Symplectic Partitioned Runge-Kutta Methods Research Articles

A Splitting Nearly-Analytic Symplectic Partitioned Runge–Kutta Method for Solving 2D Elastic Wave Equations

In this paper, we present a new splitting nearly-analytic symplectic partitioned Runge–Kutta (SNSPRK) method for the two-dimensional (2D) elastic wave equations. It is an extension to elastic wave equation of our recent work on the locally one-dimensional nearly-analytic symplectic partitioned Runge–Kutta (LOD-NSPRK) method for the 2D acoustic wave equations. The method is based on the spatial differential operator-split technique, in which the resulting spatial discrete matrices are symmetric unlike the conventional nearly-analytic symplectic partitioned Runge–Kutta (NSPRK) method. The stability condition is given, which has more relax restriction for the time step than the NSPRK schemes. To show the performance of the new method, numerical experiments are given. Numerical results illustrate that the SNSPRK method has better long-term calculation capability as time proceeds than the NSPRK method.

Optimal Control of Closed Quantum Systems via B-Splines with Carrier Waves

We consider the optimal control problem of determining electromagnetic pulses for implementing logical gates in a closed quantum system, where the Hamiltonian models the dynamics of coupled superconducting qudits. The quantum state is governed by Schrodinger's equation, which we formulate in terms of the real and imaginary parts of the state vector and solve by the Stormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method. A novel parameterization of the control functions based on B-splines with carrier waves is introduced. The carrier waves are used to trigger the resonant frequencies in the system Hamiltonian, and the B-spline functions specify their amplitude and phase. This approach allows the number of control parameters to be independent of, and significantly smaller than, the number of time steps for integrating Schrodinger's equation. We present numerical examples of how the proposed technique can be combined with an interior point L-BFGS algorithm for realizing quantum gates, and generalize our approach to calculate risk-neutral controls that are resilient to noise in the Hamiltonian model. The proposed method is also shown to compare favorably with QuTiP/pulse\_optim and Grape-Tensorflow.

Symplectic P-stable additive Runge—Kutta methods

<p style='text-indent:20px;'>Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are <i>P-stable</i>, therefore suitable for application to highly oscillatory problems.</p>

Learning Adaptive Regularization for Image Labeling Using Geometric Assignment

We study the inverse problem of model parameter learning for pixelwise image labeling, using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determines the regularization properties of the assignment flow. Using the symplectic partitioned Runge–Kutta method for numerical integration, it is shown that deriving the sensitivity conditions of the parameter learning problem and its discretization commute. A convenient property of our approach is that learning is based on exact inference. Carefully designed experiments demonstrate the performance of our approach, the expressiveness of the mathematical model as well as its limitations, from the viewpoint of statistical learning and optimal control.

A nearly analytic symplectic partitioned Runge–Kutta method based on a locally one‐dimensional technique for solving two‐dimensional acoustic wave equations

ABSTRACTIn this paper, we develop a new nearly analytic symplectic partitioned Runge–Kutta method based on locally one‐dimensional technique for numerically solving two‐dimensional acoustic wave equations. We first split two‐dimensional acoustic wave equation into the local one‐dimensional equations and transform each of the split equations into a Hamiltonian system. Then, we use both a nearly analytic discrete operator and a central difference operator to approximate the high‐order spatial differential operators, which implies the symmetry of the discretized spatial differential operators, and we employ the partitioned second‐order symplectic Runge–Kutta method to numerically solve the resulted semi‐discrete Hamiltonian ordinary differential equations, which results in fully discretized scheme is symplectic unlike conventional nearly analytic symplectic partitioned Runge–Kutta methods. Theoretical analyses show that the nearly analytic symplectic partitioned Runge–Kutta method based on locally one‐dimensional technique exhibits great higher stability limits and less numerical dispersion than the nearly analytic symplectic partitioned Runge–Kutta method. Numerical experiments are conducted to verify advantages of the nearly analytic symplectic partitioned Runge–Kutta method based on locally one‐dimensional technique, such as their computational efficiency, stability, numerical dispersion and long‐term calculation capability.

