Operator Splitting Research Articles

Controlled cyclic remote preparation

Multi-party quantum communication has gradually attracted widespread attention. To realize the perfect transmission of quantum states among multiple participants, a novel multi-party controlled cyclic remote preparation protocol for arbitrary single-qubit states with three senders is proposed. With the permission of one controller, each sender can transmit an arbitrary single-qubit state to its neighbor. In addition, we give a universal protocol for multi-party controlled cyclic remote preparation of arbitrary single-qubit states in the case of multiple senders, which can realize deterministic cyclic preparation of multiple quantum states in one direction. The scheme shows that the communication task can be successfully achieved only if all senders cooperate with the controller, and there is no need for the senders to employ information splitting and additional operations before performing measurements. Finally, we discuss the cyclic remote preparation protocol with three senders under five types of noisy environment, and the closeness between the output state and original state is measured by calculating fidelity.

Shuffle Graph Convolutional Network for Skeleton-Based Action Recognition

A shuffle graph convolutional network (Shuffle-GCN) is proposed to recognize human action by analyzing skeleton data. It uses channel split and channel shuffle operations to process multi-feature channels of skeleton data, which reduces the computational cost of graph convolution operation. Compared with the classical two-stream adaptive graph convolutional network model, the proposed method achieves a higher precision with 1/3 of the floating-point operations (FLOPs). Even more, a channel-level topology modeling method is designed to extract more motion information of human skeleton by learning the graph topology from different channels dynamically. The performance of Shuffle-GCN is tested under 56,880 action clips from the NTU RGB+D dataset with the accuracy 96.0% and the computational complexity 12.8 GFLOPs. The proposed method offers feasible solutions for developing practical applications of action recognition.

A numerical approach to the optimal control of thermally convective flows

The optimal control of thermally convective flows is usually modeled by an optimization problem with constraints of Boussinesq equations that consist of the Navier-Stokes equation and an advection-diffusion equation. This optimal control problem is challenging from both theoretical analysis and algorithmic design perspectives. For example, the nonlinearity and coupling of fluid flows and energy transports prevent direct applications of gradient type algorithms in practice. In this paper, we propose an efficient numerical method to solve this problem based on the operator splitting and optimization techniques. In particular, we employ the Marchuk-Yanenko method leveraged by the L2-projection for the time discretization of the Boussinesq equations so that the Boussinesq equations are decomposed into some easier linear equations without any difficulty in deriving the corresponding adjoint system. Consequently, at each iteration, four easy linear advection-diffusion equations and two degenerated Stokes equations at each time step are needed to be solved for computing a gradient. Then, we apply the Bercovier-Pironneau finite element method for space discretization, and design a BFGS type algorithm for solving the fully discretized optimal control problem. We look into the structure of the problem, and design a meticulous strategy to seek step sizes for the BFGS efficiently. Efficiency of the numerical approach is promisingly validated by the results of some preliminary numerical experiments.

A weak approximation for Bismut’s formula: An algorithmic differentiation method

The paper provides a novel algorithmic differentiation method by constructing a weak approximation for Bismut’s formula. A new operator splitting method based on Gaussian Kusuoka-approximation is introduced for an enlarged semigroup describing “differentiation of diffusion semigroup”. The effectiveness of the new algorithmic differentiation is checked through numerical examples.

Temple system on networks

This paper deals with the well-posedness on a network of a Temple system of nonlinear hyperbolic balance laws. Temple systems are characterized by the fact that shock and rarefaction curves coincide. This study is motivated by a model for traffic, recently proposed, inspired by kinetic considerations. The proof of the well-posedness is based on the wave-front tracking procedure, on the pseudo-polygonal technique and on the operator splitting method.

Novel operator-splitting methods for real-time dynamic substructure testing

Improved and corrected Operator-Splitting (OS) methods are proposed for Real-Time Dynamic Substructure Testing (RTDST) in this paper. Firstly, the theory and the stability of two proposed OS methods are investigated. Furthermore, the effectiveness of two proposed OS methods is validated by RTDSTs with one pure mass specimen and one mass-stiffness specimen. By using the predictor velocity and acceleration as well as the correction force approximations, the improved OS method and the corrected OS method are explicit methods for RTDST. It is shown from the analytical and experimental results that the improved OS method is unconditionally stable for linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping system if the mass ratio of the experimental substructure to the numerical substructure is less than 1. In contrast, the improved OS method is unsuitable for RTDST if the mass ratio is larger than 1. The corrected OS method is unconditionally stable for the linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping systems. The stability of the corrected OS method is better than the improved OS method, which indicates the advantage of the corrected OS method over the improved OS method.

