Symplectic Partitioned Runge Kutta 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.

A modified symplectic discontinuous Galerkin method for acoustic and elastic wave simulations

Numerically solving seismic wave equations is vital to large-scale forward modeling and full waveform inversion. In this paper, a new modified symplectic discontinuous Galerkin (MSDG) method is proposed to solve the acoustic and elastic equations. The MSDG method employs a symmetric interior penalty Galerkin formulation as the space discretization. The time discretization is based on a modified symplectic partitioned Runge–Kutta scheme with minimized phase error. Thus, the MSDG method has the advantages of high accuracy, being flexible to deal with complex geometric boundaries and internal structures, and stable for long time simulations. The numerical stability conditions, dispersion and dissipation are investigated in detail for the MSDG method. To validate the method, we carry out several numerical examples for solving the acoustic and elastic wave equations in various media. The numerical results show that the MSDG method can effectively suppress the numerical dispersion and is suitable for wavefield simulations.

A conservative fourth-order real space method for the (2+1)D Dirac equation

Modelling the time-dependent (2+1)D Dirac equation has recently gained importance since this equation effectively describes multiple condensed matter systems. To avoid the large dispersion errors of second-order real space schemes, a highly accurate method is presented here instead. The method utilises a fourth-order central difference on a staggered grid and an explicit symplectic Partitioned Runge–Kutta (PRK) time integrator. In contrast to traditional Runge–Kutta (RK) time stepping, no unphysical dissipation is introduced into the simulation. Moreover, it is demonstrated, both theoretically via Poisson maps and numerically, that the novel scheme has excellent conservation properties. Furthermore, the proposed numerical method is provably stable and the dispersion error is low and isotropic. Several interesting numerical examples are presented. Besides validating the advocated method, they also showcase its computational efficiency and low memory consumption.

Discrete LQR and ILQR methods based on high order Runge-Kutta discretizations

In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretizations are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in Hager (2000) [11], direct approaches with RK discretization are equivalent with indirect approaches based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue of continuous and discrete dynamic programming. Then, based on the equivalence, we discuss the issue that the internal-stage controls produced by direct approaches may have lower order accuracy than the RK method used. We propose order conditions for internal-stage controls and then demonstrate that third or fourth order explicit RK methods cannot avoid the order reduction phenomenon. To overcome this obstacle, we calculate node control instead of internal-stage controls in DLQR and ILQR methods. Another advantage of our methods is high computational efficiency which comes from the usage of feedback technique. A simple numerical example will illustrate the validity of ILQR in solving nonlinear optimal control problems with high-order RK discretizations. In this paper, we also demonstrate that ILQR is essentially a quasi-Newton method with linear convergence rate.

An efficient symplectic stereo-modeling method for seismic inversion by using deep learning technique

AbstractThis paper proposes an efficient symplectic stereo-modeling (SSTEM) method for full waveform inversion (FWI) by using a deep learning technique. To solve the 2D acoustic equation, the SSTEM method uses a third-order optimal symplectic partitioned Runge–Kutta approach as a time-stepping method. An eighth-order stereo-modeling operator is used for spatial discretization. The SSTEM method is then expressed with a recurrent neural network (RNN). This is realized mainly because the time advancing format of the SSTEM method is similar to that of RNN, and they both use the information from the previous time step to obtain information from the current time step. With SSTEM as the forward modeling method, FWI is implemented using Tensorflow. The well-known adaptive moment estimation (Adam) optimizer and Nesterov adaptive moment estimation (Nadam) optimizer with mini-batch are used. The applicability of the developed code is also verified on GPUs. The numerical results show that the SSTEM method is more efficient and produces less numerical dispersion than the conventional finite-difference (FD) method when the same sampling rate in a wavelength is used. We compare several loss functions. The mean square (MSE) error and absolute (ABS) error loss functions are first tested. Another loss function that adds a physical differential operator to the original loss function is then considered. The FWI results show that this loss function has some improvements. Finally, we implement FWI on the complex Marmousi and SEG/EAGE models, and the inversion results demonstrate that the proposed method is suitable for seismic imaging in complex media.

