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Wen-Wei Lin

Publications and source records attributed to Wen-Wei Lin.

At least 19 recordsLinked to original sources
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.

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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.

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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.

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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 $\Gamma$-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.

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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.

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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.

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arXiv2024

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.

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arXiv2024

An Attention-based Framework with Multistation Information for Earthquake Early Warnings

Earthquake early warning systems play crucial roles in reducing the risk of seismic disasters. Previously, the dominant modeling system was the single-station models. Such models digest signal data received at a given station and predict earth-quake parameters, such as the p-phase arrival time, intensity, and magnitude at that location. Various methods have demonstrated adequate performance. However, most of these methods present the challenges of the difficulty of speeding up the alarm time, providing early warning for distant areas, and considering global information to enhance performance. Recently, deep learning has significantly impacted many fields, including seismology. Thus, this paper proposes a deep learning-based framework, called SENSE, for the intensity prediction task of earthquake early warning systems. To explicitly consider global information from a regional or national perspective, the input to SENSE comprises statistics from a set of stations in a given region or country. The SENSE model is designed to learn the relationships among the set of input stations and the locality-specific characteristics of each station. Thus, SENSE is not only expected to provide more reliable forecasts by considering multistation data but also has the ability to provide early warnings to distant areas that have not yet received signals. This study conducted extensive experiments on datasets from Taiwan and Japan. The results revealed that SENSE can deliver competitive or even better performances compared with other state-of-the-art methods.

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arXiv2024

Isovolumetric Energy Minimization for Ball-Shaped Volume-Preserving Parameterizations of 3-Manifolds

A volume-preserving parameterization is a bijective mapping that maps a 3-manifold onto a specified canonical domain that preserves the local volume. This paper formulates the computation of ball-shaped volume-preserving parameterizations as an isovolumetric energy minimization (IEM) problem with the boundary points constrained on a unit sphere. In addition, we develop a new preconditioned nonlinear conjugate gradient algorithm for solving the IEM problem with guaranteed theoretical convergence and significantly improved accuracy and computational efficiency compared to other state-of-the-art algorithms. Applications to solid shape registration and deformation are presented to highlight the usefulness of the proposed algorithm.

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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.

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arXiv2024

Numerical Solutions for Stochastic Continuous-time Algebraic Riccati Equations

We are concerned with efficient numerical methods for stochastic continuous-time algebraic Riccati equations (SCARE). Such equations frequently arise from the state-dependent Riccati equation approach which is perhaps the only systematic way today to study nonlinear control problems. Often involved Riccati-type equations are of small scale, but have to be solved repeatedly in real time. Important applications include the 3D missile/target engagement, the F16 aircraft flight control, and the quadrotor optimal control, to name a few. A new inner-outer iterative method that combines the fixed-point strategy and the structure-preserving doubling algorithm (SDA) is proposed. It is proved that the method is monotonically convergent, and in particular, taking the zero matrix as initial, the method converges to the desired stabilizing solution. Previously, Newton's method has been called to solve SCARE, but it was mostly investigated from its theoretic aspect than numerical aspect in terms of robust and efficient numerical implementation. For that reason, we revisit Newton's method for SCARE, focusing on how to calculate each Newton iterative step efficiently so that Newton's method for SCARE can become practical. It is proposed to use our new inner-outer iterative method, which is provably convergent, to provide critical initial starting points for Newton's method to ensure its convergence. Finally several numerical experiments are conducted to validate the new method and robust implementation of Newton's method.

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arXiv2023

On Novel Fixed-Point-Type Iterations with Structure-Preserving Doubling Algorithms for Stochastic Continuous-time Algebraic Riccati equations

In this paper we mainly propose efficient and reliable numerical algorithms for solving stochastic continuous-time algebraic Riccati equations (SCARE) typically arising from the differential statedependent Riccati equation technique from the 3D missile/target engagement, the F16 aircraft flight control and the quadrotor optimal control etc. To this end, we develop a fixed point (FP)-type iteration with solving a CARE by the structure-preserving doubling algorithm (SDA) at each iterative step, called FP-CARE SDA. We prove that either the FP-CARE SDA is monotonically nondecreasing or nonincreasing, and is R-linearly convergent, with the zero initial matrix or a special initial matrix satisfying some assumptions. The FP-CARE SDA (FPC) algorithm can be regarded as a robust initial step to produce a good initial matrix, and then the modified Newton (mNT) method can be used by solving the corresponding Lyapunov equation with SDA (FPC-mNT-Lyap SDA). Numerical experiments show that the FPC-mNT-Lyap SDA algorithm outperforms the other existing algorithms.