An operator-splitting approach for variational optimal control formulations for diffeomorphic shape matching

We present formulations and numerical algorithms for solving diffeomorphic shape matching problems. We formulate shape matching as a variational problem governed by a dynamical system that models the flow of diffeomorphism ft∈diff(R3). We overview our contributions in this area, and present an improved, matrix-free implementation of an operator splitting strategy for diffeomorphic shape matching. We showcase results for diffeomorphic shape matching of real clinical cardiac data in R3 to assess the performance of our methodology.

Low-rank Tensor Restoration for ERP extraction

Event-related potential (ERP) data is essentially multi-dimensional, with correlated data in some spaces. Therefore, matrix and vector analysis results in structural information loss. Tensor decomposition can be used to explore the shared structural information of ERP signal among related conditions. Perceiving the decaying trends of the singular value changes of unfolding matrices indicate that they are low-rank matrices. Based on this assumption, in this work, a low-rank tensor restoration (LTR) method is proposed. An operator splitting method known as the alternating direction method of multipliers (ADMM) is adapted to tackle the proposed optimization problem with orthogonality and sparsity constraints. Accordingly, the problem is solved in a sequential fashion by computing the unconstrained and orthogonality constrained quadratic sub-problems with closed-form solutions. The algorithm is examined under three application areas, namely, noise removal, feature extraction and subject-to-subject transfer learning. The empirical evaluations on real P300-based ERP dataset demonstrate the robustness and effectiveness of the proposed method.

An implicit time integration approach for simulation of corona discharges

Corona discharge is a type of electric discharge that occurs when a small conductor carries a voltage strong enough to ionize the surrounding air. The resultant ions are drifted by the electric field and transfer their momentum to neutral particles via elastic collisions, creating a flow effect known as ionic wind that has promising application in flow control. Over the last two decades, a large amount of numerical works have been realized to study this effect. However, simulation of corona discharges with explicit time integration usually requires an excessive amount of time to finish, ranging from days to months in two dimensions, depending on mesh size and physical end time. The reason lies in the profound disparity between physical time scales in a discharge: for example, electron density evolves on the order of picoseconds while the time scale of airflow dynamics is on the order of milliseconds. The objective of this work is therefore to reduce the CPU time of corona discharges. We propose an implicit time strategy in the scope of the Local Field Approximation plasma model. We investigate some algorithms, namely the Lie operator splitting, the Douglas-Rachford method and a nonlinear variant of the Gauss-Seidel algorithm, to solve the system of discrete equations, which are adapted to take into consideration a constraint on the electron density. This constraint is introduced to compensate for sources of electrons from physical processes that are not accounted for in the plasma model; it is expressed by a minimum density, or floor density, imposed on the electrons. Therefore, simplified plasma models can be employed for discharge simulations, for instance a four-specie, four-reaction model is used in this article. The investigation is done on a one-dimensional wire-to-wire corona discharge. After having identified the most suitable algorithm, we carry on to two-dimensional simulations and compare the numerical results with experiment data for different levels of floor density.

RBF based some implicit–explicit finite difference schemes for pricing option under extended jump-diffusion model

In this manuscript, we presented some efficient and accurate radial basis function-based finite difference (RBF-FD) implicit–explicit (IMEX) numerical techniques for pricing the option when the underlying asset follows the jump-diffusion process with local volatility. For the time semi-discretization, we present three numerical techniques Crank–Nikolson Leap-Frog (CNLF), Crank–Nikolson AdamBashforth (CNAB), and Backward difference formula of order two (BDF2), incorporated with the radial basis function based finite difference (RBF-FD) method. The stabilities of time semi-discretized schemes are also proved. The computational methods developed for the European option are extended for the American option. We amalgamate the RBF-FD implicit–explicit methods with an operator splitting (OS) method for solving the linear complementarity problem (LCP) with variable parameters that determines the price of an American option. In order to demonstrate the effectiveness and precision of the current techniques, numerical data for European and American put options under the Merton and Kou models are presented.

Global-and-local aware network for low-light image enhancement

Photos taken under nighttime or backlit conditions often suffer from complex and unpredictable degradation, such as low visibility, messy noise, and distorted color. Previous methods mainly focused on global brightness and contrast while ignoring structural and textural details, or they handled the fusion of features without adequately considering their intrinsic association, resulting in incomplete feature representations. To address this issue, we propose a global-and-local aware network (GLAN) by projecting the features into the frequency domain and incorporating them in a knowledge-sharing manner. This method effectively integrates the global modeling capability of the transformer and the local sensitivity of the convolutional neural network to represent structure and texture. First, the global branch, which is comprised of transformer blocks, performs feature extraction under the global receptive field, while the local branch constructs multi-scale features to provide local fine-grained details. Then, we design a novel adaptive multi-scale feature block (AMSFB) that deploys channel split operation to decrease the calculation amount. To better learn the channel and spatial correlations of intermediate features, we introduce a multi-scale channel attention module (MSCAM) and a pixel attention module (PAM) into the AMSFB. Finally, a frequency-aware interaction module (FAIM) is developed for bidirectional information supplementation, which builds feature descriptors simultaneously covering low-frequency and high-frequency information based on the discrete cosine transform (DCT). Through extensive quantitative and qualitative experiments, our method can achieve competitive results compared with over ten state-of-the-art image enhancement methods on eight benchmark datasets.