Second-order symplectic PRK scheme for ground-penetrating radar simulations in dispersive materials

An accurate and effective forward model is helpful in interpreting measured ground-penetrating radar (GPR) data. In this paper, an explicit second-order symplectic partitioned Runge–Kutta scheme (SPRK) with Higdon's absorbing boundary condition is developed to simulate GPR wave propagation in dispersive media. For this purpose, the first-order Debye dispersion model is successfully applied to the second-order SPRK method. A single-layer dispersive medium model and a three-layer pavement structure model are simulated, in order to verify the accuracy and efficiency of the second-order SPRK method, as well as to clarify the effect of dispersion on the GPR echo signal. The numerical simulation results demonstrate that the proposed algorithm not only can maintain the simulation accuracy, but can also save about 19% of the calculation time, when compared with the FDTD algorithm, in a dispersive medium. We also obtain the GPR profile of a pavement structure with a circular cavity disease and an underground pipe structure in a dispersive medium, and the simulation results show that the dispersion characteristics of the materials lead to high absorption of the GPR electromagnetic waves. Finally, field experiment data are used to further verify the applicability of the algorithm. The numerical simulations and experiments indicate that the dispersion of the medium must be considered in GPR numerical simulation, and the authenticity of the simulated image must be ensured.

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.

An optimal nearly analytic splitting method for solving 2D acoustic wave equations

In this article, we suggest a new optimal nearly analytic splitting (ONAS) method for 2D acoustic wave simulation. We first split 2D acoustic wave equation into the local one-dimensional (LOD) equations. Then we solve each of these equations over half of the time step used for the complete 2D wave equation using technique similar to the optimal nearly analytic discrete method (ONADM) for the 1D acoustic wave equation. We provide the theoretical study on the properties of the ONAS method, such as stability criteria, theoretical error, numerical error, and numerical dispersion. We also compare some acoustic modeling results of this method against those of the ONADM, the fourth-order nearly analytic symplectic partitioned Runge–Kutta (NSPRK) method, the fourth-order Lax–Wendroff correction (LWC) method. Theoretical analysis and numerical tests show that the ONAS method is fourth-order accurate in time and fourth-order accurate in space, and the ONAS method is much more effective than the fourth-order LWC, the fourth-order NSPRK and the ONADM in suppressing numerical dispersion. The new ONAS method theoretically preserves the symplectic geometry structure as time proceeds. Numerical results illustrate that the ONAS method has better numerical stability and long-term calculation capability as time proceeds than the ONADM

The Conformal Finite-Difference Time-Domain Simulation of GPR Wave Propagation in Complex Geoelectric Structures

The finite-difference time-domain (FDTD) method adopts the most popular numerical model simulating ground penetrating radar (GPR) wave propagation in an underground structure. However, a staircase approximation method is usually adopted to simulate the curved boundary of an irregular object in the FDTD and symplectic partitioned Runge-Kutta (SPRK) methods. The approximate processing of rectangular mesh parameters will result in calculation errors and virtual surface waves for irregular targets of an underground structure. In this paper, we examine transverse mode (TM) electromagnetic waves with numerical models of electromagnetic wave propagation in geoelectric structures with conformal finite-difference time-domain (CFDTD) method technology in which the effective dielectric parameters are used to accurately simulate the dielectric surface and to absorb waves at the edges of the grid. The third orders of the transmission boundary are used in this paper. Additionally, three complex geocentric models of inclined layered media, spherical media, and three-layered pavement model with structural damages are set up for simulation calculations, then we carry out the actual radar wave detection in a laboratory as the fourth numerical example. Comparison of simulated reflectance waveform of FDTD, symplectic partitioned Runge-Kutta (SPRK), and CFDTD methods shows that at least 50% of the virtual waves can be reduced by using the proposed algorithm. Wiggle diagrams of FDTD and CFDTD methods show that much of the virtual waves have been reduced, and the radar image is clearer than before. This provides a method for the detection of complex geoelectric and layered structures in actual engineering.

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.

Nonsplit complex-frequency-shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 2: Wavefield simulations

The perfectly matched layer (PML) is an efficient artificial boundary condition that has been routinely implemented in seismic wave modeling. However, the effective combination of PML with symplectic numerical schemes for solving seismic wave equations has rarely been studied. In a companion paper, we have developed a complex-frequency-shifted convolutional PML (CPML) with a nonconstant compression grid parameter for solving the time-domain second-order seismic wave equation. Subsequently, we combine this CPML with two classes of symplectic methods to formulate symplectic partitioned Runge-Kutta (SPRK) + CPML and nearly analytic SPRK (NSPRK) + CPML, both of which are properly synchronized. To further investigate their validity, the two algorithms are then applied to acoustic and elastic wave simulations in typical geologic models, including a heterogeneous acoustic model, several isotropic and orthotropic elastic models, and an isotropic elastic model with a free-surface boundary. Relevant numerical results demonstrate the effectiveness of our CPML and combination algorithms. Specifically, the numerical accuracy and stability of the CPML that we develop are greatly improved compared with the classic split-field PML. Moreover, the final model with the free-surface boundary condition indicates that the nonconstant grid-compression parameter can eliminate the unstable modes at the free surface in the PML domain. The (N)SPRK + CPML that we propose is prospective for future application in other complex models and wave-equation-based migration and inversion.

