Operator Splitting Method Research Articles

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation method in space combined with the Crank–Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution, which avoids solving the nonlinear algebraic equation. Moreover, the consistency of the fully discretized scheme for the linear subproblem and error estimates of the operator splitting scheme are provided. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, mass and energy conservation of the proposed method.

A fast and efficient numerical algorithm for image segmentation and denoising

<abstract><p>Image segmentation is the process of partitioning an image into homogenous regions, and represents one of the most fundamental and important procedures in image processing. Image denoising is a process to remove unwanted noise from a digital image, enhancing its visual quality. Various algorithms, like non-local means and deep learning-based approaches, have been developed to remove noise while preserving important image details. Currently, the prevalent application of pattern recognition technology is achieved through the implementation of image segmentation algorithms. In this study, we present a new, highly efficient, and fast computational scheme specifically developed for a phase-field mathematical model of image segmentation. The numerical methodology is based on an operator splitting method (OSM). The split operators are solved by using closed-form analytic solutions and a finite difference method (FDM) with an alternating direction explicit (ADE) method. To show the notable efficiency and rapid computational performance of the proposed computational algorithm, we conduct a series of numerical experiments. Through these computational tests, we confirm a significant contribution to the advancement of methodologies employed in the critical domain of image processing.</p></abstract>

The ADMM algorithm for audio signal recovery and performance modification with the dual Douglas-Rachford dynamical system

<abstract><p>Practitioners employ operator splitting methods—such as alternating direction method of multipliers (ADMM) and its "dual" Douglas-Rachford method (DR)—to solve many kinds of optimization problems. We provide a gentle introduction to these algorithms, and illustrations of their duality-like relationship in the context of solving basis pursuit problems for audio signal recovery. Recently, researchers have used the dynamical systems associated with the iterates of splitting methods to motivate the development of schemes to improve performance. These developments include a class of methods that act by iteratively minimizing surrogates for a Lyapunov function for the dynamical system. An exemplar of this class is currently state-of-the-art for the feasibility problem of finding wavelets with special structure. Early experimental evidence has also suggested that, when implemented in a primal-dual (ADMM and DR) framework, this exemplar may provide improved performance for basis pursuit problems. We provide a reasonable way to compute the updates for this exemplar, and we study the application of this method to the aforementioned basis pursuit audio problems. We provide experimental results and visualizations of the dynamical system for the dual DR sequence. We observe that for highly structured problems with real data, the algorithmic behavior is noticeably different than for randomly generated problems.</p></abstract>

Energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn model

We present a new energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn (NtAC) model, where the standard Laplace operator is deliberately replaced with a spatial convolution term that aims at describing long-range interactions among particles and the structure-preserving scheme is obtained based on the operator splitting method. The discrete maximum bound principle and global convergence of the new scheme are analyzed rigorously. Moreover, the scheme is strictly dissipative for a modified energy, which coincides with the original energy up to O(τ), where τ is the time step. To the best of our knowledge, this is the first work to show that the operator splitting method can guarantee the maximum bound principle for a ternary Allen-Cahn model. Various 2D/3D numerical experiments including the NtAC model with mass conservation are tested to verify the bound-preserving and energy-stable properties of the resulted scheme.

Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type

Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type

Generalisation of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schrödinger type

The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schrödinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schrödinger case.Numerical illustrations for the time-dependent Gross–Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods.The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross–Pitaevskii systems in real and imaginary time.

Stability and error estimates of Strang splitting method for the nonlocal ternary conservative Allen–Cahn model

The nonlocal model has attracted a great attention in materials science for describing various types of material heterogeneities and defects. In this study, we consider a nonlocal ternary conservative Allen–Cahn model, where the standard Laplace operator is intentionally replaced with a spatial convolution term that aims at describing long-range interactions among particles. A linear energy stable scheme is developed based on the operator splitting method. The mass conservation, energy stability and global convergence of the new scheme are analyzed rigorously. Numerical stability and convergence of the present numerical scheme are analyzed theoretically. Two and three dimensional numerical experiments are performed to validate the theoretical analysis and the efficiency of the method.

