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splitting algorithms FOR NUMERICAL SOLUTION OF NAVIER - STOKES EQUATIONS IN FLUID DYNAMICS PROBLEMS

JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS 2021年第3期62卷 391-400页

作者： Kovenya, V. M. Fed Res Ctr Informat & Computat Technol Novosibirsk 630090 Russia Novosibirsk State Univ Novosibirsk 630090 Russia

Implicit finite-volume predictor-corrector algorithms based on the splitting method are proposed for the numerical solution of the Navier - Stokes equations written in integral form for a compressible gas, and the properties of these algorithms are investigated. An economical algorithm for splitting equations into physical processes and spatial variables is considered. Numerical solutions of two-dimensional and spatial fluid dynamics problems are determined and compared with the known computational results. It can be concluded on the basis of the estimates obtained and calculations performed that the proposed algorithms are effective.

关键词： Navier Stokes equations finite-volume schemes splitting algorithms supersonic flows

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splitting algorithms FOR RARE EVENT SIMULATION OVER LONG TIME INTERVALS

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ANNALS OF APPLIED PROBABILITY 2020年第6期30卷 2963-2998页

作者： Buijsrogge, Anne Dupuis, Paul Snarski, Michael Univ Twente Fac Elect Engn Math & Comp Sci Enschede Netherlands Brown Univ Div Appl Math Providence RI 02912 USA

In this paper we study the performance of splitting algorithms, and in particular the RESTART method, for the numerical approximation of the probability that a process leaves a neighborhood of a metastable point during some long time interval [0, T]. We show that, in contrast to alternatives such as importance sampling, the decay rate of the second moment does not degrade as T -> infinity. In the course of the analysis we develop some related large deviation estimates that apply when the time interval of interest depends on the large deviation parameter.

关键词： splitting algorithms RESTART Monte Carlo methods large deviations metastable points

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Linear Convergence Rate of splitting algorithms for Multi-Block Constrained Convex Minimizations

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IEEE ACCESS 2020年 8卷 120694-120700页

作者： Deng, Xiaoge Liu, Feng Huang, Feng Natl Univ Def Technol Coll Comp Natl Lab Parallel & Distributed Proc PDL Changsha 410073 Peoples R China

Multi-block linear constrained separable convex minimizations are ubiquitous and have been drawing increasing attention in recent researches. The alternating direction method of multipliers (ADMM) has been well studied and used in the literature for the two-block case. However, the direct extension of the ADMM to the multi-block case is not necessarily convergent. ADMM with Gaussian Back Substitution and ADMM with Prox-Parallel splitting are two useful schemes to deal with the multi-block situation. Nevertheless, only a sublinear convergence rate was given in previous studies. In this paper, we prove the linear convergence rate of these two schemes under some assumptions. The proofs mainly depend on the variational inequalities.

关键词： Multi-block alternating direction method of multipliers linear convergence rate variational inequalities splitting algorithms

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An inertial projective splitting method for the sum of two maximal monotone operators

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COMPUTATIONAL & APPLIED MATHEMATICS 2025年第1期44卷 1-34页

作者： Penton Machado, Majela Univ Fed Bahia Inst Matemat & Estat Campus Ondina Ave Adhemar Barros S-N BR-40170110 Salvador BA Brazil

We propose a projective splitting type method to solve the problem of finding a zero of the sum of two maximal monotone operators. Our method considers inertial and relaxation steps, and also allows inexact solutions of the proximal subproblems within a relative-error criterion. We study the asymptotic convergence of the method, as well as its iteration-complexity. We also discuss how the inexact computations of the proximal subproblems can be carried out when the operators are Lipschitz continuous. In addition, we provide numerical experiments comparing the computational performance of our method with previous (inertial and non-inertial) projective splitting methods.