Prediction of evanescent coupling efficiency in two parallel silica nanowires

In this study the efficiency of evanescent wave coupling between two air-clad silica nanowires in single-mode operation is numerically predicted in time-domain using the finite difference method. To this end, a three-dimensional scheme developed for solving the Maxwell’s equations on staggered grids will be employed. The electric and magnetic field solutions are sought subjected to the zero-divergence condition (or Gauss’s law) in discrete level. In addition, we aim to conserve Hamiltonians existing in ideal Maxwell’s equations at all times by adopting the explicit second-order accurate symplectic partitioned Runge–Kutta temporal scheme. Moreover, all the spatial derivative terms in the Faraday’s and Ampère’s equations are approximated by the proposed scheme which can render not only fourth-order spatial accuracy, but also minimize the discrepancy between the exact and the derived numerical phase velocities.

The Three Dimension First-Order Symplectic Partitioned Runge-Kutta Scheme Simulation for GPR Wave Propagation in Pavement Structure

Numerical simulation of three-layer layered electromagnetic waves is key problem for nondestructive testing of ground penetrating radar (GPR) pavement. In this paper, the difference iterative scheme of three-dimensional first-order symplectic partitioned Rung-Kutta is derived, which is applied to pavement detection of ground penetrating radar by using Higdon ABC boundary condition. Incident waves are considered as line sources. The accuracy and efficiency of the proposed algorithm are verified by the traditional 3D-FDTD algorithm. The results indicate that the accuracy and efficiency between the two methods are consistent. Unlike the traditional 3D-FDTD algorithm, the CPU time of the proposed method is reduced by 30%. The diseases location of the pavement structure is directly reflected by the numerical simulation result of the proposed method. This provides a three-dimensional symplectic partitioned Rung-Kutta algorithm, which can be applied to the forward simulation of GPR. It provides a three-dimensional symplectic partitioned Rung-Kutta algorithm, which can be applied to the forward simulation of GPR. The accurate electromagnetic response information of the target can be obtained by 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.

A discrete optimality system for an optimal harvesting problem

In this paper, we obtain the discrete optimality system of an optimal harvesting problem. While maximizing a combination of the total expected utility of the consumption and of the terminal size of a population, as a dynamic constraint, we assume that the density of the population is modeled by a stochastic quasi-linear heat equation. Finite-difference and symplectic partitioned Runge–Kutta (SPRK) schemes are used for space and time discretizations, respectively. It is the first time that a SPRK scheme is employed for the optimal control of stochastic partial differential equations. Monte-Carlo simulation is applied to handle expectation appearing in the cost functional. We present our results together with a numerical example. The paper ends with a conclusion and an outlook to future studies, on further research questions and applications.

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

Modified symplectic schemes with nearly-analytic discrete operators for acoustic wave simulations

Using a structure-preserving algorithm significantly increases the computational efficiency of solving wave equations. However, only a few explicit symplectic schemes are available in the literature, and the capabilities of these symplectic schemes have not been sufficiently exploited. Here, we propose a modified strategy to construct explicit symplectic schemes for time advance. The acoustic wave equation is transformed into a Hamiltonian system. The classical symplectic partitioned Runge–Kutta (PRK) method is used for the temporal discretization. Additional spatial differential terms are added to the PRK schemes to form the modified symplectic methods and then two modified time-advancing symplectic methods with all of positive symplectic coefficients are then constructed. The spatial differential operators are approximated by nearly-analytic discrete (NAD) operators, and we call the fully discretized scheme modified symplectic nearly analytic discrete (MSNAD) method. Theoretical analyses show that the MSNAD methods exhibit less numerical dispersion and higher stability limits than conventional methods. Three numerical experiments are conducted to verify the advantages of the MSNAD methods, such as their numerical accuracy, computational cost, stability, and long-term calculation capability.

A modified symplectic PRK scheme for seismic wave modeling

A new scheme for the temporal discretization of the seismic wave equation is constructed based on symplectic geometric theory and a modified strategy. The ordinary differential equation in terms of time, which is obtained after spatial discretization via the spectral-element method, is transformed into a Hamiltonian system. A symplectic partitioned Runge–Kutta (PRK) scheme is used to solve the Hamiltonian system. A term related to the multiplication of the spatial discretization operator with the seismic wave velocity vector is added into the symplectic PRK scheme to create a modified symplectic PRK scheme. The symplectic coefficients of the new scheme are determined via Taylor series expansion. The positive coefficients of the scheme indicate that its long-term computational capability is more powerful than that of conventional symplectic schemes. An exhaustive theoretical analysis reveals that the new scheme is highly stable and has low numerical dispersion. The results of three numerical experiments demonstrate the high efficiency of this method for seismic wave modeling.