THE UPWIND OPERATOR SPLITTING FINITE DIFFERENCE METHOD FOR COMPRESSIBLE TWO-PHASE DISPLACEMENT PROBLEM AND ANALYSIS

For compressible two-phase displacement problem, a kind of upwind operator splitting finite difference schemes is put forward and make use of operator splitting, of calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L 2 norm are derived to determine the error, in the approximate solution.

A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure.The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix.Moreover,we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations,which is unconditionally convergent for any positive constant.Meanwhile,the iteration methods introduce a new preconditioner for Krylov subspace methods.Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner,compared with the existing approaches.

The Operator Splitting Method for Black-Scholes Equation

The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.

Differentially weighted operator splitting Monte Carlo method for simulating complex aerosol dynamic processes

A differentially weighted operator splitting Monte Carlo （DWOSMC） method is developed to solve com- plex aerosol dynamic processes by coupling the differentially weighted Monte Carlo method and the operator splitting technique. This method is validated by analytical solutions and a sectional method in different aerosol dynamic processes. It is first validated in coagulation and condensation processes, and nucleation and coagulation processes, and then validated through simultaneous nucleation, coagulation, and condensation processes. The results show that the DWOSMC method is a computationally efficient and quantitatively accurate method for simulating complex aerosol dynamic processes.

Operator Splitting for Three-Phase Flow in Heterogeneous Porous Media

We describe an operator splitting technique based on physics rather than on dimension for the numerical solution of a nonlinear system of partial differential equations which models three-phase flow through heterogeneous porous media.The model for three-phase flow considered in this work takes into account capillary forces,general relations for the relative permeability functions and variable porosity and permeability fields.In our numerical procedure a high resolution,nonoscillatory,second order,conservative central difference scheme is used for the approximation of the nonlinear system of hyperbolic conservation laws modeling the convective transport of the fluid phases.This scheme is combined with locally conservative mixed finite elements for the numerical solution of the parabolic and elliptic problems associated with the diffusive transport of fluid phases and the pressure-velocity problem.This numerical procedure has been used to investigate the existence and stability of nonclassical shock waves(called transitional or undercompressive shock waves)in two-dimensional heterogeneous flows,thereby extending previous results for one-dimensional flow problems.Numerical experiments indicate that the operator splitting technique discussed here leads to computational efficiency and accurate numerical results.

Binary Level Set Methods for Dynamic Reservoir Characterization by Operator Splitting Scheme

In this paper,operator splitting scheme for dynamic reservoir characterization by binary level set method is employed.For this problem,the absolute permeability of the two-phase porous medium flow can be simulated by the constrained augmented Lagrangian optimization method with well data and seismic time-lapse data.By transforming the constrained optimization problem in an unconstrained one,the saddle point problem can be solved by Uzawas algorithms with operator splitting scheme,which is based on the essence of binary level set method.Both the simple and complicated numerical examples demonstrate that the given algorithms are stable and efficient and the absolute permeability can be satisfactorily recovered.

Propagation and Pinning of Travelling Wave for Nagumo Type Equation

In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.

THREE HIGH-ORDER SPLITTING SCHEMES FOR 3D TRANSPORT EQUATION

Two high-order splitting schemes based on the idea of the operators splitting method are given. The three-dimensional advection-diffusion equation was split into several one-dimensional equations that were solved by these two schemes, only three computational grid points were needed in each direction but the accuracy reaches the spatial fourth-order. The third scheme proposed is based on the classical ADI scheme and the accuracy of the advection term of it can reach the spatial fourth-order. Finally, two typical numerical experiments show that the solutions of these three schemes compare well with that given by the analytical solution when the Peclet number is not bigger than 5.

Connections between Operator-Splitting Methods and Deep Neural Networks with Applications in Image Segmentation

Deep neural network is a powerful tool for many tasks.Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years.In the literature of mathematical analysis of deep neural networks,a lot of works is dedicated to establishing representation theories.How to make connections between deep neural networks and mathematical algorithms is still under development.In this paper,we give an algorithmic explanation for deep neural networks,especially in their connections with operator splitting.We show that with certain splitting strategies,operator-splitting methods have the same structure as networks.Utilizing this connection and the Potts model for image segmentation,two networks inspired by operator-splitting methods are proposed.The two networks are essentially two operator-splitting algorithms solving the Potts model.Numerical experiments are presented to demonstrate the effectiveness of the proposed networks.

Numerical simulation of profile control by clay particles after polymer flooding

A three-dimensional,two-phase,five-component mathematical model has been developed to describe flow characteristics of clay particles and flocs in the profile control process,in which the clay particle suspension is injected into the formation to react with residual polymer.This model considers the reaction of clay particles with residual polymer,apparent viscosity of the mixture,retention of clay particles and flocs,as well as the decline in porosity and permeability caused by the retention of clay particles and flocs.A finite difference method is used to discretize the equation for each component in the model.The Runge-Kutta method is used to solve the polymer flow equation,and operator splitting algorithms are used to split the flow equation for clay particles into a hyperbolic equation for convection and a parabolic equation for diffusion,which effectively ensures excellent precision,high speed and good stability.The numerical simulation had been applied successfully in the 4-P1920 unit of the Lamadian Oilfield to forecast the blocking capacity of clay particle suspension and to optimize the injection parameters.

