Double Periodic Conformal Flattening Algorithm for Genus-One Surfaces

In this paper, we propose novel parameterization methods for genus-one surfaces, called the Double Periodic Conformal Flattening (DPCF) algorithm. The desired conformal map is obtained by minimizing a conformal energy functional under periodic boundary conditions, which is characterized as an easily solvable quadratic functional minimization problem, yielding a sparse linear system. The proposed DPCF algorithm offers several key advantages: (a) the optimal boundary and periodic translation vectors are obtained simultaneously with the conformal map; (b) the resulting map is independent of the chosen cutting path, thus introducing no extra conformal distortion near the cut seams; (c) bijectivity is guaranteed under the positive edge weights condition, which can be satisfied by, e.g., using an intrinsic Delaunay triangulation. Based on this guaranteeing, a simple strategy is employed to ensure bijectivity of the resulting maps for general triangulations. Numerical experiments illustrate that DPCF algorithm exhibits high accuracy and a 5 fold improvement over the state-of-the-art algorithm in terms of efficiency. Applications on texture mapping and medical imaging illustrate the practicality of our developed algorithm.

math.NA↗

arXiv2026

Null--Space--Free 6D Spectral Embedding with Local Rayleigh Quotient Recovery for 3D Quasiperiodic Maxwell's Eigenproblems

We develop a numerical framework for three-dimensional quasiperiodic Maxwell eigenvalue problems obtained through a six-dimensional periodic embedding. A projected Bloch--Fourier discretization yields a structured generalized eigenvalue problem with a large gradient-type kernel. Explicit orthonormal bases for the longitudinal and transverse subspaces remove this kernel exactly and reduce the original generalized eigenvalue problem to a null-space-free standard eigenvalue problem containing only the positive spectrum. An explicit inverse representation of the reduced operator avoids nested inner--outer linear solves and leads to an inverse Lanczos method whose main inner computation is a Hermitian positive definite conjugate-gradient solve with condition number bounded by that of the mass matrix; a residual estimate quantifies the effect of inner solves on the inverse Ritz pairs. To recover the computed modes in physical space, the six-dimensional Fourier eigenvectors are reconstructed on a three-dimensional Yee grid by a separated multi-center Taylor expansion, which avoids the dense Fourier-to-grid phase matrix and remains practical when direct dense reconstruction becomes prohibitively expensive. Local weighted Rayleigh quotients provide an independent physical-space validation, and their mass-weighted expectation is proved to equal the cropped Yee Rayleigh quotient under a partition-of-unity condition. Numerical experiments confirm the accuracy and computational effectiveness of the proposed framework and its physical-space recovery of three-dimensional quasiperiodic Maxwell modes.

math.NA↗

arXiv2026

A Unified DeepONet Framework for Logarithmically Stable Infinite-Dimensional Inverse Problems

We develop a unified DeepONet framework for logarithmically stable inverse problems between infinite-dimensional function spaces, with inverse acoustic scattering as a model application. The framework is formulated at the operator level by separating the learned inverse map into measurement encoding, finite-dimensional neural approximation, and functional reconstruction components. For inverse maps satisfying a logarithmic stability estimate, we establish quantitative a priori error bounds that separate the encoder, finite-dimensional neural approximation, and reconstruction contributions. For prescribed encoder and reconstruction ranks, we obtain a network-size-dependent bound for the finite-dimensional neural approximation error, together with rank-dependent bounds for the encoding and reconstruction errors. For comparison, we also record the corresponding Lipschitz-stable estimate arising from the same error decomposition. The quantitative inverse-scattering analysis is then specialized, in three dimensions, to the recovery of a medium contrast from fixed-frequency far-field measurements. Numerical experiments in two and three dimensions, using both function-space and finite-dimensional priors, illustrate the reconstruction performance and empirical sensitivity to synthetic measurement noise.

math.NA↗

arXiv2026

A Novel Bijective Angle and Volume-preservation Balanced Parameterization for $n$-dimensional Manifolds

We propose a unified framework for balanced and bijective parameterizations of $n$-dimensional manifolds. The proposed energy combines conformal and volume-preserving terms to control both local anisotropy and volumetric distortion. At the continuous level, both energies are nonnegative and their zero-energy mappings are characterized. At the discrete level, the conformal, volume-preserving, and logarithmic barrier energies are formulated on oriented simplicial manifolds. A key result is that all their gradients admit a unified cotangent Laplacian-type representation, enabling sparse and dimension-independent computation. Bijectivity is enforced through signed simplex Jacobians, feasibility restoration, and a strictly orientation-preserving logarithmic barrier. The framework applies uniformly to spherical boundary parameterizations and parameterizations of discrete $n$-manifolds onto ball-like canonical domains.

