arXiv subjects

Jun Lai

Publications and source records attributed to Jun Lai.

At least 19 recordsLinked to original sources

arXiv2026

Spectral Convergence of the Multipole Expansion Method for Acoustic Scattering in Three Dimensions

Multiple scattering is a fundamental wave interaction phenomenon in acoustics and electromagnetics. The multipole expansion method (MEM) is the basis of many fast algorithms, such as the fast multipole method (FMM), for such problems. However, due to the infinitely many wave reflections involved, its convergence in three dimensions remains unexplored. In this paper, we prove spectral convergence of the MEM for time-harmonic acoustic scattering by finitely many well-separated spheres in three dimensions. Using a diagonally preconditioned single-layer formulation, we analyze the degree-$N$ truncated system in a natural spherical harmonic energy space. We split the interaction truncation into target-side and source-side high-degree parts and choose a different representation for each: a projected Green-kernel representation for the former and degree-wise estimates of a translated spherical wave family for the latter. The resulting argument, based on Parseval's identity and the spherical harmonic addition theorem, provides a general framework for the convergence analysis of MEM and reveals the geometric and physical origins of the convergence factors. We also obtain a sharper estimate through the first-transfer analysis. Numerical experiments confirm the predicted spectral decay and the geometric convergence factor. This paves the way for the convergence analysis of a large class of fast algorithms for multiple scattering.

math.NA↗

arXiv2026

An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions

For wave scattering by axisymmetric surfaces, the fast Fourier transform (FFT) method provides an effective tool to accelerate standard boundary integral equation (BIE) solvers. Surface BIEs can be decoupled into a series of curve integral equations on the generating curve, due to the convolution-like integral operators. The Fourier coefficients of the three-dimensional fundamental kernels can be rapidly computed through three-term recurrence relations based on Miller's algorithm. Such well-established techniques break down for nonaxisymmetric surfaces. This paper proposes a novel FFT-accelerated boundary integral method for wave scattering by smooth surfaces of arbitrary shapes. The Fourier coefficients of the singular kernels now satisfy higher-order recurrence relations. Although they can be solved with an optimal linear complexity by the standard Olver's algorithm, it turns out that a singularity swapping approach that rewrites each kernel as the product of a smooth function and an axisymmetric-related singular factor is realistically much faster. Consequently, Miller's algorithm together with the standard FFT convolution yields an ${\cal O}(M\log M)$ approach for evaluating the ${\cal O}(M)$ Fourier coefficients of the kernels, attaining exactly the same order of complexity for axisymmetric surfaces! With such FFT-based efficient procedures, we rewrite the surface BIEs in terms of ${\cal O}(M)$ curve integrals, which are proved to exhibit logarithmic singularities, discretize them by panel-based generalized Gaussian quadratures, and obtain spectrally accurate linear systems to approximate the wavefields. Extensive numerical experiments are carried out to demonstrate the effectiveness and spectral accuracy of the new approach.

math.NA↗

arXiv2026

A Spectral Solver for Acoustic Scattering by Multiple Quasi-Axisymmetric Structures

Acoustic scattering arises in a wide range of applications, including medical imaging, geophysical exploration, acoustic metamaterials, etc. In this paper, we develop a fast and highly accurate algorithm for acoustic scattering by multiple quasi-axisymmetric objects, whose axis of rotation is an arbitrary curve. The method is based on a Nystr\"om discretization that combines Gauss-Legendre quadrature with the trapezoidal rule. To treat the singular integrals that occur when target points are close to or coincide with source points, we reformulate them as evaluations of the modal Green's function and its derivatives, which are computed efficiently using the fast Fourier transform and convolution. The multiple scattering solver is then constructed by coupling the single scatterer discretizations through inter-body boundary integral interactions. We also present a convergence analysis for scattering problems with smooth geometries. Numerical examples demonstrate the efficiency and accuracy of the proposed method for solving multiple scattering problems involving up to 1000 quasi-axisymmetric structures.

math.NA↗

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↗

arXiv2026

Beyond Sparsity: Quantum Block Encoding for Dense Matrices via Hierarchically Low Rank Compression

