Operator Splitting Research Articles

A variational driven optimization framework for pansharpening of multispectral images

ABSTRACT Pansharpening is an established remote sensing image fusion technique that yields a high-resolution multispectral (HRMS) image. Despite the fact that the advanced technologies like sparse coding and deep learning have achieved a remarkable improvement in solving the pansharpening problem, a unified model is required to further enhance the fusion quality. The variational optimisation (VO) mechanism has gained interest of most of the researchers in recent years. In this article, pansharpening is designated as a constrained optimisation problem with a data generative term and two regularisers to ameliorate spatial details and spectral information. The gradient information is exploited to impart the spatial details from the panchromatic image to the fused image. The correlation among multispectral image bands inspired to promote the spectral quality of the HRMS image as well as to reduce the distortion. Consequently, the optimisation problem is efficiently solved for the required HRMS image using the operator splitting approach. The extensive experiments performed in accordance with the well-known protocols rationalise that the proposed model outperforms most of the state-of-the-art methods in terms of objective metrics and visual outcomes.

An interwoven composite tailored finite point method for two dimensional unsteady Burgers' equation

This paper centers on constructing a lucid and utilitarian approach to tackle linear and non-linear two-dimensional partial differential equations. To test the applicability of the proposed algorithm a variant of the classical two-dimensional unsteady Burgers' equation is set up as a testing ground. The method in a nutshell reduces to solving one-dimensional Burgers' equations resulting from the application of appropriate operator splitting techniques in the temporal direction. In solving these one-dimensional Burgers' equations a refined tailored finite point method in conjunction with an apposite linearization to the purpose is employed. The conditional stability, consistency, and convergence of the method are established theoretically and the method is found to be first-order convergent in time and second-order convergent in space. To illustrate the accuracy of the scheme, divers examples have been solved and the results obtained prove that this method is top-notch in terms of cost-cutting and time efficiency through the sufficiency of coarse meshes.

Generalisation of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schrödinger type

The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schrödinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schrödinger case.Numerical illustrations for the time-dependent Gross–Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods.The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross–Pitaevskii systems in real and imaginary time.

WAVEx: Stabilized finite elements for spectral wind wave models using FEniCSx

The prediction of the wind wave spectrum of the ocean using numerical models are an important tool for researchers, engineers, and communities living in coastal areas. The governing equation of the wind wave models, the Wave Action Balance Equation, presents unique challenges for implementing reliable numerical models because it is highly advective, highly nonlinear and high dimensional. Historically, most operational models have utilized finite difference methods, others have used finite volume methods but relatively few attempts at using finite element methods. In this work, we seek to fill this gap by investigating several different finite element discretizations of the Wave Action Balance Equation. The methods, which include streamline upwind Petrov–Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented and convergence properties are examined for some simplified 2-D test cases. Then, a new spectral wind wave model, WAVEx, is formulated and implemented for the full problem setting. WAVEx uses continuous finite elements along with SUPG stabilization in geographic/spectral space that allows for fully unstructured triangular meshes in both geographic and spectral space. For propagation in time, a second order fully implicit finite difference method is used. When source terms are active, a second order operator splitting scheme is used to linearize the problem. In the splitting scheme, propagation is solved using the implicit method and the nonlinear source terms are treated explicitly. Several test cases, including analytic tests and laboratory experiments, are demonstrated and results are compared to analytic solutions, observations, as well as output from another model that is used operationally.

Orthognathic surgery to improve malocclusion in a chimpanzee (Pan troglodytes)

AbstractMalocclusion is a common finding in both companion animals and humans due to dental or maxillofacial discrepancies. Treatment depends on the complications and the species it presents in. In humans, orthognathic surgery is commonly performed to address skeletal malocclusions. A male chimpanzee born in 2002 and orphaned due to the bush meat pet trade was rescued in 2010 by Chimp Eden, a chimpanzee sanctuary. In 2017, it presented with inappetence and weight loss of 6‐month duration. After a computed tomography scan was performed and full mouth impressions were made, a diagnosis of asymmetry of the mandible with the left side markedly shorter and rotated along its long axis was made (malocclusion class IV in a side‐to‐side direction). A bilateral sagittal split operation was performed to correct the malocclusion and improve its feeding. At the time of writing this report, it was eating freely and no complications were seen on multiple post‐operative radiographs.