math.NA↗

arXiv2026

A Natural Decomposition Method for Essential Boundary Conditions in Noninterpolatory Meshfree Spaces

This paper develops a natural decomposition method (NDM)for imposing essential boundary conditions in noninterpolatory meshfree Galerkin spaces without boundary parameter tuning or auxiliary constraint construction. In such spaces, algebraic coefficients generally do not coincide with boundary values; hence coefficient assignment or nodal boundary prescription is not equivalent to imposing the continuous trace required by the variational problem. NDM introduces boundary data before discretization through a natural transfer mechanism: a source subproblem accounts for the forcing term, a weighted curl correction transfers the remaining trace mismatch, and a scalar recovery step reconstructs the solution from the corrected weighted gradient. For topologically trivial single domains with connected boundary, the reconstructed solution is equivalent, at the continuous level, to the solution satisfying the prescribed essential boundary data. The discrete analysis separates the approximation defect of the recovery space from the upstream transfer error visible to that space. Numerical experiments on benchmark problems evaluate the proposed transfer mechanism and report the associated conditioning, computational cost, and boundary perturbation behavior.

math.NA↗

arXiv2026

An Efficient Parity-Blocked Method for Band-Structure Computation of 3D Anisotropic Phononic Crystals

Band-structure calculations for three-dimensional anisotropic phononic crystals require the repeated solution of large elastic generalized eigenvalue problems along Bloch paths. In standard staggered-grid discretizations, anisotropic coupling may involve derivative components located at incompatible grid positions, so additional interpolation or averaging closures are often introduced. This paper proposes a parity-blocked rotated staggered discretization based on four Bloch-periodic body-diagonal differences. The directional derivatives are reconstructed from these diagonal differences, leading to a Hermitian $B_hC_hB_h^H$ generalized eigenvalue formulation that incorporates anisotropic derivative coupling without separate interpolation closures. On even grids, when the stiffness and mass matrices are nodewise local multiplication matrices, the body-diagonal shifts preserve two independent parity invariants. The discrete velocity space is then decomposed exactly into four mutually independent block subspaces, and the full discrete spectrum can be recovered by solving the four smaller eigenvalue problems and merging their spectra. The full and block formulations are further organized in a unified Fourier SVD framework, which supports $Γ$-point zero-mode treatment, shift-invert Krylov iteration, inner PCG solves, and GPU matrix-vector products. Numerical experiments for a three-dimensional two-phase anisotropic phononic crystal show that the block implementation preserves the full-space spectrum while substantially reducing the wall-clock time. The results demonstrate that the proposed method provides a structured and efficient solver for large-scale band-structure computations of three-dimensional anisotropic phononic crystals.

math.NA↗

arXiv2026

A Novel Computational and Analytical Framework for 2D Quasiperiodic Helmholtz Eigenvalue Problems via the Projection Method

In this paper, we propose a spectral framework that embeds 1D and 2D quasiperiodic Helmholtz eigenvalue problems into higher-dimensional (2D and 4D) periodic spaces via the projection method \cite{jiang2014numerical, jiang2024numerical}. To effectively map the elevated high-dimensional states back to the original physical space, we establish a novel validation framework based on the weighted expectation of pointwise Rayleigh quotients. Supported by comprehensive error and spectral analysis, we demonstrate that the eigenvalues derived from this expectation align more authentically with the original quasiperiodic model, ultimately yielding a more appropriate and reliable eigenpair solution. Numerical experiments on continuous media demonstrate that our approach offers an accurate, robust, and scalable tool for solving quasiperiodic Helmholtz eigenvalue problems.

math.NA↗

arXiv2026

A Unifying Framework for Doubling Algorithms

The existing doubling algorithms have been proven efficient for several important nonlinear matrix equations arising from real-world engineering applications. In a nutshell, the algorithms iteratively compute a basis matrix, in one of the two particular forms, for the eigenspace of some matrix pencil associated with its eigenvalues in certain complex region such as the left-half plane or the open unit disk, and their success critically depends on that the interested eigenspace do have a basis matrix taking one of the two particular forms. However, that requirement in general cannot be guaranteed. In this paper, a new doubling algorithm, called the $Q$-doubling algorithm, is proposed. It includes the existing doubling algorithms as special cases and does not require that the basis matrix takes one of the particular forms. An application of the $Q$-doubling algorithm to solve eigenvalue problems is investigated with numerical experiments that demonstrate its superior robustness to the existing doubling algorithms.

math.NA↗

arXiv2025

Rapid general Electromagnetic Analysis with computational conformal geometry via Conformal Energy Minimization

We recently found that the electromagnetic scattering problem can be very fast in an approach expressing the fields in terms of orthonormal basis functions. In this paper we apply computational conformal geometry with the conformal energy minimization (CEM) algorithm to make possible fast solution of finite-frequency electromagnetic problems involving arbitrarily shaped, simply-connected metallic surfaces. The CEM algorithm computes conformal maps with minimal angular distortion, enabling the transformation of arbitrary simply-connected surfaces into a disk, where orthogonal basis functions can be defined and electromagnetic analysis can be significantly simplified. We demonstrate the effectiveness and efficiency of our method by investigating the resonance characteristics of two metallic surfaces: a square plate and a four-petal plate. Compared to traditional finite element methods (e.g., COMSOL), our approach achieves a three-order-of-magnitude improvement in computational efficiency, requiring only seconds to extract resonant frequencies and fields. Moreover, it reveals low-energy, doubly degenerate resonance modes that are elusive to conventional methods. These findings not only provide a powerful tool for analyzing electromagnetic fields on complex geometries but also pave the way for the design of high-performance electromagnetic devices.

physics.optics↗

arXiv2024

$n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations

In this paper, we develop an $n$ dimensional volumetric stretch energy ($n$-VSE) functional for the volume-/mass-preserving parameterization of the $n$-manifolds topologically equivalent to $n$-ball. The $n$-VSE has a lower bound and equal to it if and only if the map is volume-/mass-preserving. This motivates us to minimize the $n$-VSE to achieve the ideal volume-/mass-preserving parameterization. In the discrete case, we also guarantee the relation between the lower bound and the volume-/mass-preservation, and propose the spherical and ball volume-/mass-preserving parameterization algorithms. The numerical experiments indicate the accuracy and robustness of the proposed algorithms. The modified algorithms are applied to the manifold registration and deformation, showing the versatility of $n$-VSE.

math.NA↗

arXiv2023

A Robust Hessian-based Trust Region Algorithm for Spherical Conformal Parameterizations

Surface parameterizations are widely applied in computer graphics, medical imaging and transformation optics. In this paper, we rigorously derive the gradient vector and Hessian matrix of the discrete conformal energy for spherical conformal parameterizations of simply connected closed surfaces of genus-$0$. In addition, we give the sparsity structure of the Hessian matrix, which leads to a robust Hessian-based trust region algorithm for the computation of spherical conformal maps. Numerical experiments demonstrate the local quadratic convergence of the proposed algorithm with low conformal distortions. We subsequently propose an application of our method to surface registrations that still maintains local quadratic convergence.

math.NA↗

arXiv2022

A mixed element scheme of Helmholtz transmission eigenvalue problem for anisotropic media

In this paper, we study the Helmholtz transmission eigenvalue problem for inhomogeneous anisotropic media with the index of refraction $n(x)\equiv 1$ in two and three dimension. Starting with a nonlinear fourth order formulation established by Cakoni, Colton and Haddar [2009], by introducing some auxiliary variables, we present an equivalent mixed formulation for this problem, followed up with the finite element discretization. Using the proposed scheme, we rigorously show that the optimal convergence rate for the transmission eigenvalues both on convex and nonconvex domains can be expected. Moreover, by this scheme, we will obtain a sparse generalized eigenvalue problem whose size is so demanding even with a coarse mesh that its smallest few real eigenvalues fail to be solved by the shift and invert method. We partially overcome this critical issue by deflating the almost all of the $\infty$ eigenvalue of huge multiplicity, resulting in a drastic reduction of the matrix size without deteriorating the sparsity. Extensive numerical examples are reported to demonstrate the effectiveness and efficiency of the proposed scheme.

math.NA↗

arXiv2020

Bifurcation Analysis of the Eigenstructure of the Discrete Single-curl Operator in Three-dimensional Maxwell's Equations with Pasteur Media

