Operator Splitting Technique Research Articles

Stabilized hybrid discontinuous Galerkin finite element method for the cardiac monodomain equation.

Numerical methods for solving the cardiac electrophysiology model, which describes the electrical activity in the heart, are proposed. The model problem consists of a nonlinear reaction-diffusion partial differential equation coupled to systems of ordinary differential equations that describes electrochemical reactions in cardiac cells. The proposed methods combine an operator splitting technique for the reaction-diffusion equation with primal hybrid methods for spatial discretization considering continuous or discontinuous approximations for the Lagrange multiplier. A static condensation is adopted to form a reduced global system in terms of the multiplier only. Convergence studies exhibit optimal rates of convergence and numerical experiments show that the proposed schemes can be more efficient than standard numerical techniques commonly used in this context when preconditioned iterative methods are used for the solution of linear systems.

A novel time efficient structure-preserving splitting method for the solution of two-dimensional reaction-diffusion systems

In this article, the first part is concerned with the important questions related to the existence and uniqueness of solutions for nonlinear reaction-diffusion systems. Secondly, an efficient positivity-preserving operator splitting nonstandard finite difference scheme (NSFD) is designed for such a class of systems. The presented formulation is unconditionally stable as well as implicit in nature and even time efficient. The proposed NSFD operator splitting technique also preserves all the important properties possessed by continuous systems like positivity, convergence to the fixed points of the system, and boundedness. The proposed algorithm is implicit in nature but more efficient in time than the extensively used Euler method.

New optimization algorithms for neural network training using operator splitting techniques

In the following paper we present a new type of optimization algorithms adapted for neural network training. These algorithms are based upon sequential operator splitting technique for some associated dynamical systems. Furthermore, we investigate through numerical simulations the empirical rate of convergence of these iterative schemes toward a local minimum of the loss function, with some suitable choices of the underlying hyper-parameters. We validate the convergence of these optimizers using the results of the accuracy and of the loss function on the MNIST, MNIST-Fashion and CIFAR 10 classification datasets.

The use of space-splitting RBF-FD technique to simulate the controlled synchronization of neural networks arising from brain activity modeling in epileptic seizures

This paper investigates the behavior and synchronization of a network of reaction–diffusion neural dynamics models using a highly efficient numerical method. In fact, the dynamical modeling, behavior analysis and controlled synchronization of a network of FitzHugh–Nagumo (FHN) neurons which promising the understanding of cognitive processing are studied by considering the unidirectional gap junctions in the medium between two distant neurons. In this study, radial basis function generated finite differences (RBF-FD) technique is employed in conjunction with a suitable operator splitting technique, which allows us to decouple the nonlinear partial differential equations of neural network models into independent linear algebraic equations of very small dimensions. The most important advantages of the proposed method can be high accuracy and high speed, very low computational complexity, and the sparsity property of the matrix of the coefficients derived from its linear systems, which distinguish the proposed method from other methods. The analyses and numerical results presented totally confirm these claims.

OSQP: an operator splitting solver for quadratic programs

We present a general-purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix at almost every iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It can be configured to be division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior-point methods, and sometimes much more when factorization caching or warm start is used. OSQP has already shown a large impact with tens of thousands of users both in academia and in large corporations.

Global Attractor of Reaction–Diffusion Gene Regulatory Networks with S-Type Delay

This paper studies the existence of the global attractor for the reaction–diffusion gene regulatory networks with s-type delay. Firstly, two Hanalay inequalities are proposed and proved, then the uniform boundedness theorem is proved by using these inequalities and semigroup theory and functional analysis technique, the operator splitting technique based on the superposition principle is also utilized to deal with the asymptotic compact of the semigroup. Then some sufficient conditions on existence of global attractor are established, which are easily verifiable and have a wider application range. At last, two examples are discussed to validate the effectiveness of our results. Moreover, the simulation is given by using Matlab.

