Discrete Gradient Method Research Articles

A novel explicit fully-discrete momentum-preserving scheme of damped nonlinear stochastic wave equation influenced by multiplicative space-time noise

In this paper, the evolution laws of global momentum and averaged global momentum for the damped nonlinear stochastic wave equation (DNSWE) influenced by multiplicative space-time noise are derived. We innovatively combine the second-order central finite difference method and discrete gradient method in space, and integrate the splitting method and Störmer-Verlet type method in time. Both the novel spatial semi-discrete scheme and fully-discrete scheme constructed in this way can successfully preserve the corresponding evolution laws for discrete global momentum and discrete averaged global momentum. Numerical experiments on DNSWE with cubic nonlinearity validate the theoretical results.

A geometric integration approach to smooth optimization: foundations of the discrete gradient method

Abstract Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we present a thorough analysis of the discrete gradient method for optimization that provides a solid theoretical foundation. We show that the discrete gradient method is well-posed by proving the existence of iterates for any positive time step, as well as uniqueness in some cases, and propose an efficient method for solving the associated discrete gradient equation. Moreover, we establish an $\text{O}(1/k)$ convergence rate for convex objectives and prove linear convergence if instead the Polyak–Łojasiewicz inequality is satisfied. The analysis is carried out for three discrete gradients—the Gonzalez discrete gradient, the mean value discrete gradient, and the Itoh–Abe discrete gradient—as well as for a randomised Itoh–Abe method. Our theoretical results are illustrated with a variety of numerical experiments, and we furthermore demonstrate that the methods are robust with respect to stiffness.

A Second-Order Length-Preserving and Unconditionally Energy Stable Rotational Discrete Gradient Method for Oseen-Frank Gradient Flows

A Second-Order Length-Preserving and Unconditionally Energy Stable Rotational Discrete Gradient Method for Oseen-Frank Gradient Flows

Parallel and energy conservative/dissipative schemes for sine–Gordon and Allen–Cahn equations

The sine–Gordon and Allen–Cahn equations are two typical models in the fields of conservative Hamiltonian systems and dissipative gradient flows, respectively. As the demand for numerical methods that respect intrinsic energy conservation/dissipation laws turns into a fundamental principle, the subsequent computational efficiency is getting more and more desired. Linearly implicit methods, which only require solutions of linear algebraic systems at each time step, have become a popular way to design efficient schemes. However, the overall computational cost of these methods depends heavily on the speed at which the linear system can be solved. In this paper, we propose an alternative approach for developing highly efficient energy-conservative or dissipative schemes for the sine–Gordon and Allen–Cahn equations. Our approach is based on spatial finite difference approximations and is fully implicit at first glance, but actually it is completely decoupled point by point, allowing for the implementation of a scalar nonlinear equation successively with a lower complexity that is comparable only to the degrees of freedom. We further establish the connections between our approach and the classic Itoh–Abe discrete gradient and the partitioned averaged vector field methods, and then propose the generalized Itoh–Abe discrete gradient method, which offers great flexibility in the ordering of updating each unknown point. One immediate benefit is that we can specifically choose an ordering to achieve a naturally parallel computation of the resulting scheme, significantly improving the computational efficiency. Various numerical experiments are presented to illustrate the performance of the proposed schemes.

A novel directly energy-preserving method for charged particle dynamics

In this paper, we apply the coordinate increment discrete gradient (CIDG) method to solve the Lorentz force system which can be written as a non-canonical Hamiltonian system. Then we can obtain a new energy-preserving CIDG-I method for the system. The CIDG-I method can combine with its adjoint method CIDG-II which is also a energy-preserving method to form a new method, namely CIDG-C method. The CIDG-C method is symmetrical and can conserve the Hamiltonian energy directly and exactly. With comparison to the well-used Boris method, numerical experiments indicate that the CIDG-C method holds advantage over the Boris method in terms of energy-conserving.