This paper focuses on studying the bifurcation analysis of the eigenstructure of the $γ$-parameterized generalized eigenvalue problem ($γ$-GEP) arising in three-dimensional (3D) source-free Maxwell's equations with Pasteur media, where $γ$ is the magnetoelectric chirality parameter. For the weakly coupled case, namely, $γ< γ_{*} \equiv$ critical value, the $γ$-GEP is positive definite, which has been well-studied by Chern et.\ al, 2015. For the strongly coupled case, namely, $γ> γ_{*}$, the $γ$-GEP is no longer positive definite, introducing a totally different and complicated structure. For the critical strongly coupled case, numerical computations for electromagnetic fields have been presented by Huang et.\ al, 2018. In this paper, we build several theoretical results on the eigenstructure behavior of the $γ$-GEPs. We prove that the $γ$-GEP is regular for any $γ> 0$, and the $γ$-GEP has $2 \times 2$ Jordan blocks of infinite eigenvalues at the critical value $γ_{*}$. Then, we show that the $2 \times 2$ Jordan block will split into a complex conjugate eigenvalue pair that rapidly goes down and up and then collides at some real point near the origin. Next, it will bifurcate into two real eigenvalues, with one moving toward the left and the other to the right along the real axis as $γ$ increases. A newly formed state whose energy is smaller than the ground state can be created as $γ$ is larger than the critical value. This stunning feature of the physical phenomenon would be very helpful in practical applications. Therefore, the purpose of this paper is to clarify the corresponding theoretical eigenstructure of 3D Maxwell's equations with Pasteur media.

math.NA↗

arXiv2019

On the solvability of an indefinite nonlinear Kirchhoff equation via associated eigenvalue problems

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}dx\right) Δu+μV\left( x\right) u=Q(x)\left\vert u\right\vert ^{p-2}u+λf\left( x\right) u\text{ in }\mathbb{R}^{N}, \\ u\in H^{1}\left( \mathbb{R}^{N}\right) ,% \end{array}% \right. \end{equation*}% where $N\geq 3,2 0\right) ,$ the potential $V$ is a nonnegative function in $\mathbb{R}% ^{N}$ and the weight function $Q\in L^{\infty }\left( \mathbb{R}^{N}\right) $ with changes sign in $\overline{Ω}:=\left\{ V=0\right\} .$ We mainly prove the existence of at least two positive solutions in the cases that $% \left( i\right) $ $2 0$ sufficiently large, where $λ_{1}\left( f_{Ω}\right) $ is the first eigenvalue of $-Δ$ in $% H_{0}^{1}\left( Ω\right) $ with weight function $f_{Ω}:=f|_{% \overline{Ω}},$ whose corresponding positive principal eigenfunction is denoted by $ϕ_{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,λ$ belongs to different intervals.

math.AP↗

arXiv2018

Solving Three Dimensional Maxwell Eigenvalue Problem with Fourteen Bravais Lattices

Calculation of band structure of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalue. In this paper, we try to address this problem in such a broad context that band structure of three dimensional isotropic photonic crystals with all 14 Bravais lattices can be efficiently computed in a unified framework. We uncover the delicate machinery behind several key results of our work and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments in our framework. In this work particular effort is made on reformulating the Bloch boundary condition for all 14 Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by systematic use of commutativity among them, which has been overlooked previously, and reducing eigen-decomposition of double-curl operator to the canonical form of a 3x3 complex skew-symmetric matrix under unitary congruence. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm.

math.NA↗

arXiv2018

Structure-Preserving ΓQR and Γ-Lanczos Algorithms for Bethe-Salpeter Eigenvalue Problems

To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called ΓQR algorithm and Γ-Lanczos algorithm, are proposed in this paper. Both algorithms preserve the special structure of the initial matrix $H=\begin{bmatrix}A & B-\overline{B} & -\overline{A}\end{bmatrix}$, resulting the computed eigenvalues and the associated eigenvectors still hold the properties similar to those of $H$. Theorems are given to demonstrate the validity of the proposed two algorithms in theory. Numerical results are presented to illustrate the superiorities of our methods.

math.NA↗

arXiv2017

Intermittent behaviors in weakly coupled map lattices

In this paper, we study intermittent behaviors of coupled piecewise-expanding map lattices with two nodes and a weak coupling. We show that the successive phase transition between ordered and disordered phases occurs for almost every orbit. That is, we prove $\liminf_{n\rightarrow \infty}| x_1(n)-x_2(n)|=0$ and $\limsup_{n\rightarrow \infty}| x_1(n)-x_2(n)|\ge c_0>0$, where $x_1(n), x_2(n)$ correspond to the coordinates of two nodes at the iterative step $n$. We also prove the same conclusion for weakly coupled tent-map lattices with any multi-nodes.

math.DS↗