ABOUT THE SPLITTING ALGORITHM FOR BOLTZMANN AND B.G.K. EQUATIONS

1996 ◽  
Vol 06 (08) ◽  
pp. 1079-1101 ◽  
Author(s):  
L. DESVILLETTES ◽  
S. MISCHLER

We prove the convergence of splitting algorithms for transport and collision operators for Boltzmann and B.G.K. equations.

Author(s):  
Minh N. Dao ◽  
Hung M. Phan

AbstractSplitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.


Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2415
Author(s):  
Jinjian Chen ◽  
Xingyu Luo ◽  
Yuchao Tang ◽  
Qiaoli Dong

This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forward–backward splitting algorithm, and the other is the forward–backward–half-forward splitting algorithm. Both algorithms have a simple calculation framework. In particular, the single-valued operators are processed via explicit steps, while the set-valued operators are computed by their resolvents. Numerical experiments on constrained image denoising problems are presented to show the performance of the proposed algorithms.


Frequenz ◽  
2015 ◽  
Vol 69 (5-6) ◽  
Author(s):  
Xiaodong Ji

AbstractIn this paper, we consider a cognitive radio network scenario, where two primary users want to exchange information with each other and meanwhile, one secondary node wishes to send messages to a cognitive base station. To meet the target quality of service (QoS) of the primary users and raise the communication opportunity of the secondary nodes, a cognitive bidirectional relaying (BDR) scheme is examined. First, system outage probabilities of conventional direct transmission and BDR schemes are presented. Next, a new system parameter called operating region is defined and calculated, which indicates in which position a secondary node can be a cognitive relay to assist the primary users. Then, a cognitive BDR scheme is proposed, giving a transmission protocol along with a time-slot splitting algorithm between the primary and secondary transmissions. Information-theoretic metric of ergodic capacity is studied for the cognitive BDR scheme to evaluate its performance. Simulation results show that with the proposed scheme, the target QoS of the primary users can be guaranteed, while increasing the communication opportunity for the secondary nodes.


2011 ◽  
Vol 115 (12) ◽  
pp. 1610-1622 ◽  
Author(s):  
Junzhou Huang ◽  
Shaoting Zhang ◽  
Hongsheng Li ◽  
Dimitris Metaxas

1979 ◽  
Vol 16 (6) ◽  
pp. 964-979 ◽  
Author(s):  
P. L. Lions ◽  
B. Mercier

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