arXiv subjects

Qinglin Tang

Publications and source records attributed to Qinglin Tang.

At least 19 recordsLinked to original sources

arXiv2026

Structure and symmetry of the Gross-Pitaevskii ground-state manifold

The structure and degeneracy of ground states of the Gross-Pitaevskii energy functional play a central role in both analysis and computation, yet a precise characterization of the ground-state manifold in the presence of symmetries remains a fundamental challenge. In this paper, we establish sharp theoretical results describing the geometric structure of local minimizers and its implications for optimization algorithms. We show that when local minimizers are non-unique, the Morse-Bott condition provides a natural and sufficient criterion under which the ground-state set partitions into finitely many embedded submanifolds, each coinciding with an orbit generated by intrinsic symmetries of the energy functional, namely phase shifts and spatial rotations. This yields a structural characterization of the ground-state manifold in terms of these symmetries. Building on this insight, we characterize the local convergence behavior of general preconditioned Riemannian gradient methods (P-RG). Under the Morse-Bott condition, we derive sharp local $Q$-linear convergence estimates and prove that the condition holds if and only if the energy sequence generated by P-RG converges locally $Q$-linearly. In particular, on the ground-state set, the Morse-Bott condition is satisfied if and only if the minimizers decompose into finitely many symmetry orbits and P-RG exhibits local linear convergence nearby. When the condition fails, we establish a local sublinear convergence rate. Taken together, these results show that the Morse-Bott condition is the exact threshold separating linear from sublinear convergence, while determining the symmetry-induced structure of the ground-state manifold, connecting geometry, symmetry, and convergence behavior in a unified framework.

math.NA↗

arXiv2026

An efficient and accurate numerical method for computing the ground states of three-dimensional rotating dipolar Bose-Einstein condensates under strongly anisotropic trap

In this article, we propose an efficient and spectrally accurate numerical method to compute the ground states of three-dimensional (3D) rotating dipolar Bose-Einstein condensates (BEC) under strongly anisotropic trapping potentials.The kernel singularity, convolution non-locality and density anisotropy together complicate the dipolar potential evaluation. The fast rotation mechanism not only induces a complicated energy landscape with many local minima, but also creates a large number of vortices in the condensates. Such factors collectively make the ground state computation challenging in terms of convergence, accuracy and efficiency, especially for 3D anisotropic systems. Coupled with Fourier spectral discretization, we proposed a preconditioned conjugate gradient method (PCG) by integrating the anisotropic truncated kernel method (ATKM) for the dipolar potential evaluation. An adaptive step size control strategy is designed and ATKM allows for a spectral accuracy without introducing any extra anisotropy-dependent memory requirement or computational time. Our algorithm is spectrally accurate, highly efficient and memory-economic. Extensive numerical results are presented to confirm the accuracy and efficiency, together with applications to study impacts of the model parameters on critical rotational frequency, energies and chemical potential. Furthermore, these simulations reveal additional novel ground state patterns, such as bent vortices.

math.NA↗

arXiv2026

Fast convolution solvers using moment-matching

We propose two easy-to-implement fast algorithms based on moment-matching to compute the nonlocal potential $\varphi(\textbf{x})=(U\ast \rho)(\textbf{x})$ on bounded domain, where the kernel $U$ is singular at the origin and the density $\rho$ is a fast-decaying smooth function. Each method requires merely minor modifications to commonly-used existing methods, i.e., the sine spectral/Fourier quadrature method, and achieves a much better convergence rate. The key lies in the introduction of a smooth auxiliary function $\rho_1$ whose moments match those of the density up to an integer order $m$. Specifically, $\rho_1$ is constructed using Gaussian function in an explicit way and the associated potential can be calculated analytically. The moments of residual density vanish up to order $m$, and the corresponding residual potential $U \ast (\rho-\rho_1)$ decays much faster than the original potential $\varphi$ at the far field. As for the residual potential evaluation, for classical kernels (e.g., the Coulomb kernel), we solve a differential/pseudo-differential equation on a rectangular domain with homogeneous Dirichlet boundary conditions via sine pseudospectral method, and achieve an arbitrary high convergence rate. While, for general kernels, the regularity of Fourier integrand increase by $m$ thanks to the moments-vanishing property, therefore, the standard trapezoidal rule/midpoint quadrature also converges much faster. To gain a better numerical performance, we utilize the domain expansion technique to obtain better accuracy, and improve the efficiency by simplifying the quadrature into one discrete convolution and applying Fast Fourier Transform (FFT) to a double-sized vector. Rigorous error estimates and extensive numerical investigations showcase the accuracy and efficiency for different nonlocal potentials.

