Operator Splitting Research Articles

An acoustic-transport splitting method for the barotropic Baer-Nunziato two-phase flow model

This work focuses on the numerical approximation of the barotropic Baer-Nunziato two-phase flow model. We propose a numerical scheme that relies on an operator splitting method corresponding to a separate treatment of the acoustic and the material transport phenomena. In the subsonic case, this also corresponds to a separate treatment of the fast and the slow propagation phenomena. This approach follows the lines of the implicit-explicit schemes developed in [8]. The operator splitting enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, while maintaining accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms that figure in the two-phase model. We prove that the proposed numerical strategy is positivity preserving for the volume fractions and the partial masses. The scheme is tested against several one-dimensional test cases including flows featuring vanishing phases.

Accelerated SPIRiT Parallel MR Image Reconstruction Based on Joint Sparsity and Sparsifying Transform Learning

Iterative self-consistent parallel imaging reconstruction (SPIRiT) was an autocalibrating model for parallel magnetic resonance imaging reconstruction, which is often formulated as a SPIRiT reconstruction problem with some regularization terms. Some methods based on the operator splitting and alternating direction method of multipliers (ADMM) have been employed to solve the formulated regularized SPIRiT problem. In this paper, we propose to combine the sparsifying transform learning and joint sparsity with Cartesian SPIRiT parallel magnetic resonance imaging, and solve the resulting reconstruction problem by using the variable splitting and ADMM techniques. Simulation experiments on four in vivo data sets demonstrate that the proposed algorithm achieves a better image reconstruction quality than that of other competing methods. In addition, the proposed algorithm is very suitable for graphics processing unit (GPU) parallel computing, and its accelerated version, achieved by using a GPU, is very fast, requiring only 6.7 s to reconstruct a 200×200 pixel image with 8 channels.

A Unified Early Termination Technique for Primal-Dual Algorithms in Mixed Integer Conic Programming

We propose an early termination technique for mixed integer conic programming within branch-and-bound based solvers. Our approach generalizes previous early termination results for ADMM-based solvers to a broader class of primaldual algorithms, including both operator splitting and interior point methods. The complexity for checking early termination is O(n) for each termination check assuming a bounded problem domain. We show that this domain restriction can be relaxed for problems whose data satisfies a simple rank condition, in which case each check requires an O(n2) solve using a linear system that is factored only once at the root node. We further show how this approach can be used in hybrid model predictive control problems with bounded inputs. Numerical results show that our method leads to a moderate reduction in the computational time required for branch-and-bound conic solvers with interior-point based subsolvers.

Fast Multi-grid Methods for Minimizing Curvature Energies.

The geometric high-order regularization methods such as mean curvature and Gaussian curvature, have been intensively studied during the last decades due to their abilities in preserving geometric properties including image edges, corners, and contrast. However, the dilemma between restoration quality and computational efficiency is an essential roadblock for high-order methods. In this paper, we propose fast multi-grid algorithms for minimizing both mean curvature and Gaussian curvature energy functionals without sacrificing accuracy for efficiency. Unlike the existing approaches based on operator splitting and the Augmented Lagrangian method (ALM), no artificial parameters are introduced in our formulation, which guarantees the robustness of the proposed algorithm. Meanwhile, we adopt the domain decomposition method to promote parallel computing and use the fine-to-coarse structure to accelerate convergence. Numerical experiments are presented on image denoising, CT, and MRI reconstruction problems to demonstrate the superiority of our method in preserving geometric structures and fine details. The proposed method is also shown effective in dealing with large-scale image processing problems by recovering an image of size 1024×1024 within 40s, while the ALM method [1] requires around 200s.

Semi-Decentralized Generalized Nash Equilibrium Seeking in Monotone Aggregative Games

We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: (i) a single-layer, fixed-step algorithm with convergence guarantee for general (non cocoercive, non-strictly) monotone aggregative games and (ii) a single-layer proximal-type algorithm for a class of monotone aggregative games with linearly coupled cost functions. We also design novel accelerated variants of the algorithms via (alternating) inertial and over-relaxation steps. Finally, we show via numerical simulations that the proposed algorithms outperform those in the literature in terms of convergence speed.