While quantum algorithms for solving large scale systems of linear equations offer potentially exponential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad class of structured dense matrices arise in potential theory, covariance modeling, and computational physics, namely, hierarchically block separable (HBS) matrices. We develop two distinct methods to make these systems amenable to quantum solvers. The first is a pre-processing approach that transforms the dense matrix into a larger but sparse format. The second is a direct block encoding scheme that recursively constructs the necessary oracles from the HBS structure. We provide a detailed complexity analysis and rigorous error bounds for both methods. Numerical experiments are presented to validate the effectiveness of our approaches.

quant-ph↗

arXiv2025

A Singularity Guided Nystr\"om Method for Elastostatics on Two Dimensional Domains with Corners

We develop a comprehensive analytical and numerical framework for boundary integral equations (BIEs) of the 2D Lam\'e system on cornered domains. By applying local Mellin analysis on a wedge, we obtain a factorizable characteristic equation for the singular exponents of the boundary densities, and clarify their dependence on boundary conditions. The Fredholm well-posedness of the BIEs on cornered domains is proved in weighted Sobolev spaces. We further construct an explicit density-to-Taylor mapping for the BIE and show its invertibility for all but a countable set of angles. Based on these analytical results, we propose a singularity guided Nystr\"om (SGN) scheme for the numerical solution of BIEs on cornered domains. The SGN uses the computed corner exponents and a Legendre-tail indicator to drive panel refinement. An error analysis that combines this refinement strategy with an exponentially accurate far-field quadrature rule is provided. Numerical experiments across various cornered geometries demonstrate that SGN obtains higher order accuracy than uniform Nystr\"om method and reveal a crowding-limited regime for domains with re-entrant angles.

math.NA↗

arXiv2025

Intrinsic Incompatibility: Why Static Droplets Cannot Exist in Cahn-Hilliard-Navier-Stokes Systems

Static droplets serve as fundamental benchmarks for interface-resolved simulations of two-phase flows. However, their accurate representation in phase-field models remains elusive due to persistent numerical artifacts. This work rigorously proves that static droplets cannot exist in phase-field models governed by the Cahn-Hilliard-Navier-Stokes equations. Through equilibrium analysis of the governing equations, we demonstrate that equilibrium necessitates uniform chemical potential, which nullifies the interfacial force, enforcing a uniform pressure field. This directly contradicts the pressure jump required by Laplace's law for a curved interface, proving mechanical equilibrium is impossible. The results reveal an intrinsic incompatibility between non-flat equilibrium interfaces and the Cahn-Hilliard-Navier-Stokes system, provides a fundamental theoretical explanation for long-standing paradoxes such as droplet shrinkage and parasitic currents. This fundamental limitation applies universally to droplets/bubbles and necessitates re-evaluation of phase-field models for multiphase systems.

physics.flu-dyn↗

arXiv2025

Selective focusing of multiple particles in a layered medium

Inverse scattering in layered media has a wide range of applications, examples including geophysical exploration, medical imaging, and remote sensing. In this paper, we develop a selective focusing method for identifying multiple unknown buried scatterers in a layered medium. The method is derived through the asymptotic analysis of the time reversal operator using the layered Green's function and limited aperture measurements. We begin by showing the global focusing property of the time reversal operator. Then we demonstrate that each small sound-soft particle gives rise to one significant eigenvalue of the time reversal operator, while each sound-hard particle gives three. The associated eigenfunction generates an incident wave focusing selectively on the corresponding unknown particle. Finally, we employ the time reversal method as an initial indicator and propose an effective Bayesian inversion scheme to reconstruct multiple buried extended scatterers for enhanced resolution. Numerical experiments are provided to demonstrate the efficiency.

math.NA↗

arXiv2025

Computation of shape Taylor expansions

Shape derivative is an important analytical tool for studying scattering problems involving perturbations in scatterers. Many applications, including inverse scattering, optimal design, and uncertainty quantification, are based on shape derivatives. However, computing high order shape derivatives is challenging due to the complexity of shape calculus. This work introduces a comprehensive method for computing shape Taylor expansions in two dimensions using recurrence formulas. The approach is developed under sound-soft, sound-hard, impedance, and transmission boundary conditions. Additionally, we apply the shape Taylor expansion to uncertainty quantification in wave scattering, enabling high order moment estimation for the scattered field under random boundary perturbations. Numerical examples are provided to illustrate the effectiveness of the shape Taylor expansion in achieving high order approximations.