SuperMann: A Superlinearly Convergent Algorithm for Finding Fixed Points of Nonexpansive Operators

Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive operators, such methods suffer many well known downsides, which include high sensitivity to ill conditioning and parameter selection, and consequent low accuracy and robustness. As universal solution we propose SuperMann, a Newton-type algorithm for finding fixed points of nonexpansive operators. It generalizes the classical Krasnosel'skii-Mann scheme, enjoys its favorable global convergence properties and requires exactly the same oracle. It is based on a novel separating hyperplane projection tailored for nonexpansive mappings which makes it possible to include steps along any direction. In particular, when the directions satisfy a Dennis-Mor\'e condition we show that SuperMann converges superlinearly under mild assumptions, which, surprisingly, do not entail nonsingularity of the Jacobian at the solution but merely metric subregularity. As a result, SuperMann enhances and robustifies all operator splitting schemes for structured convex optimization, overcoming their well known sensitivity to ill conditioning.

Numerical modelling of adsorption and desorption of water vapor in zeolite 13X using a two-temperature model and mixed-hybrid finite element method numerical solver

In this paper, we present a new two-temperature mathematical model for a thermo-chemical energy storage based on the adsorption and desorption of the water vapor in zeolite 13X. The adsorption process is modelled using the Linear Driving Force (LDF) model and the Langmuir-Freundlich isotherms. A numerical solver based on the mixed-hybrid finite element method and operator splitting technique is proposed. Furthermore, we present a computational study of the charging and discharging processes of the thermo-chemical energy storage emphasising the behaviour of the two temperatures: fluid temperature and zeolite temperature.

Splitting-based domain decomposition methods for two-phase flow with different rock types

In this paper, we are concerned with the global pressure formulation of immiscible incompressible two-phase flow between different rock types. We develop for this problem two robust schemes based on domain decomposition (DD) methods and operator-splitting techniques. The first scheme follows a sequential procedure in which the (global) pressure, the saturation-advection and the saturation-diffusion problems are fully decoupled. In this scheme, each problem is treated individually using various DD approaches and specialized numerical methods. The coupling between the different problems is explicit and the time-marching is with no iterations. To adapt to different time scales of problem components and different rock types, the novel scheme uses a multirate time stepping strategy, by taking multiple finer time steps for saturation-advection within one coarse time step for saturation-diffusion and pressure, and permits independent time steps for the advection step in the different rocks. In the second scheme, we review the classical Implicit Pressure–Explicit Saturation (IMPES) method (by decoupling only pressure and saturation) in the context of multirate coupling schemes and nonconforming-in-time DD approaches. For the discretization, the saturation-advection problem is approximated with the explicit Euler method in time, and in space with the cell-centered finite volume method of first order of Godunov type. The saturation-diffusion problem is approximated in time with the implicit Euler method and in space with the mixed finite element method, as in the pressure problem. Finally, in a series of numerical experiments, we investigate the practicality of the proposed schemes, the accuracy-in-time of the multirate and nonconforming time strategies, and compare the convergence of various DD methods within each approach.

Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization

The alternating direction method of multipliers is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well known that the algorithm generates iterates that converge to a solution, provided that it exists. If a solution does not exist, then the iterates diverge. Nevertheless, we show that they yield conclusive information regarding problem infeasibility for optimization problems with linear or quadratic objective functions and conic constraints, which includes quadratic, second-order cone, and semidefinite programs. In particular, we show that in the limit the iterates either satisfy a set of first-order optimality conditions or produce a certificate of either primal or dual infeasibility. Based on these results, we propose termination criteria for detecting primal and dual infeasibility.

A high order efficient numerical method for 4-D Wigner equation of quantum double-slit interferences

We propose a high order numerical method for computing time dependent 4-D Wigner equation with unbounded potentials and study a canonical quantum double-slit interference problem. To address the difficulties of 4-D phase space computations and higher derivatives from the Moyal expansion of the nonlocal pseudo-differential operator for unbounded potentials, an operator splitting technique is adopted to decompose the 4-D Wigner equation into two sub-equations, which can be computed either analytically or numerically with high efficiency. The first sub-equation contains only a linear convection term in (x,t)-space and can be solved with an upwinding characteristic method, while the second involves the pseudo-differential term and can be approximated by a plane wave expansion in k-space. By exploiting properties of Fourier transformations, the expansion coefficients for the second sub-equation have explicit forms and the resulting scheme is shown to be unconditionally stable for any high order derivatives in the Moyal expansion, ensuring the feasibility of 4-D Wigner numerical simulations for quantum double-slit interferences. Numerical experiments demonstrate the spectral convergence in (x,k)-space and provide highly accurate information on the number, position, and intensity of the interference fringes for different types of slits, quantum particle masses, and initial states (pure and mixed).