EFFECT OF EXTRUSION PARAMETERS ON PHYSICAL PROPERTIES OF TUR DAL (PIGEON PEA) ANALOGUES DEVELOPED FROM TUR DAL BROKENS

Dehusking and splitting are typically the steps involved in converting pulses into dal. The milling losses in commercial pulse mills can be attributed to the small brokens and fine powders found during scoring and concurrent dehusking and splitting operations. The fracture of dal kernels during the milling process is a problem in the dal milling industry, and customers don’t generally like these broken kernels. The only way to turn these unused milling fractions into profitable products is through extrusion cooking. Extrusion cooking of flour of redgram dal brokens with addition of some additives was done to prepare Tur dal analogues at various extrusion process parameters. The effect of barrel temperature (120 – 175 oC), feed moisture content (18-25%), feed rate (40-60 kg/h) and screw speed (100-220 rpm) on bulk density, expansion ratio and hardness of Tur dal analogues were investigated. To develop the prediction model, a central composite rotable design (CCRD) using response surface method (RSM) was applied. Each variable’s impact on the responses was measured via regression equations. The second-order quadratic regression model fit the variation adequately. The significance was established at p0.05. An increase in bulk density and decrease in expansion ratio with increase in feed moisture content and feed rate was observed. The bulk density and hardness increased as the barrel temperature increased when the feed moisture content was lower; conversely, when the feed moisture content was higher, the bulk density and hardness reduced as the barrel temperature increases.

STRUCTURE PRESERVING SPLITTING TECHNIQUES FOR EBOLA REACTION–DIFFUSION EPIDEMIC SYSTEM

In this paper, we deal with the numerical solution of the reaction–diffusion Ebola epidemic model. The diffusion which is an important phenomenon for the epidemic model is included in the model. This inclusion has made the model more comprehensive for studying the disease dynamics in the human population. The quantities linked with the model indicate the population sizes which are taken as absolute, therefore, the numerical schemes utilized to solve the underlying Ebola epidemic system should sustain the positivity. The numerical approaches used to solve the underlying epidemic models are explicit nonstandard finite difference operator splitting (ENSFD-OS) and implicit nonstandard finite difference operator splitting (INSFD-OS) techniques. These schemes preserve all the physical features of the state variables, i.e. projected schemes hold the positive solution acquired by the Ebola diffusive epidemic model. The underlying epidemic model illustrates two stable steady states, a virus-free state, and a virus existence state. The suggested approaches retain the stability of each of the steady states possessed by the assumed epidemic model. A numerical example and simulations for validation of all the characteristics of suggested techniques are also investigated.

Unconditionally stable monte carlo simulation for solving the multi-dimensional Allen–Cahn equation

<abstract><p>In this study, we present an efficient and novel unconditionally stable Monte Carlo simulation (MCS) for solving the multi-dimensional Allen–Cahn (AC) equation, which can model the motion by mean curvature flow of a hypersurface. We use an operator splitting method, where the diffusion and nonlinear terms are solved separately. The diffusion term is calculated using MCS for the stochastic differential equation, while the nonlinear term is locally computed for each particle in a virtual grid. Several numerical experiments are presented to demonstrate the performance of the proposed algorithm. The computational results confirm that the proposed algorithm can solve the AC equation more efficiently as the dimension of space increases.</p></abstract>

FOUR NUMERICAL SCHEMES FOR SOLUTION OF BURGERS' EQUATION VIA OPERATOR SPLITTING TRIGONOMETRIC CUBIC B-SPLINE COLLOCATION METHOD

In this study, we have used operator splitting methods for numerical solutions of the Burgers' equation by given four different numerical schemes. To set these schemes, we divide the Burgers equation into two sub-problems according to the time term, as linear $ U_{t}=\mathcal{L}(U) $ and nonlinear $ U_{t}=\mathcal{N}(U) $. Then, numerical schemes have been obtained by the finite element method using trigonometric cubic B-spline basis for each sub-problem. Splitting $ \mathcal{L}\circ\mathcal{N} $, $ \mathcal{N}\circ\mathcal{L} $ Lie-Trotter and $ \mathcal{L}\circ\mathcal{N} \circ \mathcal{L} $, $ \mathcal{N\circ L\circ N} $ Strang splitting solution schemes have been used to obtain the solution of the main equation. Numerical results calculated with these schemes have been compared among themselves in terms of $ L_{2} $, $ L_{\infty} $ error norms and CPU time. Furthermore, the numerical results have been compared with some studies that solved the equation directly with the same method. It has been observed that the numerical results obtained with the proposed schemes are in agreement with the exact solution and other studies in the literature. All calculations are obtained using Matlab Version R2015a.

Coupling local and nonlocal fourth-order evolution equations for image denoising

Local and nonlocal regularization methods have attracted an extensive interest in image denoising. In this paper, we propose two hybrid energy functionals that benefit from the local Laplacian regularization in homogeneous regions and the nonlocal Laplacian regularization in texture regions. The existence and uniqueness of solutions for the gradient flows, which are local and nonlocal coupled fourth-order evolution equations, are proven via the fixed point method. We consider nonlinear generalizations of the coupled equations for image denoising by proposing a new fourth-order operator that reduces the contrast losing effect of the second-order operator. The operator splitting method is adopted for the numerical implementation such that the local and nonlocal parts are handled separately. By numerical experiments, we illustrate that the proposed equations inherit advantages of the local and nonlocal methods for image denoising.

