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Abstract Prior work on computable defect-based local error estimators for (linear) time-reversible integrators is extended to nonlinear and nonautonomous evolution equations. We prove that the asymptotic results from the linear case (Auzinger and Koch, 2018) remain valid, i.e., the modified estimators yield an improved asymptotic order as the step size goes to zero. Typically, the computational effort is only slightly higher than for conventional defect-based estimators, and it may even be lower in some cases. We illustrate this by some examples and present numerical results for evolution equations of Schrödinger type, solved by either time-splitting or Magnus-type integrators. Finally, we demonstrate that adaptive time-stepping schemes can be successfully based on our local error estimators. [ABSTRACT FROM AUTHOR]

Calculating cost-effective solutions to particle dynamics in viscous flows is an important problem in many areas of industry and nature. We implement a second-order symmetric splitting method on the governing equations for a rigid spheroidal particle model with torques, drag and gravity. The method splits the operators into a vector field that is conservative and one that takes into account the forces of the fluid. Error analysis and numerical tests are performed on perturbed and stiff particle-fluid systems. For the perturbed case, the splitting method greatly improves the solution accuracy, when compared to a conventional multistep method, and the global error behaves as O (ε h 2) for roughly equal computational cost. For stiff systems, we show that the splitting method retains stability in regimes where conventional methods blow up. In addition, we show through numerical experiments that the global order is reduced from O (h 2 / ε) in the perturbed regime to O (h) in the stiff regime. [ABSTRACT FROM AUTHOR]

Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity (respectively, strong convexity for optimization problems). In this paper, we propose a modified Krasnosel'skiĭ–Mann algorithm in connection with the determination of a fixed point of a nonexpansive mapping and show strong convergence of the iteratively generated sequence to the minimal norm solution of the problem. Relying on this, we derive a forward–backward and a Douglas–Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. Furthermore, we formulate strong convergent primal–dual algorithms of forward–backward and Douglas–Rachford-type for highly structured monotone inclusion problems involving parallel-sums and compositions with linear operators. The resulting iterative schemes are particularized to the solving of convex minimization problems. The theoretical results are illustrated by numerical experiments on the split feasibility problem in infinite dimensional spaces. [ABSTRACT FROM AUTHOR]

It is known that operator splitting methods based on forward–backward splitting, Douglas–Rachford splitting, and Davis–Yin splitting decompose difficult optimization problems into simpler subproblems under proper convexity and smoothness assumptions. In this paper, we identify an envelope (an objective function), whose gradient descent iteration under a variable metric coincides with Davis–Yin splitting iteration. This result generalizes the Moreau envelope for proximal-point iteration and the envelopes for forward–backward splitting and Douglas–Rachford splitting iterations identified by Patrinos, Stella, and Themelis. Based on the new envelope and the stable–center manifold theorem, we further show that, when forward–backward splitting or Douglas–Rachford splitting iterations start from random points, they avoid all strict saddle points with probability one. This result extends the similar results by Lee et al. from gradient descent to splitting methods. [ABSTRACT FROM AUTHOR]

An ordinary differential equation (ODE) can be split into simpler sub equations and each of the sub equations is solved subsequently by a numerical method. Such a procedure involves splitting error and numerical error caused by the time stepping methods applied to sub equations. The aim of the paper is to present an integral formula for the global error expansion of a splitting procedure combined with any numerical ODE solver.

New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a way that each part is explicitly solvable. The methods are obtained by applying different optimization criteria and preserve geometric properties of the continuous problem by construction. Different numerical examples exhibit their improved performance with respect to previous splitting methods in the literature.

