scholarly journals Implementable tensor methods in unconstrained convex optimization

2019 ◽  
Author(s):  
Yurii Nesterov
Keyword(s):  
Convex Optimization ◽  
Auxiliary Problem ◽  
Computational Cost ◽  
Rates Of Convergence ◽  
Smoothness Condition ◽  
Third Order ◽  
Worst Case ◽  
Complexity Bounds ◽  
The Third ◽  
Tensor Methods

AbstractIn this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level $$O\left( {1 \over k^4}\right) $$O1k4, where k is the number of iterations. This is very close to the lower bound of the order $$O\left( {1 \over k^5}\right) $$O1k5, which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.

scholarly journals Superfast Second-Order Methods for Unconstrained Convex Optimization

2021 ◽  
Vol 191 (1) ◽  
pp. 1-30
Author(s):  
Yurii Nesterov
Keyword(s):  
Auxiliary Problem ◽  
Main Idea ◽  
Second Order ◽  
Third Order ◽  
The Third ◽  
Problem Class ◽  
Degeneracy Condition ◽  
Math Program ◽  
Order Scheme ◽  
Second Order Methods

AbstractIn this paper, we present new second-order methods with convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This is faster than the existing lower bound for this type of schemes (Agarwal and Hazan in Proceedings of the 31st conference on learning theory, PMLR, pp. 774–792, 2018; Arjevani and Shiff in Math Program 178(1–2):327–360, 2019), which is $$O\left( k^{-7/2} \right) $$ O k - 7 / 2 . Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from Nesterov (Math Program 186:157–183, 2021) using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relative non-degeneracy condition (Bauschke et al. in Math Oper Res 42:330–348, 2016; Lu et al. in SIOPT 28(1):333–354, 2018). During this process, the Hessian of the objective function is computed once, and the gradient is computed $$O\left( \ln {1 \over \epsilon }\right) $$ O ln 1 ϵ times, where $$\epsilon $$ ϵ is the desired accuracy of the solution for our problem.

scholarly journals A Third Order Accurate Cellwise Relaxation Implicit Discontinuous Galerkin Scheme for Unstructured Hybrid Meshes

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
Hiroyuki Asada ◽  
Yousuke Ogino ◽  
Kanako Yasue ◽  
Keisuke Sawada
Keyword(s):  
Parabolic Equation ◽  
Discontinuous Galerkin ◽  
Computational Cost ◽  
Convergence Properties ◽  
Third Order ◽  
The Third ◽  
Discontinuous Galerkin Scheme ◽  
Aerospace Applications ◽  
Hybrid Meshes ◽  
Rans Simulations

A third order accurate cellwise relaxation implicit Discontinuous Galerkin (DG) scheme for RANS simulations using unstructured hybrid meshes is presented. A scalar parabolic equation is first examined to clarify what is really important in construction of implicit matrix to keep its diagonal dominance for the third and fourth order cellwise relaxation implicit DG schemes. In addition, discussions are given to approximated construction of implicit matrix for reducing computational cost. Then, the third order accurate cellwise relaxation implicit DG scheme for RANS simulations is successfully developed utilizing the expertise learned in the study of solving the parabolic equation. Superior spatial accuracy of the third order accurate cellwise relaxation implicit DG scheme for RANS simulations, while retaining reasonable convergence properties, is demonstrated for typical aerospace applications.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Simulation of Two Phase Flow Dynamics in Flexible Riser Exit Geometries for Oil and Gas Applications

2018 ◽  
Vol 39 ◽  
pp. 1-13
Author(s):  
Ikpe E. Aniekan ◽  
Owunna Ikechukwu ◽  
Satope Paul
Keyword(s):  
Oil And Gas ◽  
Two Phase Flow ◽  
Flow Analysis ◽  
Computational Cost ◽  
Maximum Velocity ◽  
Oil Well ◽  
Phase Flow ◽  
Two Phase ◽  
The Third ◽  
Riser Pipe

Four different riser pipe exit configurations were modelled and the flow across them analysed using STAR CCM+ CFD codes. The analysis was limited to exit configurations because of the length to diameter ratio of riser pipes and the limitations of CFD codes available. Two phase flow analysis of the flow through each of the exit configurations was attempted. The various parameters required for detailed study of the flow were computed. The maximum velocity within the pipe in a two phase flow were determined to 3.42 m/s for an 8 (eight) inch riser pipe. After thorough analysis of the two phase flow regime in each of the individual exit configurations, the third and the fourth exit configurations were seen to have flow properties that ensures easy flow within the production system as well as ensure lower computational cost. Convergence (Iterations), total pressure, static pressure, velocity and pressure drop were used as criteria matrix for selecting ideal riser exit geometry, and the third exit geometry was adjudged the ideal exit geometry of all the geometries. The flow in the third riser exit configuration was modelled as a two phase flow. From the results of the two phase flow analysis, it was concluded that the third riser configuration be used in industrial applications to ensure free flow of crude oil and gas from the oil well during oil production.

scholarly journals Ultrafast Third-Order Nonlinear Optical Response Excited by fs Laser Pulses at 1550 nm in GaN Crystals

Materials ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3194
Author(s):  
Adrian Petris ◽  
Petronela Gheorghe ◽  
Tudor Braniste ◽  
Ion Tiginyanu
Keyword(s):  
Harmonic Generation ◽  
Nonlinear Optical ◽  
Laser Pulses ◽  
Third Harmonic Generation ◽  
High Repetition Rate ◽  
Third Order ◽  
Third Harmonic ◽  
The Third ◽  
High Repetition ◽  
Gan Crystal

The ultrafast third-order optical nonlinearity of c-plane GaN crystal, excited by ultrashort (fs) high-repetition-rate laser pulses at 1550 nm, wavelength important for optical communications, is investigated for the first time by optical third-harmonic generation in non-phase-matching conditions. As the thermo-optic effect that can arise in the sample by cumulative thermal effects induced by high-repetition-rate laser pulses cannot be responsible for the third-harmonic generation, the ultrafast nonlinear optical effect of solely electronic origin is the only one involved in this process. The third-order nonlinear optical susceptibility of GaN crystal responsible for the third-harmonic generation process, an important indicