The steepest descent algorithm without line search for p-Laplacian

Guang-ming, Zhou; Feng, Chunsheng

doi:10.1016/j.amc.2013.07.096

Cited by 9 publications

(7 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Several works develop and analyze numerical schemes for nonlinear elliptic equations involving the p-Laplacian operator. The works [3,4,25,29,35,40,41] are based on finite element approximations in space. Recently, the vanishing moment method for the p-Laplacian was proposed in [18].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

Feng

Salgado

Wang

et al. 2017

Journal of Computational Physics

View full text Add to dashboard Cite

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixthorder nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in generic Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general the theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems -including thin film epitaxy with slope selection and the square phase field crystal model -are carried out to verify the efficiency of the scheme.

show abstract

Section: Introductionmentioning

confidence: 99%

“…A hybridizable discontinuous Galerkin method for the p-Laplacian was proposed in [11]. Of these works, [25,40,41] are primarily focused on efficient solvers for the elliptic equations with p-Laplacian terms, rather than, say, error estimates.…”

Section: Introductionmentioning

confidence: 99%

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

Feng

Salgado

Wang

et al. 2017

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…For more information on the line search strategy, readers can refer to [1][2][3][4][5][6][7]. In addition, some researchers have introduced suitable algorithms which defined α k without a line search method that can be seen in the literature [8][9][10]. Appropriate choices for an initial starting point have a positive effect on computational cost and speed of convergence.…”

Section: Introductionmentioning

confidence: 99%

“…The overall results on the convergence of the steepest descent method can be found in [4,12,13]. More practical applications of the steepest descent method are discussed in [10,[14][15][16].…”

Section: Introductionmentioning

confidence: 99%

A New Descent Algorithm Using the Three-Step Discretization Method for Solving Unconstrained Optimization Problems

Torabi

Hosseini

2018

Mathematics

View full text Add to dashboard Cite

In this paper, three-step Taylor expansion, which is equivalent to third-order Taylor expansion, is used as a mathematical base of the new descent method. At each iteration of this method, three steps are performed. Each step has a similar structure to the steepest descent method, except that the generalized search direction, step length, and next iterative point are applied. Compared with the steepest descent method, it is shown that the proposed algorithm has higher convergence speed and lower computational cost and storage.

show abstract

“…Shi, Shen and Zhou proposed descent methods without line search in [9] and [10], respectively. Further, Zhou presented the steepest descent algorithm without line search in [11].…”

Section: Introductionmentioning

confidence: 99%

Global Convergence of an Extended Descent Algorithm without Line Search for Unconstrained Optimization

Chen¹,

Han²,

Yu³

2018

JAMP

View full text Add to dashboard Cite

show abstract

The steepest descent algorithm without line search for p-Laplacian

Cited by 9 publications

References 26 publications

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

A New Descent Algorithm Using the Three-Step Discretization Method for Solving Unconstrained Optimization Problems

Global Convergence of an Extended Descent Algorithm without Line Search for Unconstrained Optimization

Contact Info

Product

Resources

About