20,408 research outputs found
On the Optimal Linear Convergence Rate of a Generalized Proximal Point Algorithm
The proximal point algorithm (PPA) has been well studied in the literature.
In particular, its linear convergence rate has been studied by Rockafellar in
1976 under certain condition. We consider a generalized PPA in the generic
setting of finding a zero point of a maximal monotone operator, and show that
the condition proposed by Rockafellar can also sufficiently ensure the linear
convergence rate for this generalized PPA. Indeed we show that these linear
convergence rates are optimal. Both the exact and inexact versions of this
generalized PPA are discussed. The motivation to consider this generalized PPA
is that it includes as special cases the relaxed versions of some splitting
methods that are originated from PPA. Thus, linear convergence results of this
generalized PPA can be used to better understand the convergence of some widely
used algorithms in the literature. We focus on the particular convex
minimization context and specify Rockafellar's condition to see how to ensure
the linear convergence rate for some efficient numerical schemes, including the
classical augmented Lagrangian method proposed by Hensen and Powell in 1969 and
its relaxed version, the original alternating direction method of multipliers
(ADMM) by Glowinski and Marrocco in 1975 and its relaxed version (i.e., the
generalized ADMM by Eckstein and Bertsekas in 1992). Some refined conditions
weaker than existing ones are proposed in these particular contexts.Comment: 22 pages, 1 figur
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
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