Stability and error estimates of Strang splitting method for the nonlocal ternary conservative Allen–Cahn model

The nonlocal model has attracted a great attention in materials science for describing various types of material heterogeneities and defects. In this study, we consider a nonlocal ternary conservative Allen–Cahn model, where the standard Laplace operator is intentionally replaced with a spatial convolution term that aims at describing long-range interactions among particles. A linear energy stable scheme is developed based on the operator splitting method. The mass conservation, energy stability and global convergence of the new scheme are analyzed rigorously. Numerical stability and convergence of the present numerical scheme are analyzed theoretically. Two and three dimensional numerical experiments are performed to validate the theoretical analysis and the efficiency of the method.

Efficient pricing of options in jump–diffusion models: Novel implicit–explicit methods for numerical valuation

This paper presents novel implicit–explicit Runge–Kutta type methods for numerically simulating partial integro-differential equations that arise when pricing options under jump–diffusion models. These methods offer an alternative approach that avoids the need for numerical or analytical inversion of the coefficient matrix. The pricing of European options is formulated as a partial integro-differential equation, while the pricing of American options are treated as a linear complementarity problem. The developed implicit–explicit Runge–Kutta type method is combined with an operator splitting technique to efficiently solve the linear complementarity problem. Stability and convergence analysis of the proposed methods are established using discrete ℓ2-norm. To validate their efficiency and accuracy, the methods are applied to pricing European and American options under Merton’s and Kou’s models, and the computed results are compared with those reported in the literature.

DLSDHMS: Design of a deep learning-based analysis model for secure and distributed hospital management using context-aware sidechains

Designing an efficient hospital management solution requires the integration of multidomain operations that include secure storage, alert system modelling, infrastructure management, staff management, report analysis, and feedback-based learning tasks. Existing hospital management models are either highly complex or do not incorporate comprehensive deep learning analysis, which limits their deployment capabilities. Moreover, most of these models use mutable storage solutions, which restricts their trust levels under multi-patient to multi-doctor mapping scenarios. To overcome these issues, this article proposes the design of a novel deep Learning-based analysis model for secure and distributed hospital management via context-aware sidechains. The model initially collects large-scale information sets from different hospital entities via an IoT-based network and stores the information on context-sensitive sidechains. These context-sensitive sidechains store information sets related to Medicine Management, Doctor Management, Insurance and Billing Management, and Appointment Management operations. These chains are optimized via an Iterative Genetic Algorithm (IGA) that assists in improving storage and retrieval performance via intelligent merging and splitting operations. Information stored on these chains is processed via a combination of Recurrent Neural Networks (RNNs) and Convolutional Neural Networks (CNNs), that assist in identifying patient-level diseases and issues. The information obtained from these classifiers is updated on the central repository and assists in the pre-emption of diseases for other patients. Due to these integrations, the proposed model is capable of reducing computational delay by 3.5 % and reducing storage cost by 8.3 % when compared to other blockchain-based deployments. The model is also able to pre-empt patient issues with 9.3 % higher accuracy and 4.8 % higher precision, which makes it useful for real-time clinical deployments.

EPCS: Endpoint-based part-aware curve skeleton extraction for low-quality point clouds

The curve skeleton is an important shape descriptor which has been utilized in various applications in computer graphics, machine vision, and artificial intelligence. In this study, the endpoint-based part-aware curve skeleton (EPCS) extraction method for low-quality point clouds is proposed. The novel random center shift (RCS) method is first proposed for detecting the endpoints on point clouds. The endpoints are used as the initial seed points for dividing each part into layers, and then the skeletal points are obtained by computing the center points of the oriented bounding box (OBB) of the layers. Subsequently, the skeletal points are connected, thus forming the branches. Furthermore, the multi-vector momentum-driven (MVMD) method is also proposed for locating the junction points which connect the branches. Due to the shape differences between different parts on point clouds, the global topology of the skeleton is finally optimized by removing the redundant junction points, re-connecting some branches using the proposed MVMD method, and applying an interpolation method based on the splitting operator. Consequently, a complete and smooth curve skeleton is achieved. The proposed EPCS method is compared with several state-of-the-art methods, and the experimental results verify its robustness and effectiveness. Furthermore, the skeleton extraction and model segmentation results on challenging point clouds of broken Terracotta also highlight the utility of the proposed method.