A variant of the discrete gradient method for the solution of the semilinear wave equation under different boundary conditions

In this paper, energy-preserving schemes based upon the discrete gradient are used to numerically solve the semilinear wave equation under periodic boundary conditions, Dirichlet boundary conditions and Neumann boundary conditions. Both the integral form and the evolutionary behaviour of the energy depend on the associated boundary conditions. The wave equations are semi-discretized in different manners for different boundary conditions such that the Hamiltonian functions of the resulting semi-discrete Hamiltonian systems are the discrete analogue of the original energy. Furthermore, the Hamiltonian functions of the resulting semi-discrete systems have similar evolutionary behaviours as the original energy. It is noted that the semi-discrete system exhibits an oscillatory structure. Subsequently, the semi-discrete approximation of the energy evolution as well as the oscillatory structure is passed to the fully-discrete problem by applying extended discrete gradient formula in the temporal direction. Numerical experiments are carried out to show the effectiveness of the new schemes.

Energy-preserving schemes for conservative PDEs based on periodic quasi-interpolation methods

In this paper, we present a novel energy-preserving scheme for solving time-dependent partial differential equations with periodic solutions on non-uniform grids. The proposed scheme combines the periodic quasi-interpolation approaches with the discrete gradient method. One of the main advantages of our method is its ability to ensure energy conservation without relying on the requirement of an anti-symmetric differentiation matrix. We rigorously verify the energy conservation properties of the full-discrete scheme, as well as establish the convergence results. Furthermore, we extend the capabilities of our method by incorporating an adaptive strategy, allowing for the design of an adaptive energy-preserving scheme. This adaptive feature enhances the accuracy and efficiency of the numerical solution. To demonstrate the effectiveness and energy-preserving ability of the proposed scheme, we present several numerical examples. These examples showcase the superior accuracy and robustness of our method.

Fréchet Discrete Gradient and Hessian Operators on Infinite-Dimensional Spaces

Benefiting from the notion of Fréchet derivatives, we define Fréchet discrete operators, such as gradient and Hessian, on infinite-dimensional spaces. The Fréchet discrete gradient expands upon the concept of the discrete gradient of Gonzalez (1996) for finite-dimensional spaces. The Fréchet discrete Hessian elevates the property to second-order representations of the Fréchet derivative. By leveraging these operators, we offer an initial exploration of discrete gradient methods for convex optimization in infinite-dimensional spaces. Under mild conditions on the objective functional, we establish the convergence of any sequence generated by the proposed Fréchet discrete gradient method, regardless of the choice of the finite learning rate.

Nonsmooth Optimization-Based Model and Algorithm for Semisupervised Clustering.

Using a nonconvex nonsmooth optimization approach, we introduce a model for semisupervised clustering (SSC) with pairwise constraints. In this model, the objective function is represented as a sum of three terms: the first term reflects the clustering error for unlabeled data points, the second term expresses the error for data points with must-link (ML) constraints, and the third term represents the error for data points with cannot-link (CL) constraints. This function is nonconvex and nonsmooth. To find its optimal solutions, we introduce an adaptive SSC (A-SSC) algorithm. This algorithm is based on the combination of the nonsmooth optimization method and an incremental approach, which involves the auxiliary SSC problem. The algorithm constructs clusters incrementally starting from one cluster and gradually adding one cluster center at each iteration. The solutions to the auxiliary SSC problem are utilized as starting points for solving the nonconvex SSC problem. The discrete gradient method (DGM) of nonsmooth optimization is applied to solve the underlying nonsmooth optimization problems. This method does not require subgradient evaluations and uses only function values. The performance of the A-SSC algorithm is evaluated and compared with four benchmarking SSC algorithms on one synthetic and 12 real-world datasets. Results demonstrate that the proposed algorithm outperforms the other four algorithms in identifying compact and well-separated clusters while satisfying most constraints.