An ordinary differential equation (ODE) can be split into simpler sub equations and each of the sub equations is solved subsequently by a numerical method. Such a procedure involves splitting error and numerical error caused by the time stepping methods applied to sub equations. The aim of the paper is to present an integral formula for the global error expansion of a splitting procedure combined with any numerical ODE solver. [ABSTRACT FROM AUTHOR]

Most of the existing numerical schemes developed to solve Burgers’ equation cannot exhibit its correct physical behavior for very small values of viscosity. This difficulty can be overcome by using splitting methods derived for near-integrable system. This class of methods has positive real coefficients and can be used for non-reversible systems such as Burgers’ equation. It also has the advantage of being able to account small viscosity in the accuracy. The algorithm is based on the combination of implicit–explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability. The resulting algorithm is accurate, efficient and easy to implement. The new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity. [ABSTRACT FROM AUTHOR]

Splitting methods are frequently used for large scale chemical reaction systems. The main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem. However, a splitting error is introduced. The most popular splitting schemes are the first order Lie–Trotter splitting and the second order Strang splitting. However, in case of stiff differential equations the Strang splitting suffers from order reduction and both schemes have order one for stiff differential equations. Therefore, the step size restrictions due to the low order can result in a prohibitive small step size. Hence, splitting schemes with order larger than one are necessary for stiff differential equations. The Richardson extrapolation of the Lie–Trotter splitting is a second order scheme. In this paper we examine differential equations with a fast chemical source term and a slow transport term. Thus, all stiffness of the system is related to the chemical source term. We show that the Richardson extrapolation does not suffer from order reduction in this case. Furthermore, we perform a stability analysis of the extrapolated splitting scheme for a linear test problem. Thereby the operators of the test problem do not commute. [ABSTRACT FROM AUTHOR]

In this paper we investigate splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial. [ABSTRACT FROM AUTHOR]

We consider fluids composed of Coulomb-interacting particles, which aremodelled by the Fokker-Planck equation with a collision operator. Based on modelling the transport and collision of the particles, we propose new, computationally efficient, algorithms based on splitting the equations of motion into a global Newtonian transport equation, where the effects of an external electric field are considered, and a local Coulomb interaction stochastic differential equation, which determinesthe new velocities of the particle. Two different numerical schemes, one deterministic and the other stochastic, as well as an Hamiltonian splitting approach, are proposed for coupling the interactionand transport equations. Results are presented for two- and multi-particle systems with different approximations for the Coulomb interaction. Methodologically, the transport part is modelled by thekinetic equations and the collision part is modelled bythe Langevin equations with Coulomb collisions. Such splitting approaches allow concentrating on different solver methods for each different part. Further, we solve multiscale problems involving an external electrostatic field. We apply a multiscale approach so that we can decompose the different time-scales of the transport and the collision parts. We discuss the benefits of the different splitting approaches and theirnumerical analysis. [ABSTRACT FROM AUTHOR]

The alternating direction method of multipliers (ADMM) recently has found many applications in various domains whose models can be represented or reformulated as a separable convex minimization model with linear constraints and an objective function in sum of two functions without coupled variables. For more complicated applications that can only be represented by such a multi-block separable convex minimization model whose objective function is the sum of more than two functions without coupled variables, it was recently shown that the direct extension of ADMM is not necessarily convergent. On the other hand, despite the lack of convergence, the direct extension of ADMM is empirically efficient for many applications. Thus we are interested in such an algorithm that can be implemented as easily as the direct extension of ADMM, while with comparable or even better numerical performance and guaranteed convergence. In this paper, we suggest correcting the output of the direct extension of ADMM slightly by a simple correction step. The correction step is simple in the sense that it is completely free from step-size computing and its step size is bounded away from zero for any iterate. A prototype algorithm in this prediction-correction framework is proposed; and a unified and easily checkable condition to ensure the convergence of this prototype algorithm is given. Theoretically, we show the contraction property, prove the global convergence and establish the worst-case convergence rate measured by the iteration complexity for this prototype algorithm. The analysis is conducted in the variational inequality context. Then, based on this prototype algorithm, we propose a class of specific ADMM-based algorithms that can be used for three-block separable convex minimization models. Their numerical efficiency is verified by an image decomposition problem. [ABSTRACT FROM AUTHOR]

