Operator Splitting Method Research Articles

Discontinuity-scale path-following methods for the embedded discontinuity finite element modeling of failure in solids

The initiation and propagation of cracks in solids often leads to unstable structural responses characterized by snap-backs. Path-following procedures allow finding a solution to the algebraic system of equations resulting from the numerical formulation of the considered problem. Accordingly, the boundary value problem is supplemented by a novel global unknown, namely, the loading factor, which should comply with a dedicated equation, the so-called path-following constraint equation. In this contribution, path-following methods are discussed within the framework of the Embedded Finite Element Method (E-FEM). Thanks to the enhanced kinematic description provided by the E-FEM, we show that it is possible to formulate constraint equations where the prescribed quantities are directly related to the dissipative process occurring at the strong discontinuity level. After introducing the augmented E-FEM formulation, three discontinuity-scale path-following constraints and their numerical implementation (using an operator-splitting method) are described. Simple quasi-static strain localization problems characterized by unstable structural responses exhibiting multiple snap-backs are numerically simulated. A comparison with several well-known constraint equations (commonly used in non-linear finite element computations) is finally established. This allows for illustrating the main features of the proposed methods as well as their efficiency in controlling highly unstable embedded discontinuity finite element simulations.

Chordal decomposition in operator-splitting methods for sparse semidefinite programs

We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous approaches, the decomposed SDP is suitable for the application of first-order operator-splitting methods, enabling the development of efficient and scalable algorithms. In particular, we apply the alternating direction method of multipliers (ADMM) to solve decomposed primal- and dual-standard-form SDPs. Each iteration of such ADMM algorithms requires a projection onto an affine subspace, and a set of projections onto small PSD cones that can be computed in parallel. We also formulate the homogeneous self-dual embedding (HSDE) of a primal-dual pair of decomposed SDPs, and extend a recent ADMM-based algorithm to exploit the structure of our HSDE. The resulting HSDE algorithm has the same leading-order computational cost as those for the primal or dual problems only, with the advantage of being able to identify infeasible problems and produce an infeasibility certificate. All algorithms are implemented in the open-source MATLAB solver CDCS. Numerical experiments on a range of large-scale SDPs demonstrate the computational advantages of the proposed methods compared to common state-of-the-art solvers.

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.

A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations

Novel compact schemes based on half-step discretization are developed to solve two dimensional fourth order quasi-linear parabolic equations. These discretizations use a single computational cell and the boundary conditions are incorporated in a natural way without the use of any fictitious points or discretization. The schemes have second order accuracy in time and fourth order accuracy in space. The proposed methods are directly applicable to problems constituting singular terms. We also present operator splitting method for the solution of linear parabolic equations which allows the use of the one-dimensional tri-diagonal solver multiple times. It is shown that the operator splitting method is unconditionally stable. The numerical schemes have been successfully applied to the two-dimensional Extended Fisher–Kolmogorov equation, Sivashinsky equation, Boussinesq equation and vibrations problem. The illustrative results confirm the theoretical order of magnitude and accuracy of the proposed methods.

Numerical solution of the regularized logarithmic Schrödinger equation on unbounded domains

The numerical solution of the logarithmic Schrödinger equation on unbounded domains is considered in this paper. It is difficult to develop numerical methods for the logarithmic Schrödinger equation on unbounded domains, due to the blow up of the logarithmic nonlinearity and the unboundedness of the physical domain. Thus, a regularized version of the logarithmic Schrödinger equation on unbounded domains with a small regularization parameter is developed. Then, the local artificial boundary conditions for the regularized logarithmic Schrödinger equation are designed by applying the unified approach, which based on the idea of well-known operator splitting method. The regularized logarithmic Schrödinger equation defined on unbounded domains is reduced to an initial boundary value problem on the bounded computational domain, which can be solved by the finite difference method. The convergence and the stability of the reduced problem are analyzed by introducing some auxiliary variables. In order to choose the optimal absorb parameter in the local artificial boundary conditions, an adaptive algorithm is presented. Numerical results are reported to verify the accuracy and effectiveness of our proposed method.