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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.

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arXiv2022

Convergence Analysis of Volumetric Stretch Energy Minimization and its Associated Optimal Mass Transport

The volumetric stretch energy has been widely applied to the computation of volume-/mass-preserving parameterizations of simply connected tetrahedral mesh models. However, this approach still lacks theoretical support. In this paper, we provide the theoretical foundation for volumetric stretch energy minimization (VSEM) to compute volume-/mass-preserving parameterizations. In addition, we develop an associated efficient VSEM algorithm with guaranteed asymptotic R-linear convergence. Furthermore, based on the VSEM algorithm, we propose a projected gradient method for the computation of the volume/mass-preserving optimal mass transport map with a guaranteed convergence rate of $\mathcal{O}(1/m)$, and combined with Nesterov-based acceleration, the guaranteed convergence rate becomes $\mathcal{O}(1/m^2)$. Numerical experiments are presented to justify the theoretical convergence behavior for various examples drawn from known benchmark models. Moreover, these numerical experiments show the effectiveness and accuracy of the proposed algorithm, particularly in the processing of 3D medical MRI brain images.

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arXiv2022

Fundamental Theory and R-linear Convergence of Stretch Energy Minimization for Equiareal parameterizations

In this paper, we first extend the finite distortion problem from the bounded domains in $\mathbb{R}^2$ to the closed genus-zero surfaces in $\mathbb{R}^3$ by the stereographic projection. Then we derive a theoretical foundation for spherical equiareal parameterizations between a simply connected closed surface $\mathcal{M}$ and a unit sphere $\mathbb{S}^2$ via minimizing the total area distortion energy on $\overline{\mathbb{C}}$. Provided we determine the minimizer of the total area distortion energy, the minimizer composed with the initial conformal map determines the equiareal map between the extended planes. Taking the inverse stereographic projection, we can derive the equiareal map between $\mathcal{M}$ and $\mathbb{S}^2$. The total area distortion energy can be rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres, respectively, and can be decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization for the computation of the spherical equiareal parameterization between $\mathcal{M}$ and $\mathbb{S}^2$. In addition, under some mild conditions, we verify that our proposed algorithm has asymptotically R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate the assumptions for convergence always hold and indicate the efficiency, reliability and robustness of the developed modified stretch energy minimization.

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arXiv2022

Convergence Analysis of Dirichlet Energy Minimization for Spherical Conformal Parameterizations

In this paper, we first derive a theoretical basis for spherical conformal parameterizations between a simply connected closed surface $\mathcal{S}$ and a unit sphere $\mathbb{S}^2$ by minimizing the Dirichlet energy on $\overline{\mathbb{C}}$ by stereographic projection. The Dirichlet energy can be rewritten as the sum of the energies associated with the southern and northern hemispheres and can be decreased under an equivalence relation by alternatingly solving the corresponding Laplacian equations. Based on this theoretical foundation, we develop a modified Dirichlet energy minimization with nonequivalence deflation for the computation of the spherical conformal parameterization between $\mathcal{S}$ and $\mathbb{S}^2$. In addition, under some mild conditions, we verify the asymptotically R-linear convergence of the proposed algorithm. Numerical experiments on various benchmarks confirm that the assumptions for convergence always hold and indicate the efficiency, reliability and robustness of the developed modified Dirichlet energy minimization.

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arXiv2022

Nearly Optimal Stochastic Approximation for Online Principal Subspace Estimation

Principal component analysis (PCA) has been widely used in analyzing high-dimensional data. It converts a set of observed data points of possibly correlated variables into a set of linearly uncorrelated variables via an orthogonal transformation. To handle streaming data and reduce the complexities of PCA, (subspace) online PCA iterations were proposed to iteratively update the orthogonal transformation by taking one observed data point at a time. Existing works on the convergence of (subspace) online PCA iterations mostly focus on the case where sample are almost surely uniformly bounded. In this paper, we analyze the convergence of a subspace online PCA iteration under more practical assumption and obtain a nearly optimal finite-sample error bound. Our convergence rate almost matches the minimax information lower bound. We prove that the convergence is nearly global in the sense that the subspace online PCA iteration is convergent with high probability for random initial guesses. This work also leads to a simpler proof of the recent work on analyzing online PCA for the first principal component only.

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