Efficient Numerical Simulation of Biochemotaxis Phenomena in Fluid Environments.

A novel dimension splitting method is proposed for the efficient numerical simulation of a biochemotaxis model, which is a coupled system of chemotaxis-fluid equations and incompressible Navier-Stokes equations. A second-order pressure correction method is employed to decouple the velocity and pressure for the Navier-Stokes equations. Then, the alternating direction implicit scheme is used to solve the velocity equation, and the operator with dimension splitting effect is used instead of the traditional elliptic operator to solve the pressure equation. For the chemotactic equation, the operator splitting method and extrapolation technique are used to solve oxygen and cell density to achieve second-order time accuracy. The proposed dimension splitting method splits the two-dimensional problem into a one-dimensional problem by splitting the spatial derivative, which reduces the computation and storage costs. Finally, through interesting experiments, we show the evolution of the cell plume shape during the descent process. The effect of changing specific parameters on the velocity and plume shape during the descent process is also studied.

An operator splitting Legendre-tau spectral method for Maxwell’s equations with nonlinear conductivity in two dimensions

In the paper, an operator splitting Legendre-tau spectral method is proposed to solve Maxwell’s equations with nonlinear conductivity in two dimensions. By the implicit–explicit numerical scheme, the linear and nonlinear parts are treated in different ways. The linear terms are treated by the Legendre-tau spectral method implicitly, and the nonlinear terms are treated by some collocation methods explicitly. The scheme uses polynomial spaces of different degrees to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. The splitting leap frog Crank–Nicolson method is applied in time discretization, which is a three-level scheme with the nonlinear term being treated explicitly in intermediate level. With the application of inverse inequality, for the Legendre interpolation operator, optimal L2-error estimates are achieved for the proposed scheme. In numerical examples, the nonlinear terms are computed at Chebyshev–Gauss–Lobatto points by the fast Legendre transform. Numerical results validate the efficiency and spectral accuracy of the scheme.

Fast and efficient numerical method for solving the Allen–Cahn equation on the cubic surface

In this study, we present a fast and efficient finite difference method (FDM) for solving the Allen–Cahn (AC) equation on the cubic surface. The proposed method applies appropriate boundary conditions in the two-dimensional (2D) space to calculate numerical solutions on cubic surfaces, which is relatively simpler than a direct computation in the three-dimensional (3D) space. To numerically solve the AC equation on the cubic surface, we first unfold the cubic surface domain in the 3D space into the 2D space, and then apply the FDM on the six planar sub-domains with appropriate boundary conditions. The proposed method solves the AC equation using an operator splitting method that splits the AC equation into the linear and nonlinear terms. To demonstrate that the proposed algorithm satisfies the properties of the AC equation on the cubic surface, we perform the numerical experiments such as convergence test, total energy decrease, and maximum principle.

NUMERICAL SIMULATION OF A FLUID FLOW IN THE CYLINDRICAL DUCT CONTAINING TWO DIAPHRAGMS WITH ORIFICES OF DIFFERENT DIAMETERS

The flow of a viscous incompressible fluid in a cylindrical duct with two serial diaphragms with orifices of different diameters was studied based on the numerical solution of the unsteady Navier-Stokes equations. The solution algorithm was based on the finite volume method using second-order accurate difference schemes in both space and time. The TVD (Total-Variation Diminishing) form of a central-difference scheme with a flux limiter was used for the interpolation of the convective terms. The combined evaluation of the velocity and pressure fields was carried out using the PISO (Pressure Implicit Split Operator) procedure. The problem was solved using OpenFOAM (Open-source Field Operation And Manipulation) open-source toolkit libraries using the computing power of the cluster supercomputer of the V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine. It was shown that within a certain range of the diameter ratio of the orifices of the diaphragms, a circulation movement is established in the cavity between the diaphragms. The boundary layer breaks off from the surface of the first diaphragm and forms an annular shear layer. When approaching the second diaphragm, a sequence of ring vortices is formed in it, which interact with the surface of the diaphragm and lead to the emergence of tonal sound. When the ratio of the orifice diameter of the second diaphragm to the first decreases, the share of the jet’s kinetic energy, which participates in the circulation in the middle of the cavity between the diaphragms, increases. As a result, the amplitude of velocity fluctuations in the orifice of the second diaphragm decreases. When the ratio of the diameters of the orifices increases, the share of energy involved in the circulation decreases, and when a certain value of the ratio is reached, the interaction between the vortices in the shear layer and the surface of the diaphragm does not occur. As a result, the excitation of the tonal sound stops. The Strouhal number of self-oscillations practically does not change when the diameter ratio of the orifices changes. During the calculations, two different modes of self-oscillation were obtained, which is consistent with previous works.