Elastic waveform inversion in the frequency domain for an application in mechanized tunneling

The excavation process in mechanized tunneling can be improved by reconnaissance of the geology ahead. A nondestructive exploration can be achieved in means of seismic imaging. A full waveform inversion approach, which works in the frequency domain, is investigated for the application in tunneling. The approach tries to minimize the difference of seismic records from field observations and from a discretized ground model by changing the ground properties. The final ground model might be a representation of the geology. The used elastic wave modeling approach is described as well as the application of convolutional perfectly matched layers. The proposed inversion scheme uses the discrete adjoint gradient method, a multi-scale approach as well as the L-BFGS method. Numerical parameters are identified as well as a validation of the forward wave modeling approach is performed in advance to the inversion of every example. Two-dimensional blind tests with two different ground scenarios and with three different source and receiver station configurations are performed and analyzed, where only the seismic records, the source functions and the ambient ground properties are provided. Finally, an inversion for a three-dimensional tunnel model is performed and analyzed for three different source and receiver station configurations.

Numerical Methods That Preserve a Lyapunov Function for Ordinary Differential Equations

The paper studies numerical methods that preserve a Lyapunov function of a dynamical system, i.e., numerical approximations whose energy decreases, just like in the original differential equation. With this aim, a discrete gradient method is implemented for the numerical integration of a system of ordinary differential equations. In principle, this procedure yields first-order methods, but the analysis paves the way for the design of higher-order methods. As a case in point, the proposed method is applied to the Duffing equation without external forcing, considering that, in this case, preserving the Lyapunov function is more important than the accuracy of particular trajectories. Results are validated by means of numerical experiments, where the discrete gradient method is compared to standard Runge–Kutta methods. As predicted by the theory, discrete gradient methods preserve the Lyapunov function, whereas conventional methods fail to do so, since either periodic solutions appear or the energy does not decrease. Moreover, the discrete gradient method outperforms conventional schemes when these do preserve the Lyapunov function, in terms of computational cost; thus, the proposed method is promising.

Geometric Integration of ODEs Using Multiple Quadratic Auxiliary Variables

We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE with only quadratic integrals to which the midpoint rule is applied. The quadratic auxiliary variables can subsequently be eliminated yielding a midpoint-like method on the original phase space. The resulting method is shown to be a novel discrete gradient method. Furthermore, the averaged vector field method can be obtained as a special case of the proposed method. The method can be extended to higher-order through composition and is illustrated through a number of numerical examples.

Biomechanical Analysis of Touch Ball Movements in Tennis Forehand Strokes.

In order to improve the effect of certain theoretical basis for tennis coaches to correct technical movements and teaching training, a method oriented towards discrete gradient methods that can be used for computational solid mechanics is presented. The biggest feature of this method is that it can directly perform numerical simulation analysis on any point cloud model, without relying on any structured or unstructured grid model. Experimental results show that about 80% of the shots in the game are within 2.5 m of the athlete's moving distance, and the athlete needs to have 300 to 500 high-intensity exercises; the total running distance of the competition is 1 100–3 600 m. The average VO2 of athletes during the competition is 20–30 mL/min/kg (45%–55% V.O2max), the average heart rate is 135–155 beats/min (70%–85% HRmax), the mean blood lactate was <4 mmol/L, the subjective fatigue was 12–14 (moderate intensity), and the mean metabolic equivalent was 5–7 METs. It is proved that the discrete gradient method can effectively solve the biomechanical analysis problem of tennis forehand hitting the ball. Make up for the lack of action details in the differentiation stage. Improve the effect of certain theoretical basis for tennis coaches to correct technical movements and teaching training.

Energy-Preserving Continuous-Stage Exponential Runge--Kutta Integrators for Efficiently Solving Hamiltonian Systems

As one of the most important properties, energy preservation is a natural requirement for numerical integrators of Hamiltonian systems. Considering the limited second-order accuracy of the existing exponential average vector filed (or discrete gradient) method, this paper is devoted to developing high-order energy-preserving exponential integrators for general semilinear Hamiltonian systems. To this end, we first formulate and analyze the continuous-stage exponential Runge--Kutta (ERK) method. After deriving the order conditions, energy-preserving conditions, and symmetric conditions for the continuous-stage ERK method, we construct a novel class of fourth-order symmetric energy-preserving continuous-stage ERK methods with a free parameter based on the established conditions and present a convergence theorem for the fixed-point iteration associated with the continuous-stage ERK method. Furthermore, we extend the proposed continuous-stage ERK method to noncanonical Hamiltonian systems and discuss implementation issues in detail. Finally, numerical results from the FPU problem, the charged-particle dynamics, and the Klein--Gordon equation demonstrate the high efficiency, good energy-preserving behavior, and applicability of large time stepsizes of the proposed fourth-order energy-preserving continuous-stage ERK method.

Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise

In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Padé approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings.

A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation

The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh–Abe methods. These are generalisations of the Itoh–Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh–Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency.

Accelerating numerical simulation of continuous-time Boolean satisfiability solver using discrete gradient

To explore the design of analog computing devices, modeling the problem-solving process as a continuous-time dynamical system is important. Ercsey-Ravasz and Toroczkai [Nature Physics 7, 966 (2011)] proposed such a model for solving the Boolean satisfiability (SAT) problem. This system consists of a gradient system that minimizes the potential function reduced from the SAT problem and a system for achieving a temporal variation of the potential function to avoid the problem of non-solution local minima. Although the ability of the system to find a solution to the SAT problem is demonstrated, its large simulation cost hinders theoretical research towards the physical realization and limits its utility on digital computers. This is due to the necessity of small time steps to maintain the numerical stability of the simulation. In this study, we propose a fast and stable numerical simulation algorithm for this solver using the discrete gradient method to allow a larger time step. We also propose an adaptive time step control method for this system. The proposed algorithm achieves a faster simulation by a factor of approximately 100, compared to conventional methods. Although taking a large time step degrades the accuracy, we found that it does not necessarily degrade the performance as a SAT solver; this indicates the new utility of the discrete gradient apart from the conventional studies of numerical simulation algorithms that pursue accuracy as well as efficiency.

Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates

Numerical schemes that preserve the structure of the kinetic equations can provide stable simulation results over a long time. An electromagnetic particle-in-cell solver for the Vlasov--Maxwell equations that preserves at the discrete level the noncanonical Hamiltonian structure of the Vlasov--Maxwell equations has been presented in [Kraus et al., J. Plasma Phys., 83 (2017)]. While the original formulation has been obtained for Cartesian coordinates, we extend the formulation to curvilinear coordinates in this paper. For the discretization in time, we discuss several (semi-)implicit methods either based on a Hamiltonian splitting or a discrete gradient method combined with an antisymmetric splitting of the Poisson matrix and discuss their conservation properties and computational efficiency.

Energy-conserving time propagation for a structure-preserving particle-in-cell Vlasov–Maxwell solver

This paper discusses energy-conserving time-discretizations for finite element particle-in-cell discretizations of the Vlasov–Maxwell system. A geometric spatially discrete system can be obtained using a standard particle-in-cell discretization of the particle distribution and compatible finite element spaces for the fields to discretize the Poisson bracket of the Vlasov–Maxwell model (see Kraus et al. (2017) [1]). In this paper, we derive energy-conserving time-discretizations based on the discrete gradient method applied to an antisymmetric splitting of the Poisson matrix. Firstly, we propose a semi-implicit method based on a splitting that yields constant Poisson matrices in each substep. Moreover, we devise an alternative discrete gradient that yields a time discretization that can additionally conserve Gauss' law. Finally, we explain how substepping for fast species dynamics can be incorporated.

An adaptive image inpainting method based on euler's elastica with adaptive parameters estimation and the discrete gradient method

Euler's Elastica is a common approach developed based on minimizing the elastica energy. It is one of the effective approaches to solve the image inpainting problem. Nevertheless, there are two major issues: the Euler's elastica variational image inpainting model itself is multiparameter, and the performance of methods for solving the model is not high. In the article, we propose an adaptive Euler's elastica image inpainting model by combining with adaptive parameter estimation based on the smoothed structure tensor. To implement the model, a numerical algorithm based on the discrete gradient method is developed. The experiments showed that the proposed image inpainting method outperforms other state-of-the-arts methods in terms of inpainted image quality.