math.NA↗

arXiv2025

Shape Taylor expansion for wave scattering problems

The Taylor expansion of wave fields with respect to shape parameters has a wide range of applications in wave scattering problems, including inverse scattering, optimal design, and uncertainty quantification. However, deriving the high order shape derivatives required for this expansion poses significant challenges with conventional methods. This paper addresses these difficulties by introducing elegant recurrence formulas for computing high order shape derivatives. The derivation employs tools from exterior differential forms, Lie derivatives, and material derivatives. The work establishes a unified framework for computing the high order shape perturbations in scattering problems. In particular, the recurrence formulas are applicable to both acoustic and electromagnetic scattering models under a variety of boundary conditions, including Dirichlet, Neumann, impedance, and transmission types.

math.NA↗

arXiv2024

Analytical solutions of layered Poiseuille flows in the diffuse interface model

Based on the two-phase macroscopic governing equations in the phase field model, the governing equations and analytical solutions for the steady-state layered Poiseuille flows in the diffuse interface (DI) model are derived and analyzed. Then, based on three dynamic viscosity models commonly used in the literature, the corresponding analytical solutions of the velocity profile are obtained. Under the condition of high dynamic viscosity ratio, the analytical solution of DI model may be significantly different from that of the sharp interface (SI) model, and the degree of deviation depends on the dynamic viscosity model and the interface thickness. Therefore, the numerical simulation of layered Poiseuille flow with DI model should be compared with the analytical solution of DI model with the same dynamic viscosity model. A direct comparison of the numerical solution results with the SI analytical solution could misinterpret the model error with the numerical error. In addition, the direct numerical simulation data and the DI analytical solutions agree well, which validates the theoretical results. Finally, a new set of symmetrical dynamic viscosity models is proposed and recommended for the simulation of two-phase flows in the DI model, which makes both the viscosity profiles and velocity profiles close to the SI model.

physics.flu-dyn↗

arXiv2024

Adaptive Modality Balanced Online Knowledge Distillation for Brain-Eye-Computer based Dim Object Detection

Advanced cognition can be extracted from the human brain using brain-computer interfaces. Integrating these interfaces with computer vision techniques, which possess efficient feature extraction capabilities, can achieve more robust and accurate detection of dim targets in aerial images. However, existing target detection methods primarily concentrate on homogeneous data, lacking efficient and versatile processing capabilities for heterogeneous multimodal data. In this paper, we first build a brain-eye-computer based object detection system for aerial images under few-shot conditions. This system detects suspicious targets using region proposal networks, evokes the event-related potential (ERP) signal in electroencephalogram (EEG) through the eye-tracking-based slow serial visual presentation (ESSVP) paradigm, and constructs the EEG-image data pairs with eye movement data. Then, an adaptive modality balanced online knowledge distillation (AMBOKD) method is proposed to recognize dim objects with the EEG-image data. AMBOKD fuses EEG and image features using a multi-head attention module, establishing a new modality with comprehensive features. To enhance the performance and robust capability of the fusion modality, simultaneous training and mutual learning between modalities are enabled by end-to-end online knowledge distillation. During the learning process, an adaptive modality balancing module is proposed to ensure multimodal equilibrium by dynamically adjusting the weights of the importance and the training gradients across various modalities. The effectiveness and superiority of our method are demonstrated by comparing it with existing state-of-the-art methods. Additionally, experiments conducted on public datasets and system validations in real-world scenarios demonstrate the reliability and practicality of the proposed system and the designed method.

cs.CV↗

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

A robust and high precision algorithm for elastic scattering problems from cornered domains

The Navier equation is the governing equation of elastic waves, and computing its solution accurately and rapidly has a wide range of applications in geophysical exploration, materials science, etc. In this paper, we focus on the efficient and high-precision numerical algorithm for the time harmonic elastic wave scattering problems from cornered domains via the boundary integral equations in two dimensions. The approach is based on the combination of Nyström discretization, analytical singular integrals and kernel-splitting method, which results in a high-order solver for smooth boundaries. It is then combined with the recursively compressed inverse preconditioning (RCIP) method to solve elastic scattering problems from cornered domains. Numerical experiments demonstrate that the proposed approach achieves high accuracy, with stabilized errors close to machine precision in various geometric configurations. The algorithm is further applied to investigate the asymptotic behavior of density functions associated with boundary integral operators near corners, and the numerical results are highly consistent with the theoretical formulas.

