Operator Splitting Method Research Articles

A closer look at SRB souring in porous media: Higher-order finite difference scheme in radial coordinate

Microorganisms have different benefits including bioremediation of aquifers contaminated with the non-aqueous liquid phase, microbial EOR, and bacterial degradation of asphaltenes. Despite these profits, a special kind of bacteria named Sulfate Reducing Bacteria (SRB) causes H2S generation which is toxic and also results in equipment erosion and corrosion. The injected water approaches the production well in a radial form geometry. However, less is known about the H2S generation and movement in the radial coordinate. In this study, we investigated SRB souring in aquifers and oil reservoirs using a reactive numerical simulation of multicomponent and multiphase radial flow. The operator-splitting method was utilized to model this dispersion-dependent phenomenon. Furthermore, a higher-order finite difference approach was adopted to more accurately model SRB souring by avoiding the numerical dispersion. The results of the proposed approach were verified with the existing analytical radial solution by applying the simplifying assumption of the analytical radial solution. Also, the effect of numerical and physical dispersions was evaluated and it was highlighted that under certain conditions numerical dispersion can be confused with the physical one. Moreover, a comparison of the H2S profile in the radial coordinate with linear geometry showed significant differences which should be considered in SRB souring studies. This difference highlights the necessity of radial grid refinement around the wells in full-field reservoir simulation to attain more accuracy. The study of SRB souring during water injection in an oil reservoir also illustrates the same pattern as souring in aquifer but the oil presence delays mixing between sea and formation water and postpones H2S generation. The findings of this work provide a better understanding of near-wellbore SRB souring during water injection in porous media.

A Convergent Interacting Particle Method and Computation of KPP Front Speeds in Chaotic Flows

In this paper, we study the propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficients on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting methods for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical methods. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space.

An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging

Gaussian curvature is an important geometric property of surfaces, which has been used broadly in mathematical modeling. Due to the full nonlinearity of the Gaussian curvature, efficient numerical methods for models based on it are uncommon in literature. In this article, we propose an operator-splitting method for a general Gaussian curvature model. In our method, we decouple the full nonlinearity of Gaussian curvature from differential operators by introducing two matrix- and vector-valued functions. The optimization problem is then converted into the search for the steady state solution of a time dependent PDE system. The above PDE system is well-suited to time discretization by operator splitting, the sub-problems encountered at each fractional step having either a closed form solution or being solvable by efficient algorithms. The proposed method is not sensitive to the choice of parameters, its efficiency and performances being demonstrated via systematic experiments on surface smoothing and image denoising.

Lie-Trotter operator splitting spectral method for linear semiclassical fractional Schrödinger equation

In this paper the error estimates are derived for Lie-Trotter operator splitting spectral method for semiclassical linear fractional Schrödinger equation. We first establish a priori estimates for the solution in fractional Sobolev space. On the basis of these a priori estimates, we obtain local error bound for the well-known Lie-Trotter splitting operator associated with the linear fractional Schrödinger equation in the semiclassical regime by using a formula for the fractional Laplacian of the product of two functions. The convergence orders of the fully discrete scheme based on Fourier spectral methods for the space approximation are then analyzed and provided with respect to the time step-size Δt and the small (scaled) Planck constant ε. As far as we know, this is the first rigorous proof of local error estimate for an semi-discrete operator splitting method for semiclassical fractional Schrödinger equation. Computational experiments confirm the theoretical results.

Three-dimensional volume reconstruction from multi-slice data using a shape transformation

We present a computational method for the 3D volume reconstruction from cross-sectional data. The proposed method is based on the Allen–Cahn (AC) equation with a source term. The source term is related to shape transformation from a source object to a target object. Using the operator splitting method, the governing equation is solved by splitting it into three steps. The numerical solution is obtained explicitly using the Euler's methods and the separation of variables. To reconstruct the 3D object from two slice data, we set one slice as the target data and the other data as the initial data. We solve the governing equation and stack intermediate solutions based on the relative fraction of the symmetric difference of two regions occupied by the target and source data. To demonstrate that the proposed method can reconstruct a 3D model through extracted intermediate slice data during shape transformation, we perform several computational tests. Furthermore, the proposed method is applied to a 3D volume reconstruction from multi-slice data of human vertebra.