Recasting an operator splitting solver into a standard finite volume flux-based algorithm. The case of a Lagrange-projection-type method for gas dynamics

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive in the low-Mach regime, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes.

An efficient method for the anisotropic diffusion equation in magnetic fields

We solve the anisotropic diffusion equation in 2D, where the dominant direction of diffusion is defined by a vector field which does not conform to a Cartesian grid. Our method uses operator splitting to separate the diffusion perpendicular and parallel to the vector field. The slow time scale is solved using a provably stable finite difference formulation in the perpendicular to the vector field, and an integral operator for the diffusion parallel to it. Energy estimates are shown to for the continuous and semi-discrete cases. Numerical experiments are performed showing convergence of the method, and examples is given to demonstrate the capabilities of the method.

Distributed continuous-time proximal algorithm for nonsmooth resource allocation problem with coupled constraints

This paper studies the distributed resource allocation problem with nonsmooth local cost functions subject to the coupled equality and inequality constraints. In particular, each local cost function is expressed as the sum of a differentiable function and two nonsmooth functions. By using the operator splitting and primal–dual method, a continuous-time distributed proximal algorithm is developed, which can be applied to more general local cost functions that are convex but not necessarily smooth. In addition, the proposed algorithm is fully distributed in the sense that the gain parameter can be determined locally and does not require any global information of the network. By applying Lyapunov stability analysis and convex optimization theory, it is shown that the decision variables of all the agents converge to an optimal solution. Finally, a simulation example is carried out to demonstrate the effectiveness of the proposed algorithm.

Numerical solution of nonlinear Schrödinger equation with damping term on unbounded domain

The artificial boundary method is applied to numerically solve the nonlinear Schrödinger equation with damping term on unbounded domain. A novel transformation is developed to eliminate the troublesome damping term, which yields the difficulty to design the artificial boundary conditions. The artificial boundary conditions are investigated to reduce the original problem into an initial boundary value problem, based on the idea of operator splitting approach. The stability of the reduced problem is analyzed by the energy estimation. The accuracy and effectiveness of the proposed method are verified by numerical examples.

A co-evolutionary genetic algorithm with knowledge transfer for multi-objective capacitated vehicle routing problems

The capacitated vehicle routing problem (CVRP) aims to find the routes that minimize transportation costs. However, in real-world applications, the equitable workload is essential for the drivers and logistics companies. For assigning an appropriate load for each vehicle, this paper formulates the CVRP with workload balancing as a multi-objective optimization problem (MO-CVRP). The goal is to simultaneously minimize the total length of routes and the variance of loads assigned to each vehicle. Although several multi-objective evolutionary algorithms (MOEAs) have been proposed for the MO-CVRP, they often suffer from slow convergence speeds due to the complexity of the search space, which hinders the identification of clear search directions. To address this issue, the knowledge transfer method is introduced that can leverage search experiences from related problems to expedite the solution process of the current problem. Specifically, a co-evolutionary genetic algorithm with knowledge transfer is proposed, where a simplified version of the traveling salesman problem (TSP) by disregarding the vehicle capacity constraint is utilized to facilitate the quick solution of the original MO-CVRP. To enable information interaction between these two problems, the widely used denoising autoencoder is adopted to construct a transformation relation between solutions from the TSP to the MO-CVRP. Additionally, an adaptive migration strategy is proposed to adaptively select the migration methods and an improved split operator strategy is designed to ensure the feasibility of the solutions. The proposed algorithm is compared with four state-of-the-art MOEAs on MO-CVRP test sets with different scales. Experimental results show that the proposed algorithm exhibits faster convergence speed and achieves solutions with superior convergence.