Hamilton系统下基于相位误差的精细辛算法

Symplectic methods, including the generating function method, the symplectic Runge-Kutta (RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10^-13. For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.

High-order stochastic symplectic partitioned Runge-Kutta methods for stochastic Hamiltonian systems with additive noise

In this paper, a simple class of stochastic partitioned Runge–Kutta (SPRK) methods is proposed for solving stochastic Hamiltonian systems with additive noise. Firstly, the order conditions and symplectic condictions are analysised by using colored rooted tree theory. Then a family of mean-square order 1.5 diagonally implicit stochastic symplectic partitioned Runge–Kutta (SSPRK) methods is presented. Moreover, several explicit SSPRK methods are constructed for systems with a separable Hamiltonian H0=V(p)+U(t,q). Furthermore, these methods are proved to converge with mean-square order 2.0 to the solution when they are applied to second-order stochastic Hamiltonian systems with a separable Hamiltonian and additive noise. Finally, several numerical examples are performed to demonstrate efficiency of those SSPRK methods.

Two modified symplectic partitioned Runge–Kutta methods for solving the elastic wave equation

Based on a modified strategy, two modified symplectic partitioned Runge–Kutta (PRK) methods are proposed for the temporal discretization of the elastic wave equation. The two symplectic schemes are similar in form but are different in nature. After the spatial discretization of the elastic wave equation, the ordinary Hamiltonian formulation for the elastic wave equation is presented. The PRK scheme is then applied for time integration. An additional term associated with spatial discretization is inserted into the different stages of the PRK scheme. Theoretical analyses are conducted to evaluate the numerical dispersion and stability of the two novel PRK methods. A finite difference method is used to approximate the spatial derivatives since the two schemes are independent of the spatial discretization technique used. The numerical solutions computed by the two new schemes are compared with those computed by a conventional symplectic PRK. The numerical results, which verify the new method, are superior to those generated by traditional conventional methods in seismic wave modeling.

A phase-preserving and low-dispersive symplectic partitioned Runge–Kutta method for solving seismic wave equations

A phase-preserving and low-dispersive symplectic partitioned Runge–Kutta method for solving seismic wave equations

A Low‐Dispersive Symplectic Partitioned Runge–Kutta Method for Solving Seismic‐Wave Equations: II. Wavefield Simulations

In a companion paper (Ma et al. , 2014), we proposed a series of nearly analytic symplectic Runge–Kutta (NSPRK) methods with fourth‐order spatial accuracy and with second‐, third‐, and fourth‐order temporal accuracy. In this article, the NSPRK(2,4) scheme, which is temporally second order and the most efficient of the NSPRK schemes, is applied to 3D seismic modeling. We apply the NSPRK(2,4) scheme to simulate acoustic or elastic wave propagation in common 3D medium models, including a three‐layer acoustic model and a vertical transversely isotropic model. A model with a fluid‐filled fracture is used to study elastic wave propagation. When it was necessary to truncate the computational domain, we used the classical split‐field perfectly matched layer (PML) approach to eliminate reflected waves. Compared with conventional methods, the NSPRK method requires only a little additional computation and memory storage on PML domains to obtain a large absorption effect. Numerical results show no visible numerical dispersion, although all simulations were done on relatively coarse grids. These promising numerical results support the use of NSPRK‐type schemes on large‐scale practical models. The NSPRK method with symplecticity‐preserving properties can provide more accurate wave amplitudes than other methods. Online Material: MATLAB code.