An improved finite integration method for nonlocal nonlinear Schrödinger equations

In this paper, we study the nonlocal nonlinear Schrödinger equation (NNLSE), which is an important extension of the nonlinear Schrödinger equation (NLSE). Since the nonlinearity of NNLSE involves a convolution, numerical approximation of its solution is very expensive. To achieve an efficient numerical method for solving the NNLSE, according to the property of convolution, we firstly present a partial differential equation (PDE) method to transfer the NNLSE from an integro-differential equation to an equivalent or approximate system of PDEs, and the numerical solution of the NNLSE can then be obtained by solving these PDEs. Based on the recently developed finite integration method (FIM), we derive an improved method (IFIM) in this paper. The novelty of this IFIM is that the quadrature method is only used once instead of multiple times for multi-layer integral, so it consumes less running time and provides higher accuracy. In addition, we adapt a second-order operator-splitting method (OS2) at each time-step to ensure a convergent solution for long-time integration. Several numerical experiments are given to verify the efficiency and accuracy of the proposed IFIM for solving the NNLSE.

A dynamic relaxation method with operator splitting and random-choice strategy for SPH

In this paper, we propose an efficient dynamic relaxation method for smoothed particle hydrodynamics (SPH) to address the time-consuming issue when dealing with achieving the equilibrium of a dynamic system. First, an artificial-viscosity-based damping term is added into the momentum equation to dissipate the velocity gradient without losing the momentum conservation of the system. Two operator splitting methods, which update velocities implicitly, are then developed to discretize the added damping term for relaxing the time-step limit. To further improve the computational efficiency, a random-choice strategy is also introduced into the dynamic relaxation process, by which the damping is imposed randomly rather than each time step. Furthermore, a splitting cell-linked list scheme is proposed by dividing the conventional cell-linked list into several blocks to avoid the thread conflict when shared-memory parallelization is applied to accelerate the computation. A number of benchmark tests show the fast and efficient feature of the present method for dynamic systems achieving their equilibrium.

Numerical Valuation of European and American Options under Fractional Black-Scholes Model

In this paper, we investigate the numerical valuation of European and American options under the time fractional Black-Scholes model. We first apply a coordinate stretching transformation to the asset price so that the spatial region can focus on the vicinity of singularities, which are usually found in the payoff function. The radial basis function finite difference method is used for the spatial discretization, and the improved L1 method is used to deal with the reduced order of convergence for the nonsmooth initial data. We use the operator splitting method for solving the linear complementary problem of American options. The proposed scheme leads to a sparse linear system which is trivial to solve. Moreover, the stability of the proposed numerical scheme is analyzed using Fourier analysis. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.

An effective operator splitting scheme for two-dimensional conservative nonlocal Allen–Cahn equation

An effective operator splitting scheme for solving the nonlocal Allen–Cahn equation with a Lagrange multiplier is studied. Firstly, based on the operator splitting method, the original equation is discretized into a nonlinear equation, a nonlocal equation and a Lagrange multiplier equation, respectively. Then, the nonlinear equation is solved analytically, the nonlocal equation is discretized using Crank–Nicolson format, and the Lagrange multiplier equation is solved explicitly. The mass conservation and convergence analysis of the numerical algorithm are analyzed theoretically. Numerical experiments are presented to confirm the validity of the proposed method, including the convergence, energy decline and mass conservation.