PolyStokes: A Polynomial Model Reduction Method for Viscous Fluid Simulation

Standard liquid simulators apply operator splitting to independently solve for pressure and viscous stresses, a decoupling that induces incorrect free surface boundary conditions. Such methods are unable to simulate fluid phenomena reliant on the balance of pressure and viscous stresses, such as the liquid rope coil instability exhibited by honey. By contrast, unsteady Stokes solvers retain coupling between pressure and viscosity, thus resolving these phenomena, albeit using a much larger and thus more computationally expensive linear system compared to the decoupled approach. To accelerate solving the unsteady Stokes problem, we propose a reduced fluid model wherein interior regions are represented with incompressible polynomial vector fields. Sets of standard grid cells are consolidated into super-cells, each of which are modelled using a quadratic field of 26 degrees of freedom. We demonstrate that the reduced field must necessarily be at least quadratic, with the affine model being unable to correctly capture viscous forces. We reproduce the liquid rope coiling instability, as well as other simulated examples, to show that our reduced model is able to reproduce the same fluid phenomena at a smaller computational cost. Futhermore, we performed a crowdsourced user survey to verify that our method produces imperceptible differences compared to the full unsteady Stokes method.

A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling

We present a robust and efficient method for simulating Lagrangian solid-fluid coupling based on a new operator splitting strategy. We use variational formulations to approximate fluid properties and solid-fluid interactions, and introduce a unified two-way coupling formulation for SPH fluids and FEM solids using interior point barrier-based frictional contact. We split the resulting optimization problem into a fluid phase and a solid-coupling phase using a novel time-splitting approach with augmented contact proxies , and propose efficient custom linear solvers. Our technique accounts for fluids interaction with nonlinear hyperelastic objects of different geometries and codimensions, while maintaining an algorithmically guaranteed non-penetrating criterion. Comprehensive benchmarks and experiments demonstrate the efficacy of our method.

RAM: Rapid Advection Algorithm on Arbitrary Meshes

The study of many astrophysical flows requires computational algorithms that can capture high Mach number flows, while resolving a large dynamic range in spatial and density scales. In this paper we present a novel method, RAM: Rapid Advection Algorithm on Arbitrary Meshes. RAM is a time-explicit method to solve the advection equation in problems with large bulk velocity on arbitrary computational grids. In comparison with standard upwind algorithms, RAM enables advection with larger time steps and lower truncation errors. Our method is based on the operator splitting technique and conservative interpolation. Depending on the bulk velocity and resolution, RAM can decrease the numerical cost of hydrodynamics by more than one order of magnitude. To quantify the truncation errors and speed-up with RAM, we perform one- and two-dimensional hydrodynamics tests. We find that the order of our method is given by the order of the conservative interpolation and that the effective speed-up is in agreement with the relative increment in time step. RAM will be especially useful for numerical studies of disk−satellite interaction, characterized by high bulk orbital velocities and nontrivial geometries. Our method dramatically lowers the computational cost of simulations that simultaneously resolve the global disk and potential well inside the Hill radius of the secondary companion.

Distributed robust optimization with coupled constraints via Tseng’s splitting method

We consider distributed robust convex optimization with both local and coupled constraints, where coefficients with respect to local constraints are uncertain in polyhedral sets. We derive an equivalent optimization model with extended decision variables and finite constraints, and characterize its optimality condition. Also, we transform the problem as finding zeros of an operator and reveal the monotonicity of the underlying operator. By employing Tseng’s operator splitting method, we design a distributed algorithm to solve the considered robust optimization problem with guaranteed convergence, and also illustrate the efficiency in numerical simulations.

DOSnet as a non-black-box PDE solver: When deep learning meets operator splitting

Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning-based solvers utilize the representation power of DNNs to approximate the input-output relations in an automated manner. However, the lack of physics-in-the-loop often makes it difficult to construct a neural network solver that simultaneously achieves high accuracy, low computational burden, and interpretability. In this work, focusing on a class of evolutionary PDEs characterized by decomposable operators, we show that the classical “operator splitting” technique can be adapted to design neural network architectures. This gives rise to a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet). Such non-black-box network design is constructed from the physical rules and operators governing the underlying dynamics, and is more efficient and flexible than the classical numerical schemes and standard DNNs. To demonstrate the advantages of our new AI-enhanced PDE solver, we train and validate it on several types of operator-decomposable differential equations. We also apply DOSnet to nonlinear Schrödinger equations which have important applications in the signal processing for modern optical fiber transmission systems, and experimental results show that our model has better accuracy and lower computational complexity than numerical schemes and the baseline DNNs.