Application of CE/SE method to gas-particle two-phase detonations under an Eulerian-Lagrangian framework

This study aims to extend the original Eulerian space-time conservation element and solution element (CE/SE) method to the Eulerian-Lagrangian framework to solve the gas-particle two-phase detonation problems. The gas-aluminum particle two-phase detonations are numerically investigated by the developed Eulerian-Lagrangian code, in which the gas-phase compressible Euler equations are solved by our in-house CE/SE scheme based on quadrilateral meshes. Additionally, the particle-phase Lagrangian equations, together with the stiff source terms of interphase interactions and chemical reactions, are explicitly integrated via the operator-splitting technique. A dynamic data structure is introduced to store particle information to overcome the tremendous communication costs when applying message passing interface parallel to the Eulerian-Lagrangian framework. The code is shown to be of better parallel efficiency in moderate-scale computations than that uses static arrays. Comparisons with previous one-dimensional and two-dimensional simulation results and experimental observations are conducted to demonstrate the accuracy and reliability of the developed Eulerian-Lagrangian CE/SE code in gas-particle two-phase detonation simulations. Moreover, the code is also applied to simulate polydisperse gas-particle detonations which is close to a realistic scenario, and significant differences in detonation characteristics are found when compared with the monodisperse counterparts. The great demands of using the Eulerian-Lagrangian method to obtain more physics-consistent gas-particle detonation results are addressed, which the traditional Eulerian-Eulerian simulations fail to observe.

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

AbstractIn the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L2 and L∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.

Conservative domain decomposition schemes for solving two-dimensional heat equations

In this paper, by combining the operator splitting technique, a new mass-conserved domain decomposition method for two-dimensional heat equations is proposed. Along the each direction, the interface fluxes are first calculated from the explicit fluxes, then the sub-domain’s interior solutions are paralelly computed by the C–N implicit scheme. The scheme is stable under the condition $$r\le 2(\sqrt{6}-2)$$ and the corresponding convergence order of the scheme are given in $$L^2$$ -norm. Numerical results confirm the theoretical results.

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.

Modeling the spreading and interaction between wild and transgenic mosquitoes with a random dispersal.

Due to recent advances in genetic manipulation, transgenic mosquitoes may be a viable alternative to reduce some diseases. Feasibility conditions are obtained by simulating and analyzing mathematical models that describe the behavior of wild and transgenic populations living in the same geographic area. In this paper, we present a reaction–diffusion model in which the reaction term is a nonlinear function that describes the interaction between wild and transgenic mosquitoes, considering the zygosity, and the diffusive term that represents a nonuniform spatial spreading characterized by a random diffusion parameter. The resulting nonlinear system of partial differential equations is numerically solved using the sequential operator splitting technique, combining the finite element method and Runge-Kutta method. This scheme is numerically implemented considering uncertainty in the diffusion parameters of the model. Several scenarios simulating spatial release strategies of transgenic mosquitoes are analyzed, demonstrating an intrinsic association between the transgene frequency in the total population and the strategy adopted.

Reliable nonlinear hybrid simulation using modified operator splitting technique

One of the main challenges of hybrid simulation is developing integration methods that not only provide accurate and stable results but also are compatible with the hybrid simulation circumstances. This paper presents a novel enhanced integration technique for hybrid simulation termed “modified operator splitting” (MOS) method. The main aim of the MOS technique is to improve the precision of the operator splitting (OS) method by reducing the corrector step length, where initial stiffness is utilized instead of actual stiffness. For this purpose, a new algorithm is proposed, which makes a more precise estimation of the predictor displacement; thus minimizes the effect of the corrective procedure. As a result, a higher percentage of the restoring force is based on actual experimental behavior rather than being approximated, which finally leads to a very precise and reliable integration method, especially for nonlinear hybrid simulations. Performance of the MOS method is evaluated by the following: (a) Analytical studies, (b) numerical simulations, and (c) hybrid simulation. The accuracy analysis for a wide range of structures and ductility levels demonstrates the superior precision of MOS over OS, especially for severe nonlinearities and larger Δt/T0. In terms of stability, it is shown that for practical applications of civil engineering problems, MOS provides stable results. Furthermore, the application of MOS to hybrid simulation not only again verifies its higher precision over OS, but also shows that MOS minimizes the accumulation of errors during the test; the characteristic that is desirable for a method applied to feedback systems such as hybrid simulation.