Numerical Investigation on Low-Salinity Augmented Microbial Flooding within a Sandstone Core for Enhanced Oil Recovery under Nonisothermal and pH Gradient Conditions

Summary In an era of increasing energy demand, declining oil fields, and fluctuating crude oil prices globally, most oil companies are looking forward to implementing cost-effective and environmentally sustainable enhanced oil recovery (EOR) techniques such as low salinity waterflooding (LSWF) and microbial EOR (MEOR). The present study numerically investigates the combined influence of simultaneous LSWF and microbial flooding for in-situ MEOR in tertiary mode within a sandstone core under spatiotemporally varying pH and temperature conditions. The developed black oil model consists of five major coupled submodels: nonlinear heat transport model; ion transport coupled with multiple ion exchange (MIE) involving uncomplexed cations and anions; pH variation with salinity and temperature; coupled reactive transport of injected substrates, Pseudomonas putida and produced biosurfactants with microbial maximum specific growth rate varying with temperature, salinity, and pH; relative permeability and fractional flow curve variations owing to interfacial tension (IFT) reduction and wettability alteration (WA) by LSWF and biofilm deposition. The governing equations are solved using finite difference technique. Operator splitting and bisection methods are adopted to solve the MIE-transport model. The present model is found to be numerically stable and agree well with previously published experimental and analytical results. In the proposed MIE-transport mechanism, decreasing injection water (IW) salinity from 2.52 to 0.32 M causes enhanced Ca2+ desorption rendering rock surface toward more water-wet. Consequently, oil relative permeability (kro) increases with >55% reduction in water fractional flow (fw) at water saturation of 0.5 from the initial oil-wet condition. Further reducing IW salinity to 0.03 M causes Ca2+ adsorption shifting the surface wettability toward more oil-wet, thus increasing fw by 52%. Formation water (FW) salinity showed minor impact on WA with <5% decrease in fw when FW salinity is reduced from 3.15 to 1.05 M. During low-salinity augmented microbial flooding (LSAMF), biosurfactant production is enhanced by >63% on reducing IW salinity from 2.52 to 0.32 M with negligible increase on further reducing IW and FW salinities. This might be owing to limiting nonisothermal condition (40 to 55°C), dispersion, sorption, and microbial decay. During LSAMF, maximum biosurfactant production occurs at microbial maximum specific growth rate of 0.53 h-1, mean fluid velocity of 2.63×10-3 m h-1 and initial oil saturation of 0.6, thus resulting in significant WA, increase in kro by >20%, and corresponding fw reduction by >84%. Moreover, the EOR efficiency of LSAMF is marginally impacted even on increasing the minimum attainable IFT by two orders of magnitude from 10-3 to 10-1 mN m-1. Though pH increased from 8.0 to 8.9, it showed minor impact on microbial metabolism. Formation damage owing to bioplugging observed near injection point causing increase in fw by ~26% can be mitigated by adopting suitable well-stimulation strategies during the LSAMF run time. The present study is a novel attempt to show synergistic effect of LSAMF over LSWF in enhancing oil mobility and recovery at core scale by simultaneously addressing complex crude oil-brine-rock (COBR) chemistry and critical thermodynamic parameters that govern MEOR efficiency within a typical sandstone formation. The present model with relatively lower computational cost and running time improves the predictive capability to preselect potential field candidates for successful LSAMF implementation.

A high order weak approximation for jump-diffusions using Malliavin calculus and operator splitting

AbstractThe paper introduces a novel high order discretization scheme for expectation of jump-diffusion processes by using a Malliavin calculus approach and an operator splitting method. The test function of the target expectation is assumed to be only Lipschitz continuous in order to apply the method to financial problems. Then Kusuoka’s estimate is employed to justify the proposed discretization scheme. The algorithm with a numerical example is shown for implementation.

Practical finite difference method for solving multi-dimensional black-Scholes model in fractal market

In this paper, we employ a practical finite difference method to research the multi-dimensional fractional Balck-Scholes model under one asset and three assets. In the case of one asset, we establish explicit scheme and Crank-Nicolson scheme to study the effect of different Hurst exponent (H) on numerical results and Greeks. With the increase of H, the numerical figures of the finite difference scheme also change. In addition, we also verify the effectiveness of Crank-Nicolson scheme in numerical solution of Greeks. We observe that when H=0.5, the results of Delta, Gamma and Theta are consistent with the accurate results. In the case of three assets, we use operator splitting method (OSM) and establish semi-implicit scheme. We hold that H will affect the numerical results and Greeks results in fractional Black-Scholes model. If the effect of H is not considered in option hedging, the result will deviate greatly from the actual result.

