Search

The second-order mean field games system (MFGS) in a bounded domain with the lateral Cauchy data are considered. This means that both Dirichlet and Neumann boundary data for the solution of the MFGS are given. Two Hölder stability estimates for two slightly different cases are derived. These estimates indicate how stable the solution of the MFGS is with respect to the possible noise in the lateral Cauchy data. Our stability estimates imply uniqueness. The key mathematical apparatus is the apparatus of two new Carleman estimates. [ABSTRACT FROM AUTHOR]

Based on a hierarchical basis a posteriori error estimator, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the Stokes problem in two and three dimensions. In this paper, we propose two novel diagonalization techniques for velocity and pressure, respectively. Using diagonalization techniques, we need only to solve two diagonal linear algebraic systems corresponding to the degree of freedom to get the error estimator. The upper bound and lower bound of the error estimator are also shown to address the reliability of the adaptive method. Numerical simulations are provided to demonstrate the effectiveness and robustness of our algorithm. [ABSTRACT FROM AUTHOR]

We consider the optimal control problems constrained by Stokes equations. It has been shown in the literature, the problem can be discretized by the finite element method to generate a discrete system, and the error estimate has also been established. In this paper, we focus on solving the discrete system by the alternating splitting augmented Lagrangian method, which is a direct extension of alternating direction method of multipliers and possesses a global O ⁢ (1 / k) \mathcal{O}({1}/{k}) convergence rate. In addition, we propose an acceleration scheme based on the alternating splitting augmented Lagrangian method to improve the efficiency of the algorithm. The error estimates and convergence analysis of our algorithms are presented for several different types of optimization problems. Finally, numerical experiments are performed to verify the efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

To understand the modifications of lignin in the steam explosion process (209°C, 7 min), lignin samples in native poplar (LP), steam explosion solid residue (LS) and steam explosion liquid (LL) were separated and studied. The lignin samples (LP, LS and LL) were characterized and analyzed by size exclusion chromatography (SEC), Fourier transform infrared spectroscopy (FTIR), and nuclear magnetic resonance [carbon 13 numclear magnetic resonance (13C-NMR) and heteronuclear single quantum coherence or heteronuclear single quantum correlation experiment NMR (HSQC) NMR] analysis. The results revealed that the pretreatment induced reductions in amounts of β-O-4′, β-β′, and spirodienone structure, and increases in the syringyl/guaiacyl (S/G) ratio from 1.14 (LP) to 1.70 (LS) and 1.86 (LL). The HSQC NMR spectra also gave more information about the predominance of G and S units, and only small amounts of p-hydroxyphenyl (H) units. Moreover, SEC demonstrated the depolymerization and repolymerization of lignin, which were the main reasons for the increase in the average molecular weight of LS and the decrease in average molecular weight of LL, respectively. In brief, after steam explosion treatment, the lignin structure changed, but the backbone structure was not noticeably modified. [ABSTRACT FROM AUTHOR]