In this note we consider splitting methods based on linear multistep methods and stabilizing corrections. To enhance the stability of the methods, we employ an idea of Bruno and Cubillos [O. P. Bruno and M. Cubillos, J. Comput. Phys., 307 (2016), pp. 476-495], who combine a high-order extrapolation formula for the explicit term with a formula of one order lower for the implicit terms. Several examples of the obtained multistep stabilizing correction methods are presented, and results on linear stability and convergence are derived. The methods are tested in the application to the well-known Heston model arising in financial mathematics and are found to be competitive with well-established one-step splitting methods from the literature. [ABSTRACT FROM AUTHOR]

The Klein–Gordon equation on an infinite two dimensional strip is considered. Numerical computation is reduced to a finite domain by using the Hagstrom–Warburton (H–W) absorbing boundary conditions (ABCs) with free parameters in the formulation of the auxiliary variables. The spatial discretization is achieved by using fourth order finite differences and the time integration is made by means of an efficient and easy to implement fourth order exponential splitting scheme which was used in Alonso-Mallo and Portillo (2016) considering the fixed Padé parameters in the formulation of the ABCs. Here, we generalize the splitting time technique to other choices of the parameters. To check the time integrator we consider, on one hand, four types of fixed parameters, the Newmann’s parameters, the Chebyshev’s parameters, the Padé’s parameters and optimal parameters proposed in Hagstrom et al. (2007) and, on the other hand, an adaptive scheme for the dynamic control of the order of absorption and the parameters. We study the efficiency of the splitting scheme by comparing with the fourth-order four-stage Runge–Kutta method. [ABSTRACT FROM AUTHOR]

The unsteady incompressible Navier–Stokes equations in primitive variables are often numerically solved by segregating the computation of velocity and pressure, according to either functional analysis arguments following the pioneering work of A.J. Chorin and R. Temam (STD, Split-Then-Discretize paradigm) or linear algebra arguments based on the inexact block factorization of the discretized problem (DTS, Discretize-Then-Split paradigm). The presence of the time derivative allows for the calibration of an appropriate approximation of the pseudo-differential operator of the pressure problem and excellent results in terms of both accuracy and efficiency have been obtained as witnessed by the abundant literature. The extension of the same segregated approach to the steady Navier–Stokes equations is unclear, unless a pseudo-time advancing formulation is undertaken. In this paper we present a methodology for a segregated computation of the primitive variables in a genuinely steady formulation, so to avoid iterations to get to the steady limit. The approach is largely inspired by the algebraic factorization of the unsteady problem (DTS approach), yet we detail specific settings required by the absence of the velocity time-derivative. The basic idea relies on the introduction of some parameters in a modified Picard linearization. We discuss stability bounds and the convergence of the segregated method to the unsplit solution. Several numerical results on different test cases confirm the efficiency of the procedure. [ABSTRACT FROM AUTHOR]

We consider time discretizations of the Vlasov-HMF (Hamiltonian mean-field) equation based on splitting methods between the linear and nonlinear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time step size. [ABSTRACT FROM AUTHOR]

In the context of linear inverse problems, we propose and study a general iterative regularization method allowing to consider large classes of data-fit terms and regularizers. The algorithm we propose is based on a primal-dual diagonal descent method. Our analysis establishes convergence as well as stability results. Theoretical findings are complemented with numerical experiments showing state-of-the-art performances. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the different splitting approaches to numerically solvethe Gross–Pitaevskii equation (GPE). The models are motivated from spinor Bose–Einsteincondensate (BEC). This system is formed of coupled mean-field equations, which are based oncoupled Gross–Pitaevskii equations. We consider conservative finite-difference schemes and spectralmethods for the spatial discretisation. Furthermore, we apply implicit or explicit time-integrators andcombine these schemes with different splitting approaches. The numerical solutions are comparedbased on the conservation of the L2-norm with the analytical solutions. The advantages of the novelsplitting methods for large time-domains are based on the asymptotic conservation of the solution ofthe soliton’s applications. Furthermore, we have the benefit of larger local time-steps and thereforeobtain faster numerical schemes.

