Geometric Numerical Integration Research Articles

A Geometric Numerical Integration with Simple Cell Mapping for Global Analysis of Nonlinear Dynamical Systems

In order to improve the efficiency in computing the global properties of nonlinear systems, some effective methods have been proposed for the global analysis of these systems. However, there are only few investigations focusing on the numerical algorithms of the system response trajectories. This paper presents a higher-efficiency geometric numerical integration method with simple cell mapping for solving the basins of attraction of nonlinear systems. This numerical algorithm is based on the rule of Lie derivative, using it to calculate the trajectories of the nonlinear systems. Then, the global structure is studied by the numerical algorithm associated with a simple cell mapping method. Compared with the traditional Runge–Kutta methods of orders 4 and 5, it is demonstrated that the proposed Lie derivative iterative algorithm has significant advantages in efficiency.

Forty Years: Geometric Numerical Integration of Dynamical Systems in China

Forty Years: Geometric Numerical Integration of Dynamical Systems in China

A new symplectic integrator for stochastic Hamiltonian systems on manifolds

This paper presents a significant development in the field of geometric numerical stochastic integration schemes. Specifically, the Geometric Symplectic Itô-Taylor 1.0 strong numerical integration scheme, tailored for Hamiltonian systems evolving on 2-sphere or S2 manifold is proposed. The core of this proposed algorithm centers around a new weak symplectic condition. This condition ensures the numerical stability for long duration time integration that can be readily achieved through numerical steps rather than relying solely on analytical satisfaction of the condition. This proposed advancement over the state-of-the-art caters to improved accuracy and stability of numerical simulations in manifold-based dynamical systems for long duration simulations. Through extensive analysis and numerical experiments, the effectiveness and reliability of the proposed scheme and symplectic criterion are validated. The findings of this study promises to offer valuable insights for researchers and practitioners working in the field of symplectic numerical integration methods.

The flow method for the Baker-Campbell-Hausdorff formula: exact results

Leveraging techniques from the literature on geometric numerical integration, we propose a new general method to compute exact expressions for the Baker-Campbell-Hausdorff (BCH) formula. In its utmost generality, the method consists in embedding the Lie algebra of interest into a subalgebra of the algebra of vector fields on some manifold by means of an isomorphism, so that the BCH formula for two elements of the original algebra can be recovered from the composition of the flows of the corresponding vector fields. For this reason we call our method the flow method. Clearly, this method has great advantage in cases where the flows can be computed analytically. We illustrate its usefulness on some benchmark examples where it can be applied directly, and discuss some possible extensions for cases where an exact expression cannot be obtained.

Existence results on Lagrange multiplier approach for gradient flows and application to optimization

This paper deals with the geometric numerical integration of gradient flow and its application to optimization. Gradient flows often appear as model equations of various physical phenomena, and their dissipation laws are essential. Therefore, dissipative numerical methods, which are numerical methods replicating the dissipation law, have been studied in the literature. Recently, Cheng, Liu, and Shen proposed a novel dissipative method, the Lagrange multiplier approach, for gradient flows, which is computationally cheaper than existing dissipative methods. Although their efficacy is numerically confirmed in existing studies, the existence results of the Lagrange multiplier approach are not known in the literature. In this paper, we establish some existence results. We prove the existence of the solution under a relatively mild assumption. In addition, by restricting ourselves to a special case, we show some existence and uniqueness results with concrete bounds. As gradient flows also appear in optimization, we further apply the latter results to optimization problems.

Practical perspectives on symplectic accelerated optimization

Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Bregman Lagrangian and Hamiltonian systems from the variational framework introduced by Wibisono et al. In this paper, we discuss practical considerations which can significantly boost the computational performance of these optimization algorithms and considerably simplify the tuning process. In particular, we investigate how momentum restarting schemes ameliorate computational efficiency and robustness by reducing the undesirable effect of oscillations and ease the tuning process by making time-adaptivity superfluous. We also discuss how temporal looping helps avoiding instability issues caused by numerical precision, without harming the computational efficiency of the algorithms. Finally, we compare the efficiency and robustness of different geometric integration techniques and study the effects of the different parameters in the algorithms to inform and simplify tuning in practice. From this paper emerge symplectic accelerated optimization algorithms whose computational efficiency, stability and robustness have been improved, and which are now much simpler to use and tune for practical applications.

IPHFMs: Fast and accurate Polar Harmonic Fourier Moments

Polar Harmonic Fourier moments (PHFMs) are widely used in image processing due to their excellent reconstruction ability. However, PHFMs suffer from geometric errors and numerical integration errors. Also, direct computation of PHFMs from their definition is usually slow. This paper proposes a fast and accurate calculation method for PHFMs, named improved polar Harmonic Fourier moments (IPHFMs). The variable equidistant discrete approach and fast Fourier transform (FFT) reduce the geometric errors and time complexity. Extensive experimental results on a real-world application demonstrate the efficacy and superiority of the proposed IPHFMs, concerning speed and accuracy.

Stochastic dynamics on manifolds based on novel geometry preserving Ito–Taylor scheme

This Rapid Communication presents the development of a new Ito–Taylor based higher order geometric numerical integration scheme for stochastically excited systems on manifolds. For the dynamical systems evolving on manifolds, two key motivations for the proposed work are: geometry preservation and accurate solution, which are not well addressed in the current state of the art. The accuracy of the proposed method as compared to the existing scheme (geometric Euler Maruyama) is achieved by incorporating the Kolmogorov operators in the proposed formulation. Preservation of geometry is ensured by exploiting the correspondence between the manifold and the tangent space at identity (Lie algebra). The efficacy of the proposed algorithm is demonstrated for non-linear oscillators and pendulum cart system, which are canonical representations of various physical phenomena in modern engineering problems.

Optimal error estimates of a time-splitting Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation

Recently, a time-splitting Fourier pseudo-spectral (TSFP) scheme for solving numerically the Klein–Gordon–Dirac equation (KGDE) has been proposed (Yi et al., 2019). However, that paper only gives numerical experiments and lacks rigorous convergence analysis and error estimates for the scheme. In addition, the time symmetry of the scheme has not been proved. This is not satisfactory from the perspective of geometric numerical integration. In this paper, we proposed a new TSFP scheme for the KGDE with periodic boundary conditions by reformulating the Klein–Gordon part into a relativistic nonlinear Schrödinger equation. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. We make a rigorously convergence analysis and establish error estimates by comparing semi-discretization and full-discretization using the mathematical induction. The convergence rate of the scheme is proved to be second-order in time and spectral-order in space, respectively, in a generic norm under the specific regularity conditions. The numerical experiments support our theoretical analysis. The conclusion is also applicable to high-dimensional problems under sufficient regular conditions. Our scheme can also serve as a reference for solving some other coupled equations or systems such as Klein–Gordon–Schrödinger equation.

How do Monte Carlo estimates affect stochastic geometric numerical integration?

In this work, we investigate the numerical conservation of characteristic properties of stochastic Hamiltonian problems driven by additive noise under time discretizations, in the spirit of stochastic geometric numerical integration. In particular, we aim to understand how Monte Carlo and multilevel Monte Carlo estimators for the expected values eventually affect the conservation properties of the numerical scheme. Specifically, our analysis is focused on the role of random number generation in the conservation of invariant laws for stochastic Hamiltonian problems under time discretization by the drift-preserving numerical scheme introduced in C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, [Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math. 46(2) (2020), pp. 27.] and the stochastic perturbation of a symplectic Runge–Kutta method, introduced in K. Burrage and P.M. Burrage, [Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math. 236(16) (2012), pp. 3920–3930.]. The numerical evidence confirms the aforementioned theoretical analysis.

Geometric Numerical Integration

The topics of the workshop included interactions between geometric numerical integration and numerical partial differential equations; geometric aspects of stochastic differential equations; interaction with optimisation and machine learning; new applications of geometric integration in physics; problems of discrete geometry, integrability, and algebraic aspects.