Image denoising based on a new anisotropic mean curvature model

A number of variational models for image denoising have been proposed in the last few years in order to advance the denoising performance. To improve the denoising quality, it is very significant to describe the local structure of image in the proposed models. To this end, this paper proposes a novel denoising model which combines the gradient operator $ \nabla $ with the adaptive weighted matrix $ W $ in the mean curvature regularized term such that the proposed model can describe the local features in image efficiently. Since the proposed model is a high-order nonlinear and nonconvex optimization problem, we need to use the operator splitting method to tansform it into a multi-variable optimization problem and then the alternating direction method of multipliers (ADMM) can be applied to solve it. Numerical experiments demonstrate that the proposed model yields good performance compared with other well-known gradient-based models.

Operator split, Picard iteration and JFNK methods based on nonlinear CMFD for transient full core models in the coupling multiphysics environment

Operator Split (OS), Picard iteration and Jacobian-free Newton Krylov (JFNK) methods based on nonlinear coarse mesh finite difference (CMFD) with nodal expansion methods (NEM) are respectively introduced to solve the transient full core coupled models in the COupling Multiphysics Environment (COME). In the COME, in addition to the traditional OS and Picard coupled methods, we develop two different JFNK-based coupled methods with physics and algebraic-based hybrid preconditioners at each time step: OS_JFNK and JFNK coupled methods. Specifically, the efficient JFNK method is applied to solve CMFD discrete models of only neutronics in the OS framework for OS_JFNK coupled strategies or whole Neutronics-Thermal Hydraulic (N-TH) coupled models for JFNK coupled methods. Then nonlinear corrective coupling coefficients are updated by the two-node NEM every few Newton steps at each time step. Six cases of NEACRP 3D core transient benchmarks are analyzed and numerical solutions of OS, OS_JFNK, Picard and JFNK methods in the COME generally agree well with other published solutions. JFNK and Picard coupled methods can track the reference solutions better than OS and OS_JFNK coupled methods due to multiple iterations or simultaneous solutions between neutronics and TH models in the tight coupled form. Furthermore, OS_JFNK or JFNK methods can obtain higher computational efficiency than OS or Picard methods with or without the fission source extrapolation acceleration, respectively, which indicates the potential of the developed COME using different coupled methods for the transient full core multiphysics coupled problems.

Efficient numerical schemes for multidimensional population balance models

Multidimensional population balance models (PBMs) describe chemical and biological processes having a distribution over two or more intrinsic properties (such as size and age, or two independent spatial variables). The incorporation of additional intrinsic variables into a PBM improves its descriptive capability and can be necessary to capture specific features of interest. As most PBMs of interest cannot be solved analytically, computationally expensive high-order finite difference or finite volume methods are frequently used to obtain an accurate numerical solution. We propose a finite difference scheme based on operator splitting and solving each sub-problem at the limit of numerical stability that achieves a discretization error that is zero for certain classes of PBMs and low enough to be acceptable for other classes. In conjunction to employing specially constructed meshes and variable transformations, the scheme exploits the commutative property of the differential operators present in many classes of PBMs. The scheme has very low computational cost — potentially as low as just memory reallocation. Multiple case studies demonstrate the performance of the proposed scheme.

A generalized forward–backward splitting operator: degenerate analysis and applications

A generalized forward–backward splitting operator: degenerate analysis and applications

Non‐local low‐rank constraint‐based self‐consistent PMRI reconstruction using eigenvector maps

AbstractEigenvector‐based SPIRiT (ESPIRiT) can estimate multiple sets of coil sensitivity maps from the calibration matrix constructed from the auto‐calibration data. Recently, the L1 norm and total variation were combined with the ESPIRiT model to improve the reconstruction quality of magnetic resonance (MR) images. To further improve the reconstruction performance, the non‐local low‐rank regularisation term is incorporated into the ESPIRiT model (NLR‐ESPIRiT) is proposed. The proposed NLR‐ESPIRiT model takes full advantage of the non‐local self‐similarity features of MR images. The resulting optimisation problem can be transformed into a gradient problem and a denoising problem with low‐rank constraints using the operator splitting technique. The weighted nuclear norm (WNN) is applied as a surrogate of the rank. Then the denoising subproblem with the WNN can be effectively solved by using the alternating direction method of multipliers technique. For practical applications, a parameter‐selecting method is proposed to obtain almost optimal parameters for the same kind of MR images. Simulation experiments on in vivo data sets demonstrate that the proposed NLR‐ESPIRiT outperforms all competing traditional model‐based algorithms in terms of three objective metrics and visual comparison.