关键词： splitting algorithms Inertial algorithms Relative error Maximal monotone operators Complexity

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A Projective splitting Method for Monotone Inclusions: Iteration-Complexity and Application to Composite Optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第2期198卷 552-587页

作者： Machado, Majela Penton Sicre, Mauricio Romero Univ Fed Bahia Inst Matemat Campus OndinaAve Adhemar BarrosS-N Ondina BR-40170110 Salvador Ba Brazil

We propose an inexact projective splitting method to solve the problem of finding a zero of a sum of maximal monotone operators. We perform convergence and complexity analyses of the method by viewing it as a special instance of an inexact proximal point method proposed by Solodov and Svaiter in 2001, for which pointwise and ergodic complexity results have been studied recently by Sicre. Also, for this latter method, we establish convergence rates and complexity bounds for strongly monotone inclusions, from where we obtain linear convergence for our projective splitting method under strong monotonicity and cocoercivity assumptions. We apply the proposed projective splitting scheme to composite convex optimization problems and establish pointwise and ergodic function value convergence rates, extending a recent work of Johnstone and Eckstein.

关键词： Monotone inclusion problems Hybrid proximal extragradient methods splitting algorithms Iteration-complexity Convex optimization

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Reliability-Based splitting algorithms for Time-Constrained Distributed Detection in Random-Access WSNs

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2014年第21期62卷 5536-5551页

作者： Laitrakun, Seksan Coyle, Edward J. Georgia Inst Technol Atlanta GA 30309 USA Georgia Inst Technol Sch Elect & Comp Engn Atlanta GA 30332 USA

We consider distributed detection applications for a wireless sensor network (WSN) that has a limited time to collect and process local decisions to produce a global decision. When this time is not sufficient to collect decisions from all nodes in the network, a strategy is needed for collecting those with the highest reliability. This can be accomplished by incorporating a reliability-based splitting algorithm into the random access protocol of the WSN: the collection time is divided into frames and only nodes with a specified range of reliabilities compete for the channel using slotted ALOHA within each frame-nodes with the most reliable decisions attempt transmission in the first frame, nodes with the next most reliable set of decisions attempt in the next frame, etc. The detection error probability (DEP) of the proposed scheme is minimized, and the efficacy is maximized by determining the reliability intervals that define which nodes attempt to transmit in each frame. Intervals that maximize the channel throughput do not always minimize the DEP or maximize the efficacy. Because the scheme orders transmissions of the local decisions in approximately descending order of reliability but suffers collisions, it will offer better performance than a collision-free scheme with no reliability ordering when the time constraint prevents transmission of all local decisions. The transition point between the two schemes is found by deriving the asymptotic relative efficiency (ARE) of the proposed scheme relative to a TDMA-based scheme.

关键词： Binary sensor networks censoring sensors distributed detection ordered transmissions random-access protocols splitting algorithms wireless sensor networks

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A Variational Approach to the Design of Multivariable Discrete-Time Supertwisting-Like algorithms

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2025年第4期70卷 2495-2506页

作者： Miranda-Villatoro, Felix A. Univ Grenoble Alpes CNRS Grenoble INP Lab Jean KuntzmannCtr Inria F-38000 Grenoble France

In this article, a family of discrete-time, supertwisting-like algorithms is presented. The algorithms are naturally vector-valued and are described in an implicit fashion, reminiscent of backward-Euler discretization schemes. The well-posedness of the closed-loop is established and the robust stability, against a family of external disturbances, is thoroughly studied. Implementation strategies, involving splitting-algorithms from convex optimization, are also discussed and compared. Finally, numerical simulations show the performance of the proposed schemes.