On the symmetrization and composition of nonstandard finite difference schemes as an alternative to Richardson's extrapolation

From the classical explicit Euler scheme of first order, nonstandard finite difference (NSFD) schemes were envisioned to mimic the essential properties of the governing differential equation model for every time step-size. In the context of compartmental epidemiological models, these properties are generally concerned with the positivity of subpopulations, conservation laws (dynamics of the total population), and stability. However, for autonomous systems, the symmetry (self-adjoint) condition is not preserved. Compartmental epidemiological models are Poisson systems, so methods from geometric numerical integration should be applicable. It is found that symmetrization of NSFD schemes does not respect positivity for every step-size, though other characteristics are maintained and second order is reached. Composition through the Lie formalism can then be applied to obtain higher-order schemes. This is a more efficient and consistent alternative to Richardson's extrapolation, which has often been used to go beyond order one.

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with energy-preservation for long-time dynamics of the nonlinear Klein-Gordon equation

For the nonlinear Klein-Gordon (NKG) equation with weak nonlinearity characterized by a small parameter ε∈(0,1], the numerical methods for long-time dynamics have received more and more attention. For the NKG equation up to the time at O(1/ε2), the error bounds of classical numerical methods depend significantly on the small parameters ε which creates serious numerical burdens as ε→0+. Recently, an exponential wave integrator Fourier pseudo-spectral (EWIFP) method for the NKG equation has been proposed (Feng et al., 2021) which is uniformly accurate about ε and of course performs well over the classical methods. However, the EWIFP method can not preserve the discrete energy, which is an important structural feature of the true solution in the long-time dynamics from the perspective of geometric numerical integration. In addition, the EWIFP method only is conditionally stable under specific stability condition and the authors did not give a convergence analysis for the method without any CFL condition restrictions on the grid ratio. In this work, we propose an energy-preserving EWIFP (EPEWIFP) method. The proposed method is proved to be time symmetric, unconditionally stable and preserves the discrete energy. We carry out a rigorously error analysis and give uniform error bounds of the method at O(hm0+ε2−βτ2) up to the time at O(1/εβ) with β∈[0,2], mesh size h, time step τ and an integer m0 determined by the regularity conditions. In general, the NKG equation with weak nonlinearity can be converted to an oscillatory NKG equation with wavelength at O(ε2) in time. It is easy to extend the error bounds and energy-preservation properties to the oscillatory NKG equation. Numerical results confirm our error bounds and energy-preservation properties.

Accurate Quaternion Polar Harmonic Transform for Color Image Analysis

Polar harmonic transforms (PHTs) have been applied in pattern recognition and image analysis. But the current computational framework of PHTs has two main demerits. First, some significant color information may be lost during color image processing in conventional methods because they are based on RGB decomposition or graying. Second, PHTs are influenced by geometric errors and numerical integration errors, which can be seen from image reconstruction errors. This paper presents a novel computational framework of quaternion polar harmonic transforms (QPHTs), namely, accurate QPHTs (AQPHTs). First, to holistically handle color images, quaternion-based PHTs are introduced by using the algebra of quaternions. Second, the Gaussian numerical integration is adopted for geometric and numerical error reduction. When compared with CNNs (convolutional neural networks)-based methods (i.e., VGG16) on the Oxford5K dataset, our AQPHT achieves better performance of scaling invariant representation. Moreover, when evaluated on standard image retrieval benchmarks, our AQPHT using smaller dimension of feature vector achieves comparable results with CNNs-based methods and outperforms the hand craft-based methods by 9.6% w.r.t mAP on the Holidays dataset.

A novel class of explicit divergence-free time-domain methods for efficiently solving Maxwell's equations

In electrical engineering and physics, Maxwell's equations play a very important role and have a variety of applications. In this paper, we propose and analyse a novel class of time domain conservative schemes in rectangular coordinate to solve Maxwell's equations with periodic boundary conditions. This class of methods is explicit and easy to implement with low computational cost and memory storage. The error estimates presented in this paper demonstrate that the numerical solutions obtained by this class of conservative schemes can achieve arbitrarily high-order accuracy. The discrete divergences in numerical experiments motivate us to show that the conservative scheme is divergence-free by the rigorous numerical analysis, which is of great importance in the sense of geometric numerical integration. The discrete energy monitors give light to the statement that the conservative scheme can keep the linear relationship among some qualitative features of the underlying Maxwell's equations.