Stochastic symplectic partitioned Runge–Kutta methods for stochastic Hamiltonian systems with multiplicative noise

Some new stochastic partitioned Runge–Kutta (SPRK) methods are proposed for the strong approximation of partitioned stochastic differential equations (SDEs). The order conditions up to strong global order 1.0 are calculated. The SPRK methods are applied to solve stochastic Hamiltonian systems with multiplicative noise. Some conditions are captured to guarantee that a given SPRK method is symplectic. It is shown that stochastic symplectic partitioned Runge–Kutta (SSPRK) methods can be written in terms of stochastic generating functions. In addition, this paper also proves that the SSPRK methods can conserve the quadratic invariants of original stochastic systems. Based on the order and symplectic conditions, some low-stage SSPRK methods with strong global order 1.0 are constructed. Finally, some numerical results are presented to demonstrate the efficiency of the SSPRK methods.

Explicit symplectic momentum-conserving time-stepping scheme for the dynamics of geometrically exact rods

A new structure-preserving algorithm for simulating the nonlinear dynamics of geometrically exact rods is developed. The method is based on the simultaneous discretization in space and time of Hamilton׳s variational principle. The resulting variational integrator is explicit, second-order accurate and can be identified with a Lie-group symplectic partitioned Runge–Kutta method for finite element discretizations of rods involving large rotations and displacements. Numerical examples allow to verify that the algorithm presents an excellent long term energy behavior along with the exact conservation of the momenta associated to the symmetries of the system.

A Low-Dispersive Symplectic Partitioned Runge-Kutta Method for Solving Seismic-Wave Equations: I. Scheme and Theoretical Analysis

Abstract In this article, a series of nearly analytic symplectic partitioned Runge–Kutta (NSPRK) methods for the 3D seismic‐wave equation are developed. First, the Hamiltonian formulations for acoustic and elastic‐wave equations are presented, and then the spatial derivatives are discretized by a nearly analytic discrete operator to obtain a semidiscrete Hamiltonian system. The second‐, third‐, and fourth‐order symplectic partitioned Runge–Kutta schemes are then applied as the time integrator. A semianalytic procedure to facilitate the analysis of stability conditions and numerical dispersion relations is presented, and subsequent theoretical analysis shows that the NSPRK schemes preserve the wave velocity better than conventional symplectic schemes, especially on coarser grids. The numerical solutions computed by NSPRK schemes are compared with analytic solutions for the 3D acoustic and elastic cases. We implemented the NSPRK and some conventional schemes for a 3D acoustic wave propagation simulation in a parallel computer and compared their computational efficiencies. To generate comparable results, the NSPRK schemes require much less computer memory, central processing unit time, and communication time, which substantially accelerates the computation speed. The final simulation in the two‐layer acoustic model shows that the NSPRK schemes can suppress numerical dispersion and preserve the waveforms better than conventional symplectic schemes.

Application of Symplectic Partitioned Runge-Kutta Method and Total-Field Scattered-Field Formulation to Simulation of GPR Data

The second-order Lobatto IIIA-IIIB symplectic partitioned RungeKutta (SPRK) method, combining with the first-order Mur absorbing boundary condition, is developed for the simulation of ground penetrating radar wave propagation in layered pavement structure. For 2-dimetional case, a significant advantage of this method is that only two functions need to be calculated at each time step. The total-field/scattered-field technique is used for plane wave excitation. Numerical examples are presented to verify the accuracy and efficiency of the proposed algorithm. The results illustrate that the reflected signal calculated by the SPRK method is in good agreement with that obtained using the finite difference time domain (FDTD) scheme, but the CPU time consumed by proposed algorithm is reduce about 20% of the FDTD scheme. In addition, an actual field test is conducted to evaluate the further performance of the SPRK method. It is found that the simulated waveform fits well with the measured signal in many aspects, especially in the peak amplitude and time delay.

