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1.
In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice. 相似文献
2.
This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis. 相似文献
3.
4.
Mourad Kainane Mezadek Mohamed Kainane Mezadek Michael Reissig 《Mathematical Methods in the Applied Sciences》2020,43(6):3117-3147
In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity. 相似文献
5.
非线性互补问题的一种全局收敛的显式光滑Newton方法 总被引:2,自引:0,他引:2
本针对Po函数非线性互补问题,给出了一种显式光滑Newton方法,该方法将光滑参数μ进行显式迭代而不依赖于Newton方向的搜索过程,并在适当的假设条件下,证明了算法的全局收敛性。 相似文献
6.
We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point. 相似文献
7.
Lu Chuanrong 《数学年刊B辑(英文版)》1997,18(1):71-78
Let $\{\xi_{\bold t}, {\bold t}
\in {\bold Z}^d\}$ be a nonuniform $\varphi$-mixing strictly stationary
real random field with
$E\xi_{\bold 0}=0, E|\xi_{\bold 0}|^{2+\delta}<\infty$ for some
$0<\delta<1$. A sufficient
condition is given for the sequence of partial sum set-indexed process
$\{Z_n(A),\ A\in \Cal A\}$ to converge to Brownian motion. By a direct
calculation, the author shows
that the result holds for a more general class of set index ${\Cal A}$, where
${\Cal A}$ is assumed only to have the metric entropy exponent $r, 0相似文献
8.
杨晓光 《应用数学学报(英文版)》1997,13(4):337-341
1.IntroductionIntillspaperweanalyzetheconvergenceonmultiplicativeiterativealgorithmsfortheIninimizationofadiffcrentiablefunctiondefinedonthepositiveorthantofR".ThealgorithmissllggestedbyEggermolltl'],andisrelatedtotheEM[2](Expextation--Maximization)algoritllnlforPositronemissiontonlography[']andimagereconstructi..14].Wecollsidertheproblenl"linf(x)s.t.x20.Themultiplicativeiterativealgorithmshavethel'orlniforj=l,2,',n,withAhdeterminedthroughalinesearch.Whilelusem[5]establishedanelegantconv… 相似文献
9.
对由一类非线性抛物型变分不等方程所描述的无穷维动力系统,给出了存在全局吸引子及弱近似惯性流形的充分条件. 相似文献
10.
Lisa Lorentzen 《Numerical Algorithms》1995,10(1):69-111
Over the last 20 years a large number of algorithms has been published to improve the speed and domain of convergence of continued fractions. In this survey we show that these algorithms are strongly related. Actually, they essentially boil down to two main principles.We also prove some results on asymptotic expansions of tail values of limit periodic continued fractions.Dedicated to Luigi Gatteschi on his seventieth birthdayThis research was partially supported by The Norwegian Research Council and by the HMC project ROLLS, under contract CHRX-CT93-0416. 相似文献