Reduced model for fracture of geometrically exact planar beam: Non‐local variational formulation, ED‐FEM approximation and operator split solution

SummaryIn this work, we present a novel reduced model for computing the solution to a bending fracture problem of the geometrically exact beam. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model. The non‐local variational formulation is proposed to deal with the softening phenomena characteristic of fracture. The fracture energy is introduced as the main fracture parameter, rather than (in general) ill‐defined characteristic length used for previous non‐local models. The discrete approximation is built in terms of the embedded‐discontinuity finite element method (ED‐FEM), which needs no initial crack for optimal performance nor detailed crack kinematics description, such as in the X‐FEM or the phase‐field approach. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best‐approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and stress) from local (internal) variables including the crack opening. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Study on Temperature Distribution Law of Tunnel Portal Section in Cold Region Considering Fluid–Structure Interaction

The design of tunnels in cold regions contributes greatly to the feasibility and sustainability of highways. Based on the heat transfer mechanism of the tunnel surrounding rock–lining–air, this paper uses FEPG software to carry out secondary excavation and development, then the air heat convection calculation model is established by using a three-dimensional extension of the characteristic-based operator-splitting (CBOS) finite-element method and the explicit characteristic–Galerkin method. By coupling with the heat conduction model of the tunnel lining and surrounding rock, the heat conduction-thermal convection fluid–structure interaction finite-element calculation model of tunnels in cold regions is established. Relying on the Qinghai Hekashan tunnel project, the temperature field of the tunnel portal section is calculated and studied by employing the fluid–structure interaction finite-element model and then compared with the field monitoring results. It is found that the calculated values are basically consistent with the measured values over time, which proves the reliability of the model. The calculation results are threefold: (1) The temperature of the air, lining, and surrounding rock in the tunnel changes sinusoidally with the ambient temperature. (2) The temperature of each layer gradually lags behind, and the temperature variation amplitude of the extreme value of the layer temperature gradually decreases with the increase in the radial distance of the lining. (3) In the vicinity of the tunnel entrance, the lining temperature of each layer remains unchanged, and the temperature gradually decreases or increases with the increase in the depth. The model can be used to study and analyze the temperature field distribution law of the lining and surrounding rock under different boundary conditions, and then provide a calculation model with both research and practical value for the study of the temperature distribution law of tunnels in cold regions in the future.

Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems

A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.

A weak approximation for Bismut’s formula: An algorithmic differentiation method

The paper provides a novel algorithmic differentiation method by constructing a weak approximation for Bismut’s formula. A new operator splitting method based on Gaussian Kusuoka-approximation is introduced for an enlarged semigroup describing “differentiation of diffusion semigroup”. The effectiveness of the new algorithmic differentiation is checked through numerical examples.

Temple system on networks

This paper deals with the well-posedness on a network of a Temple system of nonlinear hyperbolic balance laws. Temple systems are characterized by the fact that shock and rarefaction curves coincide. This study is motivated by a model for traffic, recently proposed, inspired by kinetic considerations. The proof of the well-posedness is based on the wave-front tracking procedure, on the pseudo-polygonal technique and on the operator splitting method.

Novel operator-splitting methods for real-time dynamic substructure testing

Improved and corrected Operator-Splitting (OS) methods are proposed for Real-Time Dynamic Substructure Testing (RTDST) in this paper. Firstly, the theory and the stability of two proposed OS methods are investigated. Furthermore, the effectiveness of two proposed OS methods is validated by RTDSTs with one pure mass specimen and one mass-stiffness specimen. By using the predictor velocity and acceleration as well as the correction force approximations, the improved OS method and the corrected OS method are explicit methods for RTDST. It is shown from the analytical and experimental results that the improved OS method is unconditionally stable for linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping system if the mass ratio of the experimental substructure to the numerical substructure is less than 1. In contrast, the improved OS method is unsuitable for RTDST if the mass ratio is larger than 1. The corrected OS method is unconditionally stable for the linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping systems. The stability of the corrected OS method is better than the improved OS method, which indicates the advantage of the corrected OS method over the improved OS method.

Low-rank Tensor Restoration for ERP extraction

Event-related potential (ERP) data is essentially multi-dimensional, with correlated data in some spaces. Therefore, matrix and vector analysis results in structural information loss. Tensor decomposition can be used to explore the shared structural information of ERP signal among related conditions. Perceiving the decaying trends of the singular value changes of unfolding matrices indicate that they are low-rank matrices. Based on this assumption, in this work, a low-rank tensor restoration (LTR) method is proposed. An operator splitting method known as the alternating direction method of multipliers (ADMM) is adapted to tackle the proposed optimization problem with orthogonality and sparsity constraints. Accordingly, the problem is solved in a sequential fashion by computing the unconstrained and orthogonality constrained quadratic sub-problems with closed-form solutions. The algorithm is examined under three application areas, namely, noise removal, feature extraction and subject-to-subject transfer learning. The empirical evaluations on real P300-based ERP dataset demonstrate the robustness and effectiveness of the proposed method.