Stability of Explicit Symplectic Partitioned Runge-Kutta Methods

A numerical method for solving Hamiltonian equations is said to be symplectic if it preserves the symplectic structure associated with the equations. Various symplectic methods are widely used in many fields of science and technology. A symplectic method preserves an approximate Hamiltonian perturbed from the original Hamiltonian. It theoretically supports the effectiveness of symplectic methods for long-term integration. Although it is also related to long-term integration, numerical stability of symplectic methods have received little attention. In this paper, we consider explicit symplectic methods defined for Hamiltonian equations with Hamiltonians of the special form, and study their numerical stability using the harmonic oscillator as a test equation. We propose a new stability criterion and clarify the stability of some existing methods that are visually based on the criterion. We also derive a new method that is better than the existing methods with respect to a Courant-Friedrichs-Lewy condition for hyperbolic equations; this new method is tested through a numerical experiment with a nonlinear wave equation.

Symplectic partitioned Runge-Kutta method based on the eighth-order nearly analytic discrete operator and its wavefield simulations

We propose a symplectic partitioned Runge-Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic wave equation. Known as the eighth-order NSPRK method, this technique uses an eighth-order accurate nearly analytic discrete (NAD) operator to discretize high-order spatial differential operators and employs a second-order SPRK method to discretize temporal derivatives. The stability criteria and numerical dispersion relations of the eighth-order NSPRK method are given by a semi-analytical method and are tested by numerical experiments. We also show the differences of the numerical dispersions between the eighth-order NSPRK method and conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction (LWC) method and the eighth-order staggered-grid (SG) method. The result shows that the ability of the eighth-order NSPRK method to suppress the numerical dispersion is obviously superior to that of the conventional numerical methods. In the same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 times faster than the fourth-order SPRK, and the memory requirement is only approximately 47.17% of the fourth-order NSPRK method and 49.41 % of the fourth-order SPRK method, which indicates the highest computational efficiency. Modeling examples for the two-layer models such as the heterogeneous and Marmousi models show that the wavefields generated by the eighth-order NSPRK method are very clear with no visible numerical dispersion. These numerical experiments illustrate that the eighth-order NSPRK method can effectively suppress numerical dispersion when coarse grids are adopted. Therefore, this method can greatly decrease computer memory requirement and accelerate the forward modeling productivity. In general, the eighth-order NSPRK method has tremendous potential value for seismic exploration and seismology research.

Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method

To preserve the symplecticity property, it is natural to require numerical integration of Hamiltonian systems to be symplectic. As a famous numerical integration, it is known that the 2-stage RadauIA method is not symplectic. With the help of symplectic conditions of Runge-Kutta method and partitioned Runge-Kutta method, a symplectic partitioned Runge-Kutta method and a symplectic Runge-Kutta method are constructed on the basis of 2-stage RadauIA method in this paper.

Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method

Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method

A class of symplectic partitioned Runge–Kutta methods

This paper deals with some relevant properties of Runge–Kutta (RK) methods and symplectic partitioned Runge–Kutta (PRK) methods. First, it is shown that the arithmetic mean of a RK method and its adjoint counterpart is symmetric. Second, the symplectic adjoint method is introduced and a simple way to construct symplectic PRK methods via the symplectic adjoint method is provided. Some relevant properties of the adjoint method and the symplectic adjoint method are discussed. Third, a class of symplectic PRK methods are proposed based on Radau IA, Radau IIA and their adjoint methods. The structure of the PRK methods is similar to that of Lobatto IIIA–IIIB pairs and is of block forms. Finally, some examples of symplectic partitioned Runge–Kutta methods are presented.

Parametric symplectic partitioned Runge–Kutta methods with energy-preserving properties for Hamiltonian systems

Based on W-transformation, some parametric symplectic partitioned Runge–Kutta (PRK) methods depending on a real parameter α are developed. For α=0, the corresponding methods become the usual PRK methods, including Radau IA–IA¯ and Lobatto IIIA–IIIB methods as examples. For any α≠0, the corresponding methods are symplectic and there exists a value α∗ such that energy is preserved in the numerical solution at each step. The existence of the parameter and the order of the numerical methods are discussed. Some numerical examples are presented to illustrate these results.