math.NA↗

arXiv2025

Energy dissipation and global convergence of a discrete normalized gradient flow for computing ground states of two-component Bose-Einstein condensates

The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. Numerous numerical experiments have shown that the energy dissipation holds when calculating the ground states of multicomponent Bose-Einstein condensates (MBECs) with GFSI, while rigorous proof remains an open challenge. By introducing a Lagrange multiplier, we reformulate the GFSI into an equivalent form and thereby prove the energy dissipation for GFSI in two-component scenario with Josephson junction and rotating term, which is one of the most important and topical model in MBECs. Based on this, we further establish the global convergence to stationary states. Also, the numerical results of energy dissipation in practical experiments corroborate our rigorous mathematical proof, and we numerically verified the upper bound of time step that guarantees energy dissipation is indeed related to the strength of particle interactions.

math.NA↗

arXiv2025

On preconditioned Riemannian gradient methods for minimizing the Gross-Pitaevskii energy functional: algorithms, global convergence and optimal local convergence rate

In this article, we propose a unified framework for preconditioned Riemannian gradient (P-RG) methods to minimize Gross-Pitaevskii (GP) energy functionals with rotation on a Riemannian manifold. This framework enables comprehensive analysis of existing projected Sobolev gradient methods and facilitates the construction of highly efficient P-RG algorithms. Under mild assumptions on the preconditioner, we prove energy dissipation and global convergence. Local convergence is more challenging due to phase and rotational invariances. Assuming the GP functional is Morse-Bott, we derive a sharp Polyak-\L ojasiewicz (PL) inequality near minimizers. This allows precise characterization of the local convergence rate via the condition number $\mu/L$, where $\mu$ and $L$ are the lower and upper bounds of the spectrum of a combined operator (preconditioner and Hessian) on a closed subspace. By combining spectral analysis with the PL inequality, we identify a quasi-optimal preconditioner achieving the best possible local convergence rate: $(L-\mu)/(L+\mu)+\varepsilon$ ($\varepsilon>0$ small). To our knowledge, this is the first rigorous derivation of the local convergence rate for P-RG methods applied to GP functionals with two symmetry structures. Numerical experiments on rapidly rotating Bose-Einstein condensates validate the theoretical results and compare the performance of different preconditioners.

math.NA↗

arXiv2024

A normalized gradient flow method for computing ground states of spin-2 Bose-Einstein condensates

We propose and analyze an efficient and accurate numerical method for computing ground states of spin-2 Bose-Einstein condensates (BECs) by using the normalized gradient flow (NGF). In order to successfully extend the NGF to spin-2 BECs which has five components in the vector wave function but with only two physical constraints on total mass conservation and magnetization conservation, two important techniques are introduced for designing the proposed numerical method. The first one is to systematically investigate the ground state structure and property of spin-2 BECs within a spatially uniform system, which can be used on how to properly choose initial data in the NGF for computing ground states of spin-2 BECs. The second one is to introduce three additional projection conditions based on the relations between the chemical potentials, together with the two existing physical constraints, such that the five projection parameters used in the projection step of the NGF can be uniquely determined. Then a backward-forward Euler finite difference method is adapted to discretize the NGF. We prove rigorously that there exists a unique solution of the nonlinear system for determining the five projection parameters in the full discretization of the NGF under a mild condition on the time step size. Extensive numerical results on ground states of spin-2 BECs with different types of phases and under different potentials are reported to show the efficiency and accuracy of the proposed numerical method and to demonstrate several interesting physical phenomena on ground states of spin-2 BECs.

