Classical Splitting Method Research Articles

Multilabel Genre Prediction Using Deep-Learning Frameworks

In this study, transfer learning has been used to overcome multilabel classification tasks. As a case study, movie genre classification by using posters has been chosen. Six state-of-the-art pretrained models, VGG16, ResNet, DenseNet, Inception, MobileNet, and ConvNeXt, have been employed for this experiment. The movie posters have been obtained from Internet Movie Database (IMDB). The dataset has been divided using an iterative stratification technique. A sequence of dense layers has been added on top of each model and these models have been trained and fine-tuned. All the results of the models compared considered accuracy, loss, Hamming loss, F1-score, precision, and AUC metrics. When the metrics used were evaluated, the most successful result regarding accuracy has been obtained from the modified DenseNet architecture at 90%. Also, the ConvNeXt, which is the newest model among all, performed quite satisfactorily, reaching over 90% accuracy. This study uses an iterative stratification method to split an unbalanced dataset which provides more reliable results than the classical splitting method which is the common method in the literature. Also, the feature extraction capabilities of the six pretrained models have been compared. The outcome of this study shows promising results regarding multilabel classification. As for future work, it is planned to enhance this study by using natural language processing and ensemble methods.

A Dynamical Splitting Method for Minimizing the Sum of Three Convex Functions

In this paper, we develop a dynamical splitting method based on a nonlinear dynamical system with two proximal operators for solving the minimum value problem of the sum of three convex functions. The method endows some existing classical splitting methods such as the Forward-Backward splitting method and the Douglas-Rachford splitting method with continuous time behaviors. We demonstrate that the proposed dynamical system converges globally to a point whose image, imaged by each of the two proximal operators, is a solution to the minimum value problem. We also shows that both the residual of the two proximal operators and the objective error enjoy the convergence rates of where t denotes time. This rate can be improved to by using an ergodic average technique. Moreover, under certain smoothness and metric subregularity conditions, we further establish the global exponential convergence of the dynamical splitting method. Finally, we apply the dynamical splitting method to deal with the problem of compressive sensing.

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

The average method is much better than average

Operator splitting is a powerful method for the numerical investigation of complex time-dependent models, where the stationary (elliptic) part consists of a sum of several structurally simpler sub-operators. As an alternative to the classical splitting methods, a new splitting scheme is proposed here, the Average Method with sequential splitting. In this method, a decomposition of the original problem is sought in terms of commuting matrices. Wedemonstrate that third-order accuracy can be achieved with the Average Method. The computational performance of the method is investigated, yielding run times 1-2 orders of magnitude faster than traditional methods.

Exact Splitting Methods for Kinetic and Schrödinger Equations

In (Bernier in Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators. arXiv:1912.13219, (2019)), some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, Fokker–Planck equations, and Schrödinger type equations with an angular momentum rotation term. In this work, these exact splittings are used combined with pseudo-spectral methods in space. High accuracy and efficiency of exact splitting methods are illustrated and comparison are performed with the numerical methods in literature. We show that our methods can be used to improve significantly some classical splitting methods for some nonlinear or non-quadratic equations.

Non-existence of generalized splitting methods with positive coefficients of order higher than four

We prove that generalized exponential splitting methods making explicit use of commutators of the vector fields are limited to order four when only real coefficients are admitted. This generalizes the restriction to order two for classical splitting methods with only positive coefficients.

Long Time Behavior and Global Dynamics of Simplified Von Karman Plate Without Rotational Inertia Driven by White Noise

Without the assumption that the coefficient of weak damping is large enough, the existence of the global random attractors for simplified Von Karman plate without rotational inertia driven by either additive white noise or multiplicative white noise are proved. Instead of the classical splitting method, the techniques to verify the asymptotic compactness rely on stabilization estimation of the system. Furthermore, a clear relationship between in-plane components of the external force that act on the edge of the plate and the expectation of radius of the global random attractors can be obtained from the theoretical results. Based on the relationship between global random attractor and random probability invariant measure, the global dynamics of the plates are analyzed numerically. With increasing the in-plane components of the external force that act on the edge of the plate, global D-bifurcation, secondary global D-bifurcation and complex local dynamical behavior occur in motion of the system. Moreover, increasing the intensity of white noise leads to the dynamical behavior becoming simple. The results on global dynamics reveal that random snap-through which seems to be a complex dynamics intuitively is essentially a simple dynamical behavior.

A survey on operator splitting and decomposition of convex programs

Many structured convex minimization problems can be modeled by the search of a zero of the sum of two monotone operators. Operator splitting methods have been designed to decompose and regularize at the same time these kind of models. We review here these models and the classical splitting methods. We focus on the numerical sensitivity of these algorithms with respect to the scaling parameters that drive the regularizing terms, in order to accelerate convergence rates for different classes of models.

Projective splitting methods for sums of maximal monotone operators with applications

Splitting methods for sums of maximal monotone operators are studied in this paper. By formulating the classical splitting methods, including the Douglas/Peaceman–Rachford splitting and the forward–backward splitting, into fixed point iterations, general splitting methods are considered via parameterizing fixed point formulas. Weak convergence results for these general splitting methods are derived with the help of a demiclosedness principle. Moreover, by employing the Haugazeau-like projective method, the weak convergence splitting algorithms are forced to be strongly convergent. Finally, applications to the convex feasibility and best approximation problems are made from the viewpoint of convex optimization; new and improved convergence results for solving these two problems are obtained.