An error analysis of symmetric trigonometric integrators applied to highly oscillatory linear second-order differential equations is given. Second-order convergence is shown uniformly in the high frequencies under a finite-energy condition on the exact solution. The main novelty is the concept of the proof itself, which is based on the interpretation of trigonometric integrators as splitting methods for averaged differential equations. This allows one to combine techniques for splitting methods with those for trigonometric integrators. For the bound of the global error, cancellations in the error accumulation have to be studied carefully. [ABSTRACT FROM AUTHOR]

Fuzziness based divide and conquer (D&C) is a recently proposed strategy for promoting the classifiers (i.e., fuzzy classifiers) performance, where the amount of fuzziness quantity associated with each data point (i.e., both labeled and unlabeled) is considered as an important avenue to the empire for instance selection problem. This technique is regarded as a semi-supervised learning (SSL) technique, where different categories of instances are obtained by using fuzziness measure, and then the instances having less amount of fuzziness are incorporated into training set for improving the generalization ability of a classifier. This study proposes some effective methods and presents a novel algorithm for categorizing the instances into three groups that can effectively integrate with D&C strategy. It is observed by the experimental validation that considering the splitting criteria for instances categorization can lead the classifier to perform better on withheld set. Results on different classification data sets prove the effectiveness of proposed algorithm. [ABSTRACT FROM AUTHOR]

We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure seems promising as we can treat the nonlinearity separately. This has some computational advantages, since we only have to solve a linear constrained system and a nonlinear ordinary differential equation. However, Strang splitting suffers from order reduction which limits its efficiency. This reduction is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. The incorporation of an additional correction term resolves this problem without increasing the computational cost significantly. Numerical examples including a coupled mechanical system illustrate the proved convergence results. [ABSTRACT FROM AUTHOR]

The division of tablets and adequate methods of splitting them are a complex problem in all sectors of health care. Although tablet-splitting is often required, this procedure can be difficult for patients. Four tablets were investigated with different external features (shape, score-line, film-coat and size). The influencing effect of these features and the splitting methods was investigated according to the precision and “weight loss” of splitting techniques. All four types of tablets were halved by four methods: by hand, with a kitchen knife, with an original manufactured splitting device and with a modified tablet splitter based on a self-developed mechanical model. The mechanical parameters (harness and friability) of the products were measured during the study. The “weight loss” and precision of splitting methods were determined and compared by statistical analysis. On the basis of the results, the external features (geometry), the mechanical parameters of tablets and the mechanical structure of splitting devices can influence the “weight loss” and precision of tablet-splitting. Accordingly, a new decision-making scheme was developed for the selection of splitting methods. In addition, the skills of patients and the specialties of therapy should be considered so that pharmaceutical counselling can be more effective regarding tablet-splitting. [ABSTRACT FROM AUTHOR]

The treatment of the stochastic linear quadratic optimal control problem with finite time horizon requires the solution of stochastic differential Riccati equations. We propose efficient numerical methods, which exploit the particular structure and can be applied for large-scale systems. They are based on numerical methods for ordinary differential equations such as Rosenbrock methods, backward differentiation formulas, and splitting methods. The performance of our approach is tested in numerical experiments. [ABSTRACT FROM AUTHOR]

This article deals with the numerical integration in time of nonlinear Schrödinger equations. The main application is the numerical simulation of rotating Bose-Einstein condensates. The authors perform a change of unknown so that the rotation term disappears and they obtain as a result a nonautonomous nonlinear Schrödinger equation. They consider exponential integrators such as exponential Runge-Kutta methods and Lawson methods. They provide an analysis of the order of convergence and some preservation properties of these methods in a simplified setting and they supplement their results with numerical experiments with realistic physical parameters. Moreover, they compare these methods with the classical split-step methods applied to the same problem. [ABSTRACT FROM AUTHOR]

The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii-Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii-Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas-Rachford splitting method, the Fermat-Weber location problem as well as the alternating projection method by John von Neumann. [ABSTRACT FROM AUTHOR]

We present modifications of the second-order Douglas stabilizing corrections method, which is a splitting method based on the implicit trapezoidal rule. Inclusion of an explicit term in a forward Euler way is straightforward, but this will lower the order of convergence. In the modifications considered here, explicit terms are included in a second-order fashion. For these modified methods, results on linear stability and convergence are derived. Stability holds for important classes of reaction-diffusion equations, and for such problems the modified Douglas methods are seen to be often more efficient than related methods from the literature. [ABSTRACT FROM AUTHOR]

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First, we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray-Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators. [ABSTRACT FROM AUTHOR]

We consider the L 1 regularized iterative forward backward splitting (IFBS) algorithm for image restoration. The main aim of the paper is to develop a fast and adaptive method with an automatic selection of regularization parameter. The regularization parameter is updated in each iteration without requiring the initial value of the parameter. We establish analytically monotonicity results of the adaptive parameter. Such an algorithm corresponds to solving an adaptive iterative thresholding process. Experimental results demonstrate that the adaptive parameter method is efficient and provide competitive performance. [ABSTRACT FROM AUTHOR]

We derive a new algorithm for computing the action ƒ(A)V of the cosine, sine, hyperbolic cosine, and hyperbolic sine of a matrix A on a matrix V, without first computing ƒ(A). The algorithm can compute cos(A)V and sin(A)V simultaneously, and likewise for cosh(A)V and sinh(A)V, and it uses only real arithmetic when A is real. The algorithm exploits an existing algorithm expmv of Al-Mohy and Higham for eAV and its underlying backward error analysis. Our experiments show that the new algorithm performs in a forward stable manner and is generally significantly faster than alternatives based on multiple invocations of expmv through formulas such as cos(A)V = (eiAV + e-iAV)/2. [ABSTRACT FROM AUTHOR]

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed. [ABSTRACT FROM AUTHOR]

We consider numerical modeling of the propagation of pollution in the air on the basis of geometrical splitting method for three-dimensional nonstationary convection diffusion equations. Splitting difference schemes in the form of explicit computing schemes are proposed to solve the obtained one-dimensional problems. The approximation, monotonicity, and stability of the proposed difference schemes are investigated. [ABSTRACT FROM AUTHOR]

We compare the stability preserving properties of the Lie–Trotter, Strang–Marchuk, and symmetrically weighted sequential splitting schemes for a simple 2-dimensional linear system. We evaluate the trace and determinant of the split systems in terms the trace and determinant of the continuous system to establish stability criteria. We find that the stability region has a fingered structure, implying that stability is not a monotonic property of the splitting timestep. We provide estimates for the thickness of the stability fingers as well as the gaps between them. Counterintuitively, both the thickness and the size of gaps grow with decreasing splitting time step. [ABSTRACT FROM AUTHOR]

In this paper, we review recent progress on two related issues. Firstly, the discretisation of partial differential equations of quantum mechanics in a semiclassical regime. Due to the presence of a small parameter, such equations exhibit high oscillations and multiscale behaviour, rendering them difficult to discretise. We describe a methodology, using symmetric Zassenhaus splittings in a free Lie algebra, which allows for their exceedingly fast and accurate numerics. The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper, approximation of derivatives by skew-symmetric matrices. Here, we identify a gap in the elementary theory of finite-difference approximations: in the presence of Dirichlet boundary conditions, it is impossible to approximate the derivative to order higher than two on a uniform grid! This motivates the investigation of skew symmetry on non-uniform grids, an endeavour which, although still in its infancy, is already replete with interesting results. We conclude by discussing a number of generalisations and open problems. [ABSTRACT FROM AUTHOR]

This study concerns the asymptotic stability of an eikonal, or ray, transformation based Peaceman–Rachford splitting method for solving the paraxial Helmholtz equation with high wave numbers. Arbitrary nonuniform grids are considered in transverse and beam propagation directions. The differential equation targeted has been used for modeling propagations of high intensity laser pulses over a long distance without diffractions. Self-focusing of high intensity beams may be balanced with the de-focusing effect of created ionized plasma channel in the situation, and applications of grid adaptations are frequently essential. It is shown rigorously that the fully discretized oscillation-free decomposition method on arbitrary adaptive grids is asymptotically stable with a stability index one. Simulation experiments are carried out to illustrate our concern and conclusions. [ABSTRACT FROM AUTHOR]