math.NA↗

arXiv2022

On optimal zero-padding of kernel truncation method

The kernel truncation method (KTM) is a commonly-used algorithm to compute the convolution-type nonlocal potential $\Phi(x)=(U\ast \rho)(x), ~x \in {\mathbb R^d}$, where the convolution kernel $U(x)$ might be singular at the origin and/or far-field and the density $\rho(x)$ is smooth and fast-decaying. In KTM, in order to capture the Fourier integrand's oscillations that is brought by the kernel truncation, one needs to carry out a zero-padding of the density, which means a larger physical computation domain and a finer mesh in the Fourier space by duality. The empirical fourfold zero-padding [ Vico et al J. Comput. Phys. (2016) ] puts a heavy burden on memory requirement especially for higher dimension problems. In this paper, we derive the optimal zero-padding factor, that is, $\sqrt{d}+1$, for the first time together with a rigorous proof. The memory cost is greatly reduced to a small fraction, i.e., $(\frac{\sqrt{d}+1}{4})^d$, of what is needed in the original fourfold algorithm. For example, in the precomputation step, a double-precision computation on a $256^3$ grid requires a minimum $3.4$ Gb memory with the optimal threefold zero-padding, while the fourfold algorithm requires around $8$ Gb where the reduction factor is $\frac{37}{64}\approx \frac{3}{5}$. Then, we present the error estimates of the potential and density in $d$ dimension. Next, we re-investigate the optimal zero-padding factor for the anisotropic density. Finally, extensive numerical results are provided to confirm the accuracy, efficiency, optimal zero-padding factor for the anisotropic density, together with some applications to different types of nonlocal potential, including the 1D/2D/3D Poisson, 2D Coulomb, quasi-2D/3D Dipole-Dipole Interaction and 3D quadrupolar potential.

math.NA↗

arXiv2020

Error estimates of local energy regularization for the logarithmic Schrodinger equation

The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schr\"odinger equation (LogSE)as a prototype model. Instead of regularizing $f(\rho)=\ln \rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(\rho)=\rho\ln \rho -\rho$ locally near $\rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schr\"odinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0<\ep\ll1$. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improvesthe linear convergence rate of the regularization method in the literature. Error estimates are alsopresented for solving the ERLogSE by using Lie-Trotter splittingintegrators. Numerical results are reported to confirm our errorestimates of the LER and of the time-splitting integrators for theERLogSE. Finally our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.

math.NA↗

arXiv2018

Regularized numerical methods for the logarithmic Schrodinger equation

We present and analyze two numerical methods for the logarithmic Schr{\"o}dinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank-Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schr{\"o}dinger equation (RLogSE) with a small regularized parameter 0 < $\epsilon$ $\ll$ 1 is adopted to approximate the LogSE with linear convergence rate O($\epsilon$). Then we use the Lie-Trotter splitting integrator to solve the RLogSE and establish its error bound O($\tau$ 1/2 ln($\epsilon$ --1)) with $\tau$ > 0 the time step, which implies an error bound at O($\epsilon$ + $\tau$ 1/2 ln($\epsilon$ --1)) for the LogSE by the Lie-Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE.

math.NA↗

arXiv2018

Error estimates of a regularized finite difference method for the logarithmic Schr\"odinger equation

We present a regularized finite difference method for the logarithmic Schr\"odinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. $\ln \rho\to -\infty$ when $\rho\rightarrow 0^+$ with $\rho=|u|^2$ being the density and $u$ being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schr\"odinger equation (RLogSE) is proposed with a small regularization parameter $0<\varepsilon\ll 1$ and linear convergence is established between the solutions of RLogSE and LogSE in term of $\varepsilon$. Then a semi-implicit finite difference method is presented for discretizing the RLogSE and error estimates are established in terms of the mesh size $h$ and time step $\tau$ as well as the small regularization parameter $\varepsilon$. Finally numerical results are reported to confirm our error bounds.

math.NA↗

arXiv2016

Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial dis-cretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank-Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned gradient (PG) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement.

math.NA↗

arXiv2016

A robust and efficient numerical method to compute the dynamics of the rotating two-component dipolar Bose-Einstein condensates