Unconditionally stable second-order accurate scheme for a parabolic sine-Gordon equation

In this study, we propose an unconditionally stable temporally second-order accurate scheme for a parabolic sine-Gordon equation. The proposed scheme is based on an operator splitting method. We solve linear and nonlinear equations using a Fourier spectral method and a closed-form solution, respectively. The proposed numerical method is temporally second-order accurate and unconditionally stable. To verify the superior efficiency and accuracy of the proposed scheme, we conduct various numerical tests. Computational tests validate the accuracy, efficiency, and simplicity of the proposed scheme.

The multiscale perturbation method for two-phase reservoir flow problems

In this work we formulate and test a new procedure, the Multiscale Perturbation Method for Two-Phase Flows (MPM-2P), for the fast, accurate and naturally parallelizable numerical solution of two-phase, incompressible, immiscible displacement in porous media approximated by an operator splitting method. The proposed procedure is based on domain decomposition and combines the Multiscale Perturbation Method (MPM) [Ali, et al., Appl. Math. and Comput., 387 (2020) pp. 125023] with the Multiscale Robin Coupled Method (MRCM) [Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21]. When an update of the velocity field is called for by the operator splitting algorithm, the MPM-2P may provide, depending on the magnitude of a dimensionless algorithmic parameter, an accurate and computationally inexpensive approximation for the velocity field by reusing previously computed multiscale basis functions. Thus, a full update of all multiscale basis functions required by the MRCM for the construction of a new velocity field is avoided.There are two main steps in the formulation of the MPM-2P. Initially, for each subdomain one local boundary value problem with trivial Robin boundary conditions is solved (instead of a full set of multiscale basis functions, that would be required by the MRCM). Then, the solution of an inexpensive interface problem provides the velocity field on the skeleton of the decomposition of the domain. The resulting approximation for the velocity field is obtained by downscaling.We consider challenging two-phase flow problems, with high-contrast permeability fields and water-oil finger growth in homogeneous media. Our numerical experiments show that the use of the MPM-2P gives exceptional speed-up - almost 90% of reduction in computational cost - of two-phase flow simulations. Hundreds of MRCM solutions can be replaced by inexpensive MPM-2P solutions, and water breakthrough can be simulated with very few updates of the MRCM set of multiscale basis functions.

Fully Distributed Optimization-Based CAV Platooning Control Under Linear Vehicle Dynamics

This paper develops distributed optimization-based, platoon-centered connected and autonomous vehicle (CAV) car-following schemes, motivated by the recent interest in CAV platooning technologies. Various distributed optimization or control schemes have been developed for CAV platooning. However, most existing distributed schemes for platoon centered CAV control require either centralized data processing or centralized computation in at least one step of their schemes, referred to as partially distributed schemes. In this paper, we develop fully distributed optimization based, platoon centered CAV platooning control under the linear vehicle dynamics via the model predictive control approach with a general prediction horizon. These fully distributed schemes do not require centralized data processing or centralized computation through the entire schemes. To develop these schemes, we propose a new formulation of an objective function and a decomposition method that decomposes a densely coupled central objective function into the sum of multiple locally coupled functions whose coupling satisfies the network topology constraint. We then exploit locally coupled optimization and operator splitting methods to develop fully distributed schemes. Control design and stability analysis is carried out to achieve desired traffic transient performance and asymptotic stability. Numerical tests demonstrate the effectiveness of the proposed fully distributed schemes and CAV platooning control.