A quadrature framework for solving Lyapunov and Sylvester equations

This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low rank approximations of their solutions are produced by Runge-Kutta methods. Appropriate Runge-Kutta methods are identified following the idea of geometric numerical integration to preserve a geometric property, namely a low rank residual. For both types of equations we prove the equivalence of one particular instance of the resulting algorithm to the well known ADI iteration.

A review of structure-preserving numerical methods for engineering applications

Accurate numerical simulation of dynamical systems is essential in applications ranging from particle physics to geophysical fluid flow to space hazard analysis. However, most traditional numerical methods do not account for the underlying geometric structure of the physical system, leading to simulation results that may suggest nonphysical behavior. The field of geometric numerical integration (GNI) is concerned with numerical methods that respect the fundamental physics of a problem by preserving the geometric properties of the governing differential equations. Research over the past two decades has produced GNI methods that are so accurate that they are now used for benchmarking purposes for long-time simulation of conservative dynamical systems. However, their utility for large-scale engineering problems is still an open question. This paper presents a review of structure-preserving numerical methods with focus on their engineering applications. The purpose of this paper is to provide an overview of different classes of GNI methods for mechanical systems while providing a survey of practical examples from numerical simulation of realistic engineering problems.

Accurate quaternion radial harmonic Fourier moments for color image reconstruction and object recognition

Orthogonal moments have become a powerful tool for object representation and image analysis. Radial harmonic Fourier moments (RHFMs) are one of such image descriptors based on a set of orthogonal projection bases, which outperform other moments because of their computational efficiency. However, the conventional computational framework of RHFMs produces geometric error and numerical integration error, which will affect the accuracy of RHFMs, thus degrading the image reconstruction performance. To overcome this shortcoming, we propose a new computational framework of RHFMs, namely accurate quaternion radial harmonic Fourier moments (AQRHFMs), for color image processing, and also analyze the properties of AQRHFMs. Firstly, we propose a precise computation method of RHFMs to reduce the geometric and numerical errors. Secondly, by using the algebra of quaternions, we extend the accurate RHFMs to AQRHFMs in order to deal with the color images in a holistic manner. Experimental results show the proposed AQRHFMs achieve promising performance in image reconstruction and object recognition in both noise-free and noisy conditions.

Geometric numerical integration of the assignment flow

The assignment flow is a smooth dynamical system that evolves on an elementary statistical manifold and performs contextual data labeling on a graph. We derive and introduce the linear assignment flow that evolves nonlinearly on the manifold, but is governed by a linear ODE on the tangent space. Various numerical schemes adapted to the mathematical structure of these two models are designed and studied, for the geometric numerical integration of both flows: embedded Runge–Kutta–Munthe–Kaas schemes for the nonlinear flow, adaptive Runge–Kutta schemes and exponential integrators for the linear flow. All algorithms are parameter free, except for setting a tolerance value that specifies adaptive step size selection by monitoring the local integration error, or fixing the dimension of the Krylov subspace approximation. These algorithms provide a basis for applying the assignment flow to machine learning scenarios beyond supervised labeling, including unsupervised labeling and learning from controlled assignment flows.

Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a random forcing term, we show that a probability measure over the numerical solution of ODEs can be obtained by introducing suitable random time-steps in a classical time integrator. This intrinsic randomization allows for the conservation of geometric properties of the underlying deterministic integrator such as mass conservation, symplecticity or conservation of first integrals. Weak and mean-square convergence analysis are derived. We also analyse the convergence of the Monte Carlo estimator for the proposed random time step method and show that the measure obtained with repeated sampling converges in the mean-square sense independently of the number of samples. Numerical examples including chaotic Hamiltonian systems, chemical reactions and Bayesian inferential problems illustrate the accuracy, robustness and versatility of our probabilistic numerical method.