RBF based some implicit–explicit finite difference schemes for pricing option under extended jump-diffusion model

In this manuscript, we presented some efficient and accurate radial basis function-based finite difference (RBF-FD) implicit–explicit (IMEX) numerical techniques for pricing the option when the underlying asset follows the jump-diffusion process with local volatility. For the time semi-discretization, we present three numerical techniques Crank–Nikolson Leap-Frog (CNLF), Crank–Nikolson AdamBashforth (CNAB), and Backward difference formula of order two (BDF2), incorporated with the radial basis function based finite difference (RBF-FD) method. The stabilities of time semi-discretized schemes are also proved. The computational methods developed for the European option are extended for the American option. We amalgamate the RBF-FD implicit–explicit methods with an operator splitting (OS) method for solving the linear complementarity problem (LCP) with variable parameters that determines the price of an American option. In order to demonstrate the effectiveness and precision of the current techniques, numerical data for European and American put options under the Merton and Kou models are presented.

Efficient Numerical Simulation of Biochemotaxis Phenomena in Fluid Environments.

A novel dimension splitting method is proposed for the efficient numerical simulation of a biochemotaxis model, which is a coupled system of chemotaxis-fluid equations and incompressible Navier-Stokes equations. A second-order pressure correction method is employed to decouple the velocity and pressure for the Navier-Stokes equations. Then, the alternating direction implicit scheme is used to solve the velocity equation, and the operator with dimension splitting effect is used instead of the traditional elliptic operator to solve the pressure equation. For the chemotactic equation, the operator splitting method and extrapolation technique are used to solve oxygen and cell density to achieve second-order time accuracy. The proposed dimension splitting method splits the two-dimensional problem into a one-dimensional problem by splitting the spatial derivative, which reduces the computation and storage costs. Finally, through interesting experiments, we show the evolution of the cell plume shape during the descent process. The effect of changing specific parameters on the velocity and plume shape during the descent process is also studied.

Uncoupling Techniques for Multispecies Diffusion–Reaction Model

We consider the multispecies model described by a coupled system of diffusion–reaction equations, where the coupling and nonlinearity are given in the reaction part. We construct a semi-discrete form using a finite volume approximation by space. The fully implicit scheme is used for approximation by time, which leads to solving the coupled nonlinear system of equations at each time step. This paper presents two uncoupling techniques based on the explicit–implicit scheme and the operator-splitting method. In the explicit–implicit scheme, we take the concentration of one species in coupling term from the previous time layer to obtain a linear uncoupled system of equations. The second approach is based on the operator-splitting technique, where we first solve uncoupled equations with the diffusion operator and then solve the equations with the local reaction operator. The stability estimates are derived for both proposed uncoupling schemes. We present a numerical investigation for the uncoupling techniques with varying time step sizes and different scales of the diffusion coefficient.

Fast and efficient numerical method for solving the Allen–Cahn equation on the cubic surface

In this study, we present a fast and efficient finite difference method (FDM) for solving the Allen–Cahn (AC) equation on the cubic surface. The proposed method applies appropriate boundary conditions in the two-dimensional (2D) space to calculate numerical solutions on cubic surfaces, which is relatively simpler than a direct computation in the three-dimensional (3D) space. To numerically solve the AC equation on the cubic surface, we first unfold the cubic surface domain in the 3D space into the 2D space, and then apply the FDM on the six planar sub-domains with appropriate boundary conditions. The proposed method solves the AC equation using an operator splitting method that splits the AC equation into the linear and nonlinear terms. To demonstrate that the proposed algorithm satisfies the properties of the AC equation on the cubic surface, we perform the numerical experiments such as convergence test, total energy decrease, and maximum principle.

Distributed robust optimization with coupled constraints via Tseng’s splitting method

We consider distributed robust convex optimization with both local and coupled constraints, where coefficients with respect to local constraints are uncertain in polyhedral sets. We derive an equivalent optimization model with extended decision variables and finite constraints, and characterize its optimality condition. Also, we transform the problem as finding zeros of an operator and reveal the monotonicity of the underlying operator. By employing Tseng’s operator splitting method, we design a distributed algorithm to solve the considered robust optimization problem with guaranteed convergence, and also illustrate the efficiency in numerical simulations.

Tracking Contaminant Transport Backwards with an Operator-Splitting Method

Recovering the past movement of a contaminant plume from measurements of its current values is a challenging problem in hydrology. Moreover, modeling the movement of a contaminant plume backwards is an ill-posed problem due to the unstable and non-unique nature of the resulting solution. Therefore, standard numerical methods become unstable, making it impossible to simulate existing contaminant transport models with reversed time. This paper presents two major contributions to solve the backward problem. Firstly, a stable and consistent numerical method based on an operator-splitting concept which is effective in tracking back the contaminant movement, and secondly, an optimal condition for the choice of mesh width that enables the error during computer simulation to stay within a reasonable bound. The numerical method was validated by introducing errors of varied strengths at the starting point and reconstructing the contaminant profiles backwards at any given time.