Numerical simulation of the advection-diffusion-reaction equation using finite difference and operator splitting methods: Application on the 1D transport problem of contaminant in saturated porous media

The combined advection-diffusion-reaction (ADR) equation, which describe the transport problem of a contaminant in porous medium, does not generally admit an analytical solution. In general, when solving the ADR equation, the numerical methods (such as finite differences, finite elements, splitting), for most practical problems, the ADR equation is too difficult to solve analytically. The finite difference method is the oldest and most commonly used method for the numerical solution of this kind of equation. Although newer techniques, such as those based on finite elements and splitting are appropriate for the solution of equilibrium-type problems, the finite difference remains the most appropriate for the solution of time-dependent phenomena. The transport of a contaminant can also be written by the ADR equation; hence, our objective is to choose the most efficient method to study the 1D transport problem of a contaminant and its evolution in a porous medium. In this work, we will simulate the ADR equation using two different methods: those of finite difference and splitting ones. The numerical result will be compared with the analytical solution in order to discuss the stability and the convergence of each of them using those two different methods. In the end, we will show that the splitting technical method is more efficient for solving this kind of problems in comparison with the finite difference method despite the fact that the latter is the most widely used by researchers. The validation of the efficiency of this method, implemented in this simulation, is tested on a 1D-transport problem of contaminant in a saturated porous medium.

DeepSplit: Scalable Verification of Deep Neural Networks via Operator Splitting

Analyzing the worst-case performance of deep neural networks against input perturbations amounts to solving a large-scale non-convex optimization problem, for which several past works have proposed convex relaxations as a promising alternative. However, even for reasonably-sized neural networks, these relaxations are not tractable, and so must be replaced by even weaker relaxations in practice. In this work, we propose a novel operator splitting method that can directly solve a convex relaxation of the problem to high accuracy, by splitting it into smaller sub-problems that often have analytical solutions. The method is modular, scales to very large problem instances, and compromises operations that are amenable to fast parallelization with GPU acceleration. We demonstrate our method in bounding the worst-case performance of large convolutional networks in image classification and reinforcement learning settings, and in reachability analysis of neural network dynamical systems.

An efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère equation

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Amp\`{e}re operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the {M}onge-{A}mp{\`e}re eigenvalue problem, {\it Proceedings of the American Mathematical Society}, 148 (2020), no. 11, 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Amp\`{e}re operator by solving a constrained Monge-Amp\`{e}re equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Amp\`{e}re eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy.

Advances in simulations of heat exchange in fluid flows by high-order methods

The article presents high-order simulations of heat transfer in fluid flow. Heat induced changes in material properties (density, viscosity and heat conductivity) substantially influence flow structures. The change of fluid density implies naturally the thermal expansion what results in non-homogeneous divergence of velocity and the model of evolutionary Navier-Stokes-Fourier equations is not strictly incompressible in this case. Proposed algorithm treats the model as a fully coupled system of equations. Applied method of operator splitting is promissing to become a more general concept for numerical modelling of active quantities advected by incompressible fluids. This work follows the 2D results presented during EFM 2019 and extends to 3D together with advances of the code reimplementation to current version of the Nektar++ library (open-source framework for spectral/hp finite elements). Results for various settings for both the forced/mixed and natural convection are shown.

Fast Operator Splitting Methods for Obstacle Problems

The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator-splitting methods to solve the linear and nonlinear obstacle problems. The proposed methods have three ingredients: (i) Utilize an indicator function to formularize the constrained problem as an unconstrained problem, and associate it to an initial value problem. The obstacle problem is then converted to solving for the steady state solution of an initial value problem. (ii) An operator-splitting strategy to time discretize the initial value problem. After splitting, a heat equation with obstacles is solved and other subproblems either have explicit solutions or can be solved efficiently. (iii) A new constrained alternating direction explicit method, a fully explicit method, to solve heat equations with obstacles. The proposed methods are easy to implement, do not require to solve any linear systems and are more efficient than existing numerical methods while keeping similar accuracy. Extensions of the proposed methods to related free boundary problems are also discussed.