An operator splitting approach for distributed generalized Nash equilibria computation

In this paper, we propose a distributed algorithm for computation of a generalized Nash equilibrium (GNE) in noncooperative games over networks. We consider games in which the feasible decision sets of all players are coupled together by a globally shared affine constraint. Adopting the variational GNE as a refined solution, we reformulate the problem as that of finding the zeros of a sum of monotone operators through a primal–dual analysis and an augmentation of variables. Then we introduce a distributed algorithm based on forward–backward operator splitting methods. Each player only needs to know its local objective function, local feasible set, and a local block of the affine constraint, and share information with its neighbours. However, each player also needs to observe the decisions that its objective function directly depends on to evaluate its local gradient. We show convergence of the proposed algorithm for fixed step-sizes under some mild assumptions. Moreover, a distributed algorithm with inertia is also introduced and analysed for distributed variational GNE seeking. Numerical simulations are given for networked Cournot competition with bounded market capacities, to illustrate the algorithm efficiency and performance.

Numerical solution of a parabolic optimal control problem in economics

In this paper, we construct and study finite difference approximations of optimal control problems arising in the mathematical modeling of certain management and financial problems. For the definiteness we consider two-dimensional in space variables problems, although all the theoretical results remain valid for any dimension. The numerical calculation results are presented for 1D and 2D problems. To approximate the state equations we use implicit (backward Euler) and fractional steps (operator splitting) methods. The solutions of the constructed mesh state equations are strictly positive and keep an analogue of the mass balance condition. We provide the existence results and first order optimality conditions for the corresponding mesh optimal control problems. We use several iterative methods to implement the constructed nonlinear optimization problems and compare their effectiveness.

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.

An Envelope for Davis–Yin Splitting and Strict Saddle-Point Avoidance

It is known that operator splitting methods based on forward–backward splitting, Douglas–Rachford splitting, and Davis–Yin splitting decompose difficult optimization problems into simpler subproblems under proper convexity and smoothness assumptions. In this paper, we identify an envelope (an objective function), whose gradient descent iteration under a variable metric coincides with Davis–Yin splitting iteration. This result generalizes the Moreau envelope for proximal-point iteration and the envelopes for forward–backward splitting and Douglas–Rachford splitting iterations identified by Patrinos, Stella, and Themelis. Based on the new envelope and the stable–center manifold theorem, we further show that, when forward–backward splitting or Douglas–Rachford splitting iterations start from random points, they avoid all strict saddle points with probability one. This result extends the similar results by Lee et al. from gradient descent to splitting methods.

A novel approach to rigid spheroid models in viscous flows using operator splitting methods

Calculating cost-effective solutions to particle dynamics in viscous flows is an important problem in many areas of industry and nature. We implement a second-order symmetric splitting method on the governing equations for a rigid spheroidal particle model with torques, drag and gravity. The method splits the operators into a vector field that is conservative and one that takes into account the forces of the fluid. Error analysis and numerical tests are performed on perturbed and stiff particle-fluid systems. For the perturbed case, the splitting method greatly improves the solution accuracy, when compared to a conventional multistep method, and the global error behaves as $\mathcal {O}(\varepsilon h^{2})$ for roughly equal computational cost. For stiff systems, we show that the splitting method retains stability in regimes where conventional methods blow up. In addition, we show through numerical experiments that the global order is reduced from $\mathcal {O}(h^{2}/\varepsilon )$ in the perturbed regime to $\mathcal {O}(h)$ in the stiff regime.

Multiplicative Noise Removal Based on the Smooth Diffusion Equation

The multiplicative noise removal problem is of momentous significance in various image processing applications. In this paper, a nonlinear diffusion equation with smooth solution is proposed to remove multiplicative Gamma noise. The diffusion coefficient takes full advantage of two features of multiplicative noise image, namely, gradient information and gray level information, which makes the model has the ability to remove high level noise effectively and protect the edges. The existence of the solution has been analyzed by Schauder’s fixed-point theorem. Some other theoretical properties such as the maximum principle are also presented in the paper. In the numerical aspect, the explicit finite difference method, fast explicit diffusion method, additive operator splitting method and Krylov subspace spectral method are employed to implement the proposed model. Experimental results show that the fast explicit diffusion method achieves a better trade-off between computational time and denoising performance, and the Krylov subspace spectral method gets better restored results in the visual aspect. In addition, the capability of the proposed model for denoising is illustrated by comparison with other denoising models.

Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

Efficient time integration methods for Gross-Pitaevskii equations with rotation term

The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrodinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.