In this paper, we propose a robust and efficient numerical method to compute the dynamics of the rotating two-component dipolar Bose-Einstein condensates (BEC). Using the rotating Lagrangian coordinates transform \cite{BMTZ2013}, we reformulate the original coupled Gross-Pitaevskii equations (CGPE) into new equations where the rotating term vanishes and the potential becomes time-dependent. A time-splitting Fourier pseudospectral method is proposed to simulate the new equations where the nonlocal Dipole-Dipole Interactions (DDI) are computed by a newly-developed Gaussian-sum (GauSum) solver \cite{EMZ2015} which helps achieve spectral accuracy in space within $O(N\log N)$ operations ($N$ is the total number of grid points). The new method is spectrally accurate in space and second order accurate in time, and the accuracies are confirmed numerically. Dynamical properties of some physical quantities, including the total mass, energy, center of mass and angular momentum expectation, are presented and confirmed numerically. Interesting dynamics phenomena that are peculiar to the rotating two-component dipolar BECs, such as dynamics of center of mass, quantized vortex lattices dynamics and the collapse dynamics of 3D cases, are presented.

physics.comp-ph↗

arXiv2015

On the ground states and dynamics of space fractional nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions

In this paper, we propose some efficient and robust numerical methods to compute the ground states and dynamics of Fractional Schr\"{o}dinger Equation (FSE) with a rotation term and nonlocal nonlinear interactions. In particular, a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal interaction evaluation \cite{EMZ2015}. To compute the ground states, we integrate the preconditioned Krylov subspace pseudo-spectral method \cite{AD1} and the GauSum solver. For the dynamics simulation, using the rotating Lagrangian coordinates transform \cite{BMTZ2013}, we first reformulate the FSE into a new equation without rotation. Then, a time-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the new FSE.

math.NA↗

arXiv2015

A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its limiting models when $\varepsilon\to0^+$.

math.NA↗

arXiv2015

Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon$. Based on the error bounds, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the FDTD methods share the same $\varepsilon$-scalability on time step: $\tau=O(\varepsilon^3)$. Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability on time step is improved to $\tau=O(\varepsilon^2)$ when $0<\varepsilon\ll 1$. Extensive numerical results are reported to support our error estimates.

math.NA↗

arXiv2015

Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT

In this paper, we propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.

physics.comp-ph↗

arXiv2014

Computing the ground state and dynamics of the nonlinear Schr\"odinger equation with nonlocal interactions via the nonuniform FFT

We present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear Schr\"odinger equation (NLSE) with nonlocal interactions based on a fast and accurate evaluation of the long-range interactions via the nonuniform fast Fourier transform (NUFFT). We begin with a review of the fast and accurate NUFFT based method in \cite{JGB} for nonlocal interactions where the singularity of the Fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates. We then extend the method to compute other nonlocal interactions whose Fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform. Many of these interactions do not decay at infinity in the physical space, which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain. The performance of our method against other existing methods is illustrated numerically, with particular attention on the effect of the size of the computational domain in the physical space. Finally, to study the ground state and dynamics of the NLSE, we propose efficient and accurate numerical methods by combining the NUFFT method for potential evaluation with the normalized gradient flow using backward Euler Fourier pseudospectral discretization and time-splitting Fourier pseudospectral method, respectively. Extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the NLSE with various nonlocal interactions. Numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency.

math.NA↗

arXiv2013

An efficient spectral method for computing dynamics of rotating two-component Bose--Einstein condensates via coordinate transformation

In this paper, we propose an efficient and accurate numerical method for computing the dynamics of rotating two-component Bose--Einstein condensates (BECs) which is described by coupled Gross--Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field. By introducing rotating Lagrangian coordinates, we eliminate the angular momentum rotation term from the CGPEs, which allows us to develop an efficient numerical method. Our method has spectral accuracy in all spatial dimensions and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in literature. Numerical results show that to achieve the same accuracy, our method needs much shorter computing time. We also applied our method to study the dynamic properties of rotating two-component BECs. Furthermore, we generalize our method to solve the vector Gross--Pitaevskii equations (VGPEs) which is used to study rotating multi-component BECs.

math.NA↗