Cascaded lattice Boltzmann method based on central moments for axisymmetric thermal flows including swirling effects

A cascaded lattice Boltzmann (LB) approach based on central moments and multiple relaxation times to simulate thermal convective flows, which are driven by buoyancy forces and/or swirling effects, in the cylindrical coordinate system with axial symmetry is presented. In this regard, the dynamics of the axial and radial momentum components along with the pressure are represented by means of the 2D Navier-Stokes equations with geometric mass and momentum source terms in the pseudo Cartesian form, while the evolutions of the azimuthal momentum and the temperature field are each modeled by an advection-diffusion type equation with appropriate local source terms. Based on these, cascaded LB schemes involving three distribution functions are formulated to solve for the fluid motion in the meridian plane using a D2Q9 lattice, and to solve for the azimuthal momentum and the temperature field each using a D2Q5 lattice. The geometric mass and momentum source terms for the flow fields and the energy source term for the temperature field are included using a new symmetric operator splitting technique, via pre-collision and post-collision source steps around the cascaded collision step for each distribution function. These result in a particularly simple and compact formulation to directly represent the effect of various geometric source terms consistently in terms of changes in the appropriate zeroth and first order moments. Simulations of several complex buoyancy-driven thermal flows and including rotational effects in cylindrical geometries using the new axisymmetric cascaded LB schemes show good agreement with prior benchmark results for the structures of the velocity and thermal fields as well as the heat transfer rates given in terms of the Nusselt numbers. Furthermore, the method is shown to be second order accurate and significant improvements in numerical stability with the use of the cascaded LB formulation when compared to other collision models for axisymmetric flow simulations are demonstrated.

Simulation of spatial systems with demographic noise.

The demographic (shot) noise in population dynamics scales with the square root of the population size. This process is very important, as it yields an absorbing state at zero field, but simulating it, especially on spatial domains, is a nontrivial task. Here, we analyze two similar methods that were suggested for simulating the corresponding Langevin equation, one by Pechenik and Levine and the other by Dornic, Chaté, and Muñoz (DCM). These methods are based on operator-splitting techniques and the essential difference between them lies in which terms are bundled together in the splitting process. Both these methods are first order in the time step so one may expect that their performance will be similar. We find, surprisingly, that when simulating the stochastic Ginzburg-Landau equation with two deterministic metastable states, the DCM method exhibits two anomalous behaviors. First, the stochastic stall point moves away from its deterministic counterpart, the Maxwell point, when decreasing the noise. Second, the errors induced by the finite time step are larger by a significant factor (i.e., >10×) in the DCM method. We show that both these behaviors are the result of a finite-time-step induced shift in the deterministic Maxwell point in the DCM method, due to the particular operator splitting employed. In light of these results, care must be exercised when computing quantities like phase-transition boundaries (as opposed to universal quantities such as critical exponents) in such stochastic spatial systems.

The new mass-conserving S-DDM scheme for two-dimensional parabolic equations with variable coefficients

In the article, a new and efficient mass-conserving operator splitting domain decomposition method (S-DDM) is proposed and analyzed for solving two dimensional variable coefficient parabolic equations with reaction term. The domain is divided into multiple non-overlapping block-divided subdomains. On each block-divided subdomain, the interface fluxes are first computed explicitly by local multi-point weighted schemes and the solutions in the interior of subdomain are computed by the one-directional operator splitting implicit schemes at each time step. The scheme is proved to satisfy mass conservation over the whole domain of domain decomposition. By combining with some auxiliary lemmas and applying the energy method, we analyze theoretically the stability of our scheme and prove it to have second order accuracy in space step in the L2 norm. Numerical experiments are performed to illustrate its accuracy, conservation, stability, efficiency and parallelism. Our scheme not only keeps the excellent advantages of the non-overlapping domain decomposition and the operator splitting technique, but also preserves the global mass.