关键词： Stability analysis Emulation Robustness Digital computers Thermal stability Numerical stability Context modeling Asymptotic stability Wind energy conversion Trajectory tracking Convex optimization discrete-time systems proximal-point algorithms robustness sliding-mode control splitting algorithms

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A quantum lattice algorithm with fourth-order accuracy for the one-dimensional Dirac equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2025年 533卷

作者： Dellar, Paul J. Univ Oxford Math Inst Radcliffe Observ Quarter Oxford OX2 6GG England

The discrete time quantum walk is a quantum cellular automaton whose wavefunction comprises pairs of complex numbers assigned to uniformly spaced points on a line. The wavefunction evolves through the application of an alternating sequence of unitary operators: streaming of wavefunction values to adjacent points, and a Hadamard-type unitary matrix to blend pairs of values at individual points. Each operator generates the exact evolution due to part of the Hamiltonian for the one-dimensional Dirac equation over a finite time step. Composing these operators thus creates a discrete approximation to the Dirac equation. However, the composition of two non-commuting operators creates a global splitting error proportional to the length of the time step. The global error can be reduced from first order to second order in the time step by a unitary pre-and post-processing of the initial conditions and final output. The algorithm then becomes equivalent to a symmetric composition, a Strang splitting, between the two operators. This paper describes a fourth-order accurate composition scheme using nine stages, the fewest possible when the lengths of the time steps employed in the different stages are constrained to be integer multiples of some base time step. Each stage is itself a symmetric composition between two operators. This fourth-order scheme produces quantitatively smaller errors for a typical benchmark problem on spatial lattices with 1024 or more points, and shows the expected fourth-order convergence on sufficiently fine lattices. It has greater accuracy, over sufficiently long times, than three better-known fourth-order composition schemes using fewer stages, but with lengths related by irrational coefficients. The truncation error for plane-wave solutions is due to an operator that separates into a resonant part proportional to the Hamiltonian, and a non-resonant part orthogonal to the Hamiltonian. The resonant part commutes with the exact evolution operator, so i

关键词： Lattice algorithms splitting algorithms Composition schemes Discrete invariants Resonant interactions

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SOLUTION OF MISMATCHED MONOTONE plus LIPSCHITZ INCLUSION PROBLEMS

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SIAM JOURNAL ON OPTIMIZATION 2024年第4期34卷 3564-3591页

作者： Chouzenoux, Emilie Pesquet, Jean-Christophe Roldan, Fernando Univ Paris Saclay Cent Supelec CVN Inria F-91190 Gif Sur Yvette France Univ Concepcion Dept Ingn Matemat Concepcion Chile

In this article, we study the convergence of algorithms for solving monotone inclusions in the presence of adjoint mismatch. The adjoint mismatch arises when the adjoint of a linear operator is replaced by an approximation, due to computational or physical issues. This occurs in inverse problems, particularly in computed tomography. In real Hilbert spaces, monotone inclusion problems involving a maximally \rho -monotone operator, a cocoercive operator, and a Lipschitzian operator can be solved by the forward-backward-half-forward and the forward-Douglas--Rachford-forward methods. We investigate the case of a mismatched Lipschitzian operator. We propose variants of the two aforementioned methods to cope with the mismatch, and establish conditions under which the weak convergence to a solution is guaranteed for these variants. The proposed algorithms hence enable each iteration to be implemented with a possibly iteration-dependent approximation to the mismatch operator, thus allowing this operator to be modified at each iteration. Finally, we present numerical experiments on a computed tomography example in material science, showing the applicability of our theoretical findings.

关键词： splitting algorithms convergence analysis fixed point theory convex optimization adjoint mismatch

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Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第4期30卷 1485-1502页

作者： Briceno-Arias, Luis Chen, Jinjian Roldan, Fernando Tang, Yuchao Univ Tecn Federico Santa Maria Dept Math Santiago Chile Nanchang Univ Dept Math Nanchang Peoples R China

In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian, and a cocoercive operator. The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forwardbackward-half forward splitting, among other methods. At each iteration, our algorithm needs two computations of the Lipschitzian operator while the cocoercive operator is activated only once. By using product space techniques, we derive a method for solving a composite monotone primaldual inclusions including linear operators and we apply it to solve constrained composite convex optimization problems. Finally, we apply our algorithm to a constrained total variation least-squares problem and we compare its performance with efficient methods in the literature.

关键词： splitting algorithms Monotone operator theory Partial inverse Convex optimization

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