Infragravity Waves Produced by Wave Groups on Beaches

The generation of low frequency waves by a single or double wave groups incident upon two Plane beaches with the slope of 1/40 and 1/100 is investigated experimentally and numerically. A new type of wave maker signal is used to generate the groups, allowing the bound long wave (set-down) to be included in the group. The experiments show that the low frequency wave is generated during breaking and propagation to the shoreline of the wave group. This process of generation and propagation of low frequency waves is simulated numerically by solving the short-wave averaged mass and momentum conservation equations. The computed and measured results are in good agreement. The mechanism of generation of low frequency waves in the surf zone is examined and discussed.

An Efficient Operator-Splitting Method for Noise Removal in Images

In this work,noise removal in digital images is investigated.The importance of this problem lies in the fact that removal of noise is a necessary pre-processing step for other image processing tasks such as edge detection,image segmentation,image compression,classification problems,image registration etc.A number of different approaches have been proposed in the literature.In this work,a non-linear PDE-based algorithm is developed based on the ideas proposed by Lysaker,Osher and Tai[IEEE Trans.Image Process.,13(2004),1345-1357].This algorithm consists of two steps:flow field smoothing of the normal vectors,followed by image reconstruction.We propose a finite-difference based additive operator-splitting method that allows for much larger time-steps.This results in an efficient method for noise-removal that is shown to have good visual results.The energy is studied as an objective measure of the algorithm performance.

Efficient Splitting Methods Based on Modified Potentials:Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrodinger Equation

We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular,for the time dependent Schrodinger equation,both in real and imaginary time.They are based on the use of a double commutator and a modified processor,and are more efficient than other widely used schemes found in the literature.Moreover,for certain potentials,they achieve order six.Several examples in one,two and three dimensions clearly illustrate the computational advantages of the new schemes.

Proximity point algorithm for low-rank matrix recovery from sparse noise corrupted data

The method of recovering a low-rank matrix with an unknown fraction whose entries are arbitrarily corrupted is known as the robust principal component analysis （RPCA）. This RPCA problem, under some conditions, can be exactly solved via convex optimization by minimizing a combination of the nuclear norm and the 11 norm. In this paper, an algorithm based on the Douglas-Rachford splitting method is proposed for solving the RPCA problem. First, the convex optimization problem is solved by canceling the constraint of the variables, and ~hen the proximity operators of the objective function are computed alternately. The new algorithm can exactly recover the low-rank and sparse components simultaneously, and it is proved to be convergent. Numerical simulations demonstrate the practical utility of the proposed algorithm.

ON THE ERROR ESTIMATES OF A NEW OPERATE SPLITTING METHOD FOR THE NAVIER-STOKES EQUATIONS

In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.

An Inertial Semi-forward-reflected-backward Splitting and Its Application

Inertial methods play a vital role in accelerating the convergence speed of optimization algorithms.This work is concerned with an inertial semi-forward-reflected-backward splitting algorithm of approaching the solution of sum of a maximally monotone operator,a cocoercive operator and a monotone-Lipschitz continuous operator.The theoretical convergence properties of the proposed iterative algorithm are also presented under mild conditions.More importantly,we use an adaptive stepsize rule in our algorithm to avoid calculating Lipschitz constant,which is generally unknown or difficult to estimate in practical applications.In addition,a large class of composite monotone inclusion problem involving mixtures of linearly composed and parallel-sum type monotone operators is solved by combining the primal-dual approach and our proposed algorithm.As a direct application,the obtained inertial algorithm is exploited to composite convex optimization problem and some numerical experiments on image deblurring problem are also investigated to demonstrate the efficiency of the proposed algorithm.

New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations

We present a new splitting method for time-dependent convention-dominated diffusion problems.The original convention diffusion system is split into two sub-systems:a pure convection system and a diffusion system.At each time step,a convection problem and a diffusion problem are solved successively.A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme;while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers.The scheme can be extended for solving the Navier-Stokes equations,where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step,while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process.Numerical simulations are presented to demonstrate the stability,convergence and performance of the single-step and multistep variants of the new scheme.

On Newton’s Method for Solving Nonlinear Equations and Function Splitting

We provided in[14]and[15]a semilocal convergence analysis for Newton’s method on a Banach space setting,by splitting the given operator.In this study,we improve the error bounds,order of convergence,and simplify the sufficient convergence conditions.Our results compare favorably with the Newton-Kantorovich theorem for solving equations.

Structure-Preserving Finite-Element Schemes for the Euler-Poisson Equations

We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and selfgravitation modeling.The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law,as well as hyperbolic invariant domain properties,such as positivity of the density and a minimum principle of the specific entropy.A detailed discussion of algorithmic details is given,as well as proofs of the claimed properties.We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme.

RESERVOIR DESCRIPTION BY USING A PIECEWISE CONSTANT LEVEL SET METHOD

We consider the permeability estimation problem in two-phase porous media flow. We try to identify the permeability field by utilizing both the production data from wells as well as inverted seismic data. The permeability field is assumed to be piecewise constant, or can be approximated well by a piecewise constant function. A variant of the level set method, called Piecewise Constant Level Set Method is used to represent the interfaces between the regions with different permeability levels. The inverse problem is solved by minimizing a functional, and TV norm regularization is used to deal with the ill-posedness. We also use the operator-splitting technique to decompose the constraint term from the fidelity term. This gives us more flexibility to deal with the constraint and helps to stabilize the algorithm.