math.NA↗

arXiv2023

Singularity swapping method for nearly singular integrals based on trapezoidal rule

Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global trapezoidal rule and trigonometric interpolation, resulting in an explicit quadrature formula. The method achieves spectral accuracy for nearly singular integrals on closed analytic curves. In order to extract the singularity from the complexified distance function, an efficient root finding method is proposed based on contour integration. Through the change of variables, we also extend the quadrature method to integrals on the piecewise analytic curves. Numerical examples for Laplace's and Helmholtz equations show that high order accuracy can be achieved for arbitrarily close field evaluation.

math.NA↗

arXiv2023

Selective focusing of elastic cavities based on the time reversal far field model

This paper is concerned with the inverse time harmonic elastic scattering of multiple small and well-resolved cavities in two dimensions. We extend the so-called DORT method to the inverse elastic scattering so that selective focusing can be achieved on each cavity with far field measurements. A rigorous mathematical justification that relates the corresponding eigenfunctions of the time reversal operator to the locations of cavities is presented based on the asymptotic analysis of the far field operator and decaying property of oscillatory integrals. We show that in the regime $a\ll k^{-1}\ll L$, where $a$ denotes the size of cavity, $k$ is the compressional wavenumber $\kp$ or shear wavenumber $\ks$, and $L$ is the minimal distance between the cavities, each cavity gives rise to five significant eigenvalues and the corresponding eigenfunction generates an incident wave focusing selectively on that cavity. Numerical experiments are given to verify the theoretical result.

math.NA↗

arXiv2022

Simulation of two-phase flows at large density ratios and high Reynolds numbers using a discrete unified gas kinetic scheme

In order to treat immiscible two-phase flows at large density ratios and high Reynolds numbers, a three-dimensional code based on the discrete unified gas kinetic scheme (DUGKS) is developed, incorporating two major improvements. First, the particle distribution functions at cell interfaces are reconstructed using a weighted essentially non-oscillatory scheme. Second, the conservative lower-order Allen-Cahn equation is chosen, instead of the higher-order Cahn-Hilliard equation, to evolve the free-energy based phase field governing the dynamics of two-phase interfaces. Five benchmark problems are simulated to demonstrate the capability of the approach in treating two phase flows at large density ratios and high Reynolds numbers, including three two dimensional problems (a stationary droplet, Rayleigh-Taylor instability, and a droplet splashing on a thin liquid film) and two three-dimensional problems (binary droplets collision and Rayleigh-Taylor instability). All results agree well with the previous numerical and the experimental results. In these simulations, the density ratio and Reynolds number can reach a large value of O(1000). Our improved approach sets the stage for the DUGKS scheme to handle realistic two-phase flow problems.

physics.flu-dyn↗

arXiv2022

Study of a droplet breakup process in decaying homogeneous isotropic turbulence based on the phase-field DUGKS approach

The breakup of a spherical droplet in a decaying homogeneous isotropic turbulence is studied by solving the Cahn-Hilliard-Navier-Stokes equations, using the discrete unified gas kinetic scheme combined with the free-energy-based phase-field model. We focus on the combined effects of turbulence and surface tension on the breakup process by assuming that the two fluid phases have the same density and same viscosity. The key physical parameters of the system include the volume fraction (6.54%), the initial Weber number (21.7), and the initial Taylor microscale Reynolds number (58). Three distinct stages of droplet evolution are identified, namely, the deformation stage when the initially spherical droplet evolves into an irregular geometric shape with complex structures, the breakup stage when many daughter droplets are formed, and the restoration stage when the droplets relax towards spherical shape. These three stages are analyzed systematically from several perspectives: (1) a geometric perspective concerning the maximum equivalent diameter, the total number of droplets, total interface area, and probability distribution of droplet diameters, (2) a dynamic perspective concerning the evolution of local velocity and vorticity at the fluid-fluid interface, (3) a global perspective concerning the evolution of average kinetic energy / dissipation rate and their Fourier spectra, (4) spherical harmonics based energetics concerning simultaneous transfer of kinetic energy across different length scales and different radii relative to initial droplet center, and (5) the time evolution of global kinetic energy and free energy of the system.

physics.flu-dyn↗