Reduced model for fracture of geometrically exact planar beam: Non‐local variational formulation, ED‐FEM approximation and operator split solution

SummaryIn this work, we present a novel reduced model for computing the solution to a bending fracture problem of the geometrically exact beam. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model. The non‐local variational formulation is proposed to deal with the softening phenomena characteristic of fracture. The fracture energy is introduced as the main fracture parameter, rather than (in general) ill‐defined characteristic length used for previous non‐local models. The discrete approximation is built in terms of the embedded‐discontinuity finite element method (ED‐FEM), which needs no initial crack for optimal performance nor detailed crack kinematics description, such as in the X‐FEM or the phase‐field approach. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best‐approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and stress) from local (internal) variables including the crack opening. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

A Time Splitting Method for the Three-Dimensional Linear Pauli Equation

Abstract We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrödinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrödinger equation. We use a four term operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schrödinger equation.

Computational aeroacoustics and phase-conjugation technique for localizing flow-induced sources

This paper presents a framework comprising computational aeroacoustics (CAA) solvers and a numerical phase-conjugation technique to localize flow-induced noise sources generated by bodies immersed in low Mach number flows. Two test-cases were considered, namely, flow over a two-dimensional (a) circular cylinder at Reynolds number Re =150 and Mach number M = 0.2 and (b) NACA-0012 airfoil at Re = 5000 and angle of attack (AoA) α = 5 deg. The CAA simulations were carried out in OpenFOAM wherein the two-dimensional unsteady, incompressible Navier–Stokes equations was solved using either the pressure-implicit splitting of operators (PISO) algorithm or large eddy simulation (LES). The near-field aerodynamic sources were extracted in terms of the unsteady Lighthill stress tensor terms, following which the far-field acoustic data were computed on boundary nodes using different methods which include the numerical solution of the Lighthill equation, Curle’s analogy, and Ffowcs Williams and Hawkings (FWH) analogy as well as linearized Euler equations (LEE). Next, the frequency-domain phase-conjugation (PC) method was implemented in the finite-element (FE) based COMSOL software to reconstruct the radiated acoustic pressure field, whereby the dipole source location was readily identified by pair of focal spots at the cylinder or at the airfoil trailing-edge (TE).

Sensitivity of He Flames in X-Ray Bursts to Nuclear Physics

Through the use of axisymmetric 2D hydrodynamic simulations, we further investigate laterally propagating flames in X-ray bursts (XRBs). Our aim is to understand the sensitivity of a propagating helium flame to different nuclear physics. Using the Castro simulation code, we confirm the phenomenon of enhanced energy generation shortly after a flame is established by adding 12C(p, γ)13N(α, p)16O to the network, in agreement with the past literature. This sudden outburst of energy leads to a short accelerating phase, causing a drastic alteration in the overall dynamics of the flame in XRBs. Furthermore, we investigate the influence of different plasma screening routines on the propagation of the XRB flame. We finally examine the performance of simplified spectral deferred correction, a novel approach to hydrodynamics and reaction coupling incorporated in Castro, as an alternative to operator splitting.

A study of non-equilibrium wave groups in two-phase flow in high-contrast porous media with relative permeability hysteresis

Non-equilibrium models are applicable in various physical situations, including phase transitions, hysteresis, and chemical reactions, among others. To model the dynamics of such phenomena, partial differential equations are employed with source terms, as seen in Eq. (1.1). This work focuses on situations where we connect states in equilibrium while allowing for non-equilibrium times. In our model, we model non-equilibrium through relaxation. The source term, in this case, acts as an attractor, and the model’s dynamics allow only a small-time far from equilibrium. We use this theory to model a two-equation system in a two-phase fluid in porous media, where different permeabilities of the flow lead to hysteresis phenomena. We show several examples and study the Riemann solution to gain a better understanding of the solution. We investigate the interplay between nonlinear wave groups of water–oil saturation with different permeabilities in the non-wetting phase and the high-contrast heterogeneity of the porous medium in two-space dimensions. This phenomenon is described by a coupled set of time-dependent partial differential equations of hyperbolic–parabolic–elliptic mixed type. We present and discuss a new projection method with relaxation to obtain 1D-analytical Riemann solutions in an idealized homogenized medium, and we use this solution to validate our numerical method. The primary components of obtaining these solutions are shocks, rarefactions, and bifurcations. We obtain 2D-numerical solutions using an operator splitting approach based on physics, highlighting the discretization methods used. We provide robust analytical–numerical examples to verify the theory and illustrate the approach’s capabilities. In particular, we discuss the 1D-2D behavior of wave groups in situations driven by water–oil infiltration problems of permeable reservoir rocks.

Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems

A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.