In this paper, we employ Tseng's extragradient method with the self-adaptive stepsize to solve variational inequality problems involving non-Lipschitz continuous and quasimonotone operators in real Hilbert spaces. The convergence of the proposed method is analyzed under some mild assumptions. The key advantages of the method are that it does not require the operator associated with the variational inequality to be Lipschitz continuous and that it adopts the self-adaptive stepsize. Numerical experiments are also provided to illustrate the effectiveness and superiority of the method.

We compare the transport properties of a well-characterized hydrogen plasma for low and high current ion beams. The energy-loss of low current beams can be well understood, within the framework of current stopping power models. However, for high current proton beams, significant energy-loss reduction and collimation is observed in the experiment. We have developed a new particle-in-cell code, which includes both collective electromagnetic effects and collisional interactions. Our simulations indicate that resistive magnetic fields, induced by the transport of an intense proton beam, act to collimate the proton beam and simultaneously deplete the local plasma density along the beam path. This in turn causes the energy-loss reduction detected in the experiment.

This paper is based on Tseng's exgradient algorithm for solving variational inequality problems in real Hilbert spaces. Under the assumptions that the cost operator is quasimonotone and Lipschitz continuous, we establish the strong convergence, sublinear convergence, and Q-linear convergence of the algorithm. The results of this paper provide new insights into quasimonotone variational inequality problems, extending and enriching existing results in the literature. Finally, we conduct numerical experiments to illustrate the effectiveness and implementability of our proposed condition and algorithm.

This paper is devoted to the design of efficient primal-dual algorithm (PDA) for solving convex optimization problems with known saddle-point structure. We present a new PDA with larger acceptable range of parameters and correction, which result in larger step sizes. The step sizes are predicted by using a local information of the linear operator and corrected by linesearch to satisfy a very weak condition, even weaker than the boundedness of sequence generated. The convergence and ergodic convergence rate are established for general cases, and in case when one of the prox-functions is strongly convex. The numerical experiments illustrate the improvements in efficiency from the larger step sizes and acceptable range of parameters.

The energy deposition and the atomic processes, such as the electron-capture, ionization, excitation and radiative-decays for slow heavy ions in plasma remains an unsolved fundamental problem. Here we investigate, both experimentally and theoretically, the stopping of 100 keV=u helium ions in a well-defined hydrogen plasma. Our precise measurements show a much higher energy loss than the predictions of the semi-classical approaches with the commonly used effective charge. By solving the Time Dependent Rate Equation (TDRE) with all the main projectile states and for all relevant atomic processes, our calculations are in remarkable agreement with the experimental data. We also demonstrated that, acting as a bridge for electron-capture and ionization, the projectile excited states and their radiative decays can remarkably influence the equilibrium charge states and consequently lead to a substantial increasing of the stopping of ions in plasma.