A numerical efficient splitting method for the solution of HIV time periodic reaction–diffusion model having spatial heterogeneity

This study examines a novel reaction–diffusion model for the existence and treatment of acquired immunodeficiency syndrome. This model is a spatial extension of the recent HIV model and human immunodeficiency viruses cause this disorder. The most significant barrier to eradicating this virus is latency and the virus’ subsequent viral reservoir in CD4+ T cells. A nonstandard operator splitting strategy is proposed to approximate the solution of the time-periodic reaction–diffusion model. The main advantages of employing this approach over other techniques are its low computational costs, high accuracy and ease of implementation. The results are truly solid and match those available in the literature. The nature of the solution for the threshold parameter is demonstrated graphically using numerical results. Finally, the M-matrix theory and the positivity of the proposed scheme are discussed.

VMD–RP–CSRN Based Fault Diagnosis Method for Rolling Bearings

In response to the problems of low accuracy and poor noise immunity of the traditional fault diagnosis method for rolling bearing fault diagnosis due to the complex and variable operating conditions of rolling bearings and the large noise interference during bearing signal acquisition, a rolling bearing fault diagnosis model based on VMD–RP–CSRN is proposed. Firstly, the initial feature extraction of the bearing signal is carried out by variational modal decomposition (VMD), which is then converted into a two-dimensional image with fault features by recurrent plot (RP) coding, and then the feature images are input to a channel split residual network (CSRN) for feature extraction and fault classification. In order to verify the accuracy and noise immunity of the proposed method for the diagnosis of bearing faults under complex working conditions, experiments on the selection of parameters in the CSRN model were conducted on the bearing dataset of Jiangnan University, and experiments on the diagnosis of bearing faults under complex working conditions and noise immunity of CSRN were carried out and compared with other commonly used methods. The proposed bearing fault diagnosis method based on VMD–RP–CSRN combines VMD and RP to retain the fault features in the original signal to the maximum extent and stress the hidden features in the signal. The proposed channel split operation realizes the extraction of hidden features by selecting the main operating channel of the three-channel feature image, and makes more fault features participate in the feature extraction of the diagnosis model. The experimental results demonstrate that the proposed method is at least 1.2% better than the comparison method, and has better noise immunity. In addition, experiments on the fault diagnosis capability of the model with different data set sizes and the diagnosis of variable speed bearing data by the model show that the proposed method has better generalization performance and diagnosis capability.

American options pricing under regime-switching jump-diffusion models with meshfree finite point method

In an incomplete market construction and by no-arbitrage assumption, the American options pricing problem under the jump-diffusion regime-switching process is formulated by a variational form of coupled partial integro-differential equations. In this paper, a valuation algorithm is developed for American options when the dynamics of underlying assets follow the regime-switching jump-diffusion processes. Using the fact that the price of an American option under jump-diffusion regime-switching processes is formulated by a collection of coupled variational partial integro-differential equations with the free boundary characteristic, we combine the moving least-squares approximation with an operator splitting method to treat American constraints. Numerical experiments with American options under three, five, and seven regimes demonstrate the efficiency and effectiveness of our computational scheme for pricing American options under the regime-switching models.

Coupled viscosity-damage model for concrete under high strain rate

A coupled viscosity-damage model of concrete under high strain rate is proposed based on the bi-scalar plastic-damage model. A rheological model is provided to explain the time delay mechanism of the damage under dynamic loading. A rate-dependent damage evolution is established by introducing the Duvaut–Lions type of the damage driving stress rate, and damage and viscosity are coupled. The operator splitting method is used to design an explicit calculation procedure. It is illustrated how the damage evolution predicted by the model differs for dynamic loading and quasi-static loading. Excellent comparisons between the simulated and experimental results show that the model can be used to reflect the strain rate effect of concrete under impact.

A Review of Connected and Automated Vehicle Platoon Merging and Splitting Operations

Connected and automated vehicle (CAV) platoons have drawn much attention in the past decades, given their potential to reduce fuel consumption, elevate roadway capacity, and enhance traffic safety. As two basic platoon operations, platoon merging and splitting have been widely investigated. This study provides an overview of theoretical models and field experiments of CAV platoon merging and splitting operations. A three-step framework, including protocol design, trajectory planning, and vehicle control, is proposed to unify existing representative studies. Methodological techniques in each step are summarized and discussed. Finally, future research directions are discussed. This study contributes to the literature by providing a framework that categorizes relevant literature and guides the successful development of platoon merging and splitting operations. More importantly, it offers researchers and practitioners a rich reference for further investigations.