Numerical simulations of three-dimensional nonlinear acoustical waves: Application to the modeling of helicoïdal beams

A numerical method for the simulation of three-dimensional nonlinear acoustical wave propagation through a homogeneous or weakly heterogeneous medium is presented. This method is based on the resolution a nonlinear wave equation taking into account diffraction, nonlinearities and weak heterogeneities exact up to second order. It is numerically solved in a one-way manner by using a classical splitting method in three steps: angular spectrum method for diffraction, implicit finite differences method for heterogeneities, and semi-analytical Burgers-Hayes method for nonlinearities. Simulations of propagation of nonlinear helicoïdal beams will illustrate the capacities of the method. This kind of acoustical beams featuring radial, azimuthal and axial variations of the field is intrinsically three-dimensional. Furthermore, they have properties of potential interest in biomedical acoustics either for imaging purposes (dynamics of the so-called topological charge) or for therapy purposes (generation of helical shock front or acoustical tweezers with radiation force).

An iterative splitting method via waveform relaxation

This paper explores a new numerical strategy for a closed formulation of iterative splitting methods and their embedding in classical waveform-relaxation methods. Since iterative splitting has been developed in several papers, an abstract framework that relates these methods to other classical splitting methods would be useful and is needed. Here, we present an embedding of the iterative splitting method in the waveform-relaxation and exponential splitting methods. While we can use the theoretical background of the classical schemes, a simpler iterative splitting analysis is obtained. This is achieved by basing the analysis on semigroup and fixpoint schemes. Our approach is illustrated with numerical results obtained on differential equations with constant and time-dependent coefficients.

Efficient Monte Carlo simulation via the generalized splitting method

We describe a new Monte Carlo algorithm for the consistent and unbiased estimation of multidimensional integrals and the efficient sampling from multidimensional densities. The algorithm is inspired by the classical splitting method and can be applied to general static simulation models. We provide examples from rare-event probability estimation, counting, and sampling, demonstrating that the proposed method can outperform existing Markov chain sampling methods in terms of convergence speed and accuracy.

An algorithm for the solution of a tidal dynamics problem on a sphere

We consider a method for solving the system of equations of tidal dynamics based on approaches of classical splitting method and the theory of iterative processes. The statement of the problem is discussed when ‘pole points’ belong to computational domain. The numerical experiments are presented taking into account the real domain of the World Ocean, bottom topography, and tidal potential.

Additive and iterative operator splitting methods and their numerical investigation

As an alternative to the classical splitting methods, two new splitting schemes have been developed recently: the additive and the iterative splitting. In this paper we discuss the most important properties, the advantages and disadvantages of these schemes, and investigate their performance on simple examples as well as on more complex physical models.

Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a photonic crystal with combined nonlinearity

A conservative finite-difference scheme is constructed for the problem of propagation of a light pulse in a one-dimensional nonlinear photonic crystal with combined nonlinearity. The invariants of the corresponding differential problem and their difference analogues are given. The scheme is compared with those based on the widespread splitting method. For combined cubic and quadratic nonlinearity in photonic crystal layers, it is shown that the classical splitting method is ineffective, since it requires time steps that are smaller by one or more orders of magnitude. The finite-difference scheme proposed conserves the propagation invariants, which cannot be achieved for splitting schemes even on considerably finer grids. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes as applied to the simulation of complex nonlinear effects in photonic crystals, which require much smaller steps in space and time than those used in the case of linear propagation. The simulation is based on the approach proposed by the authors for the given class of problems.

Numerical Solution of a Hyperbolic-parabolic System by Splitting Methods and Optimal Control Approaches

AbstractA new approach has been developed for solving the tide dynamics problem, based on the splitting methods and the optimal control theory. We first apply the classical splitting method for solving the problem. Then we use the optimal control theory approaches to the system of equations obtained after the splitting. An optimal control problem has been formulated for realizing a splitting method step. We prove that the optimal control problem is well-posed and we propose an iterative process of the minimization problem. The results of the numerical experiments are presented.

Efficient removal of boundary-divergence errors in time-splitting methods

A normal mode analysis is presented and numerical tests are performed to assess the effectiveness of a new time-splitting algorithm proposed recently in Karniadakiset al. (1990) for solving the incompressible Navier-Stokes equations. This new algorithm employs high-order explicit pressure boundary conditions and mixed explicit/implicit stiffly stable time-integration schemes, which can lead to arbitrarily high-order accuracy in time. In the current article we investigate both the time accuracy of the new scheme as well as the corresponding reduction in boundary-divergence errors for two model flow problems involving solid boundaries. The main finding is that time discretization errors, induced by the nondivergent splitting mode, scale with the order of the accuracy of the integration rule employed if a proper rotational form of the pressure boundary condition is used; otherwise a first-order accuracy in time similar to the classical splitting methods is achieved. In the former case the corresponding errors in divergence can be completely eliminated, while in the latter case they scale asO(vΔt)1/2.