Numerical Simulation of Ship Motions in Regular and Irregular Waves

AbstractA numerical simulation method is presented in this study to predict ship resistance and motion responses in regular and irregular waves. The unsteady RANS (Reynolds Average Navier-Stokes) method is selected as the governing equation, and a volume of fluid (VoF) model is used to capture the free surface, combining the k-ε equations. A finite volume method (FVM) is utilized to discretize both the RANS equations and VoF transport equation. The pressure implicit split operator (PISO) method is set as the velocity-pressure coupling equation. The overset mesh technique is utilized to simulate ship motions in waves. A DTMB5415 ship is selected as a case study to predict its pitch and heave responses in regular and irregular waves at different wave length and wave steepness. The ship is free to move in the pitch and heave directions. The CFD (Computational Fluid Dynamics) results are found to be in good agreement with the strip theory and experimental data. It can be found that the CFD method presented in this study can provide a theoretical basis and technical support for green design and manufacture of ships.

An alternating direction explicit method for time evolution equations with applications to fractional differential equations

We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive operator splitting (AOS) method, which has been developed for treating a wide range of linear and nonlinear time evolution equations with the zero Dirichlet boundary condition. For linear equations, it has been shown to achieve the second order accuracy in time yet is unconditionally stable for an arbitrary time step size. For the boundary conditions considered in this work, we carefully construct the updating formula at grid points near the boundary of the computational domain and show that these formulas maintain the desired accuracy and the property of unconditional stability. We also construct numerical methods based on the ADE scheme for two classes of fractional differential equations. We will give numerical examples to demonstrate the simplicity and the computational efficiency of the method.

An inertial three-operator splitting algorithm with applications to image inpainting

The three-operators splitting algorithm is a popular operator splitting method for finding the zeros of the sum of three maximally monotone operators, with one of which is cocoercive operator. In this paper, we propose a class of inertial three-operator splitting algorithm. The convergence of the proposed algorithm is proved by applying the inertial Krasnoselskii-Mann iteration under certain conditions on the iterative parameters in real Hilbert spaces. As applications, we develop an inertial three-operator splitting algorithm to solve the convex minimization problem of the sum of three convex functions, where one of them is differentiable with Lipschitz continuous gradient. Finally, we conduct numerical experiments on a constrained image inpainting problem with nuclear norm regularization. Numerical results demonstrate the advantage of the proposed inertial three-operator splitting algorithms.

Operator splitting and discontinuous Galerkin methods for advection-reaction-diffusion problem. Application to plant root growth

The time dependant advection-reaction-diffusion equation is used in the C-Root model to simulate root growth. This equation can also be applied in many others applications in life sciences. In this context the unknown is related to densities and one of the important property of the problem is that the solution is non-negative for positive initial conditions. One of the difficulty at the discrete level is to preserve the positivity of the approximated solution during the simulations. In this work we solved the model using Discontinuous Galerkin elements combined with an operator splitting technique. The DG method is briefly presented then we motivated the use of the operator splitting technique by doing some numerical experiments. Those experiments showed that the same time approximation scheme may not be suitable for all the operators of the model. We validated our implementation of the splitting technique in a simple test case. Then we performed a simulation of a plagiotropic root of Eucalyptus.

A new operator splitting method for American options under fractional Black–Scholes models

A new operator splitting method is proposed for American options under time-fractional Black–Scholes models. The fractional linear complementarity problem is split into two easy sub-problems, with the leading coefficients separated from the convolution sum and matched through a general correction step. The method is implementation friendly in the sense that one can easily modify a fractional European solver to obtain the proposed method, since the correction step is decoupled and is trivial to solve. The method is validated through numerical experiments and demonstrated to be superior to the traditional approach. The paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models.

Tight Global Linear Convergence Rate Bounds for Operator Splitting Methods

In this paper, we establish necessary and sufficient conditions for global linear convergence rate bounds in operator splitting methods for a general class of convex optimization problems, where the associated fixed-point operator is strongly quasi-nonexpansive. We also provide a tight bound on the achievable convergence rate. Most existing results establishing global linear convergence in such methods require restrictive assumptions regarding strong convexity and smoothness of the constituent functions in the optimization problem. However, there are several examples in the literature showing that linear convergence is possible even when these properties do not hold. We provide a unifying analysis method for establishing global linear convergence based on linear regularity and show that many existing results are special cases of our approach. Moreover, we propose a novel linearly convergent splitting method for linear programming.

A New Fast Algorithm Based on Half-Step Discretization for 3D Quasilinear Hyperbolic Partial Differential Equations

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.