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We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward–backward-type algorithm and a Douglas–Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward–backward-type primal–dual algorithm and a Douglas–Rachford-type primal–dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg–Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called μ -mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg–Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge–Kutta integrator, for stringent accuracies. • Lawson and splitting methods of order 4 for Complex Ginzburg–Landau equations. • Efficient implementation by tensor-matrix products or FFT techniques. • Physical 2D and 3D numerical examples (cubic, cubic-quintic, and coupled CGL). [ABSTRACT FROM AUTHOR]

We propose an efficient approach for time integration of Klein–Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes. [ABSTRACT FROM AUTHOR]

PEPit is a python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. To do that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modeling parts, and the worst-case analysis is performed numerically via standard solvers. [ABSTRACT FROM AUTHOR]

The numerical solution of the generalized nonlinear Schrödinger equation by simple splitting methods can be disturbed by so-called spurious instabilities. We analyze these numerical instabilities for an arbitrary splitting method and apply our results to several well-known higher-order splittings. We find that the spurious instabilities can be suppressed to a large extent. However, they never disappear completely if one keeps the integration step above a certain limit and applies what is considered to be a more accurate higher-order method. The latter can be used to make calculations more accurate with the same numerically stable step, but not to make calculations faster with a much larger step. [ABSTRACT FROM AUTHOR]

B-methods are numerical methods which are especially tailored to solve non-linear partial differential equation that have blow up solutions. We have presented in Part I a systematic construction of B-methods based on the variation of constants formula. Here, we use splitting methods as a second way to construct B-methods, and we prove several special properties of such methods. We illustrate our analysis with numerical experiments. [ABSTRACT FROM AUTHOR]

We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation. [ABSTRACT FROM AUTHOR]

This paper introduces novel bulk–surface splitting schemes of first and second order for the wave equation with kinetic and acoustic boundary conditions of semi-linear type. For kinetic boundary conditions, we propose a reinterpretation of the system equations as a coupled system. This means that the bulk and surface dynamics are modeled separately and connected through a coupling constraint. This allows the implementation of splitting schemes, which show first-order convergence in numerical experiments. On the other hand, acoustic boundary conditions naturally separate bulk and surface dynamics. Here, Lie and Strang splitting schemes reach first- and second-order convergence, respectively, as we reveal numerically. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the nonlinear Schrödinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations. [ABSTRACT FROM AUTHOR]

This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e. the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form |$\tau \leqslant c h$| for some constant |$c>0$|⁠. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type. [ABSTRACT FROM AUTHOR]

Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE‐ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy, and efficiency. Simulations were performed for a benchmark problem with the Beeler–Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this configuration, we provide a reference solution for the transmembrane potential. We found a semi‐implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi‐ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time. [ABSTRACT FROM AUTHOR]

We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the primal-dual hybrid gradient (PDHG) method, which is being widely used to solve a broad spectrum of saddle-point problems. Despite of its wide applications in different areas, the study of inexact versions of PDHG still seems to be in its infancy. We investigate how to design implementable inexactness criteria for solving the subproblems in PDHG scheme so that the convergence of an inexact PDHG can be guaranteed. We propose two specific inexactness criteria and accordingly some inexact PDHG methods for saddle-point problems. The convergence of both inexact PDHG methods is rigorously proved, and their convergence rates are estimated under different scenarios. Moreover, some numerical results on image restoration problems are reported to illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms. [ABSTRACT FROM AUTHOR]

We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schrödinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group $ \mathrm{SU}(2) $. In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space. [ABSTRACT FROM AUTHOR]

We develop non-overlapping domain decomposition methods for the Biot system of poroelasticity in a mixed form. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. In the monolithic method, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. We show that the resulting interface operator is positive definite and analyze the convergence of the iteration. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. Numerical experiments are presented to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency. [ABSTRACT FROM AUTHOR]

This work deals with a broad class of convex optimization problems under uncertainty. The approach is to pose the original problem as one of finding a zero of the sum of two appropriate monotone operators, which is solved by the celebrated Douglas-Rachford splitting method. The resulting algorithm, suitable for risk-averse stochastic programs and distributionally robust optimization with fixed support, separates the random cost mapping from the risk function composing the problem's objective. Such a separation is exploited to compute iterates by alternating projections onto different convex sets. Scenario subproblems, free from the risk function and thus parallelizable, are projections onto the cost mappings' epigraphs. The risk function is handled in an independent and dedicated step consisting of evaluating its proximal mapping that, in many important cases, amounts to projecting onto a certain ambiguity set. Variables get updated by straightforward projections on subspaces through independent computations for the various scenarios. The investigated approach enjoys significant flexibility and opens the way to handle, in a single algorithm, several classes of risk measures and ambiguity sets. [ABSTRACT FROM AUTHOR]

We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker–Planck equations. [ABSTRACT FROM AUTHOR]

We extend the methods from [ 39 , 37 ] to a class of non-potential mean-field game (MFG) systems with mixed couplings. Up to now, splitting methods have been applied to potential MFG systems that can be cast as convex-concave saddle-point problems. Here, we show that a class of non-potential MFG can be cast as primal-dual pairs of monotone inclusions and solved via extensions of convex optimization algorithms such as the primal-dual hybrid gradient (PDHG) algorithm. A critical feature of our approach is in considering dual variables of nonlocal couplings in Fourier or feature spaces. [ABSTRACT FROM AUTHOR]

The present paper deals with the numerical solution of non-Hermitian positive definite tensor equation A ∗ N X = B under the Einstein product. Firstly, we extend the Hermitian and skew-Hermitian splitting (HSS) method to solve the tensor equation. Then we propose a new Hermitian splitting (NHS) method under some certain conditions, which is expected to converge faster than the HSS iteration. We also present the optimal parameters of both the HSS and NHS methods. Moreover, we apply the Smith technique to give two modified methods which can greatly accelerate the convergence rate. The performed numerical examples illustrate that the proposed methods are feasible and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we combine deterministic splitting methods with a polynomial chaos expansion method for solving stochastic parabolic evolution problems. The stochastic differential equation is reduced to a system of deterministic equations that we solve efficiently by splitting methods. The method can be applied to a wide class of problems where the related stochastic processes are given uniquely in terms of stochastic polynomials. A comprehensive convergence analysis is provided and numerical experiments validate our approach. [ABSTRACT FROM AUTHOR]

We study numerical methods for dissipative particle dynamics (DPD), which is a system of stochastic differential equations and a popular stochastic momentum-conserving thermostat for simulating complex hydrodynamic behavior at mesoscales. We propose a new splitting method that is able to substantially improve the accuracy and efficiency of DPD simulations in a wide range of friction coefficients, particularly in the extremely large friction limit that corresponds to a fluidlike Schmidt number, a key issue in DPD. Various numerical experiments on both equilibrium and transport properties are performed to demonstrate the superiority of the newly proposed method over popular alternative schemes in the literature. [ABSTRACT FROM AUTHOR]

In this article, we present a direction splitting method, combined with a nonlinear iteration, for the compressible Navier‐Stokes equations in spherical coordinates. The method is aimed at solving the equations on the sphere, and can be used for a regional geophysical simulations as well as simulations on the entire sphere. The aim of this work was to develop a method that would work efficiently in the limit of very small to vanishing Mach numbers, and we demonstrate here, using a numerical example, that the method shows good convergence and stability at Mach numbers in the range [10−2, 10−6]. We also demonstrate the effect of some of the parameters of the model on the solution, on a common geophysical test case of a rising thermal bubble. The algorithm is particularly suitable for a massive parallel implementation, and we show below some results demonstrating its excellent weak scalability. [ABSTRACT FROM AUTHOR]

This paper is concerned with solving the multilinear systems A x m - 1 = b whose coefficient tensors are mth-order and n-dimensional symmetric tensors. We first extend the accelerated overrelaxation (AOR) splitting method to solve the tensor equation. To improve the convergence, we develop a Newton-AOR (NAOR) method that hybridizes the Newton method and the accelerated overrelaxation scheme. Convergence analysis shows that the proposed methods converge under appropriate assumptions. Finally, some numerical examples are provided to show the effectiveness of the methods proposed. [ABSTRACT FROM AUTHOR]

We propose a methodology for studying the performance of common splitting methods through semidefinite programming. We prove tightness of the methodology and demonstrate its value by presenting two applications of it. First, we use the methodology as a tool for computerassisted proofs to prove tight analytical contraction factors for Douglas--Rachford splitting that are likely too complicated for a human to find bare-handed. Second, we use the methodology as an algorithmic tool to computationally select the optimal splitting method parameters by solving a series of semidefinite programs. [ABSTRACT FROM AUTHOR]

Given the success of Douglas–Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there other 2 operator resolvent-splittings sharing the favorable properties of DRS? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is impossible without lifting, where lifting roughly corresponds to enlarging the problem size. The impossibility result further raises a question. How much lifting is necessary to generalize DRS to 3 operators? This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS. [ABSTRACT FROM AUTHOR]

We analyze the existence of dark solitons in nonlinear complex Ginz- burg-Landau equations. We prove existence results concerned with the initial value problem for these equations in Zhidkov spaces using a new approach with splitting methods. [ABSTRACT FROM AUTHOR]

In this work, a diagonal splitting idea is presented for solving linear systems of ordinary differential equations. The resulting methods are specially efficient for solving systems which have arisen from semidiscretization of parabolic partial differential equations (PDEs). Unconditional stability of methods for heat equation and advection–diffusion equation is shown in maximum norm. Generalization of the methods in higher dimensions is discussed. Some illustrative examples are presented to show efficiency of the new methods. [ABSTRACT FROM AUTHOR]

We prove mean-square convergence of a exponential tamed method, for a generalized Ait-Sahalia interest rate model. The method is based on a Lamperti transform, splitting and applying a tamed numerical method for the nonlinearity. The main difficulty in the analysis is caused by the non-globally Lipschitz drift coefficients of the model. We consider the existence, uniqueness of the solution and boundedness of moments for the transformed SDE before proving bounded moments and inverse moment bounds for the numerical approximation. The exponential tamed method is a hybrid method in the sense that a backstop method is invoked to prevent solutions from overshooting zero and becoming negative. We successfully recover the strong convergence rate of order one for the exponential tamed method. In addition we prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. In our numerical experiments we compare to other numerical methods in the literature for realistic parameter values. • We recover the strong convergence rate of order one for the numerical method. • The probability that numerical solution is negative is arbitrarily small. • The method is based on a Lamperti transform, splitting and tamed method. [ABSTRACT FROM AUTHOR]

In this paper, two structure-preserving nonstandard finite difference (NSFD) operator splitting schemes are designed for the solution of reaction diffusion epidemic models. The proposed schemes preserve all the essential properties possessed by the continuous systems. These schemes are applied on a diffusive SEIQV epidemic model with a saturated incidence rate to validate the results. Furthermore, the stability of the continuous system is proved, and the bifurcation value is evaluated. A comparison is also made with the existing operator splitting numerical scheme. Simulations are also performed for numerical experiments.

The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

We propose new local error estimators for splitting and composition methods. They are based on the construction of lower order schemes obtained at each step as a linear combination of the intermediate stages of the integrator, so that the additional computational cost required for their evaluation is almost insignificant. These estimators can be subsequently used to adapt the step size along the integration. Numerical examples show the efficiency of the procedure. [ABSTRACT FROM AUTHOR]

We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). At each iteration, the algorithms require inexact evaluations of the proximal operator, as well as, approximate subgradients of the functions (namely: the ϵ -subgradients). The methods use different error criteria for approximating the proximal operators. We provide an analysis of the convergence and rate of convergence properties of these methods, considering various stepsize rules, including both, diminishing and constant stepsizes. For the case where one of the functions is smooth, we propose an inexact accelerated version of the proximal gradient method, and prove that the optimal convergence rate for the function values can be achieved. Moreover, we provide some numerical experiments comparing our algorithm with similar recent ones. [ABSTRACT FROM AUTHOR]

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the 'Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra X, spanned by 'modified' potential energies and isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with X. We calculate the dimensions cn of its homogeneous subspaces and determine the value of its entropy limn→∞c1/nn. It is 1.8249..., a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., that the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics. [ABSTRACT FROM AUTHOR]

In this paper, we propose generalized splitting methods for solving a class of nonconvex optimization problems. The new methods are extended from the classic Douglas–Rachford and Peaceman–Rachford splitting methods. The range of the new step-sizes even can be enlarged two times for some special cases. The new methods can also be used to solve convex optimization problems. In particular, for convex problems, we propose more relax conditions on step-sizes and other parameters and prove the global convergence and iteration complexity without any additional assumptions. Under the strong convexity assumption on the objective function, the linear convergence rate can be derived easily. [ABSTRACT FROM AUTHOR]

The problem of mathematical modeling and optimization of nonstationary diffusion and heat conduction processes is considered. An approach that uses the idea of splitting and computation of the obtained difference schemes using explicit schemes of point to point computing is proposed for numerical solution of multidimensional diffusion and heat conduction initial–boundary-value problems. Construction of difference splitting schemes, approximation and stability on initial data are investigated. Differential properties of the quality functional are analyzed for the numerical solution of the optimal control problem for a parabolic equation. An iterative algorithm for finding the optimal control is proposed. [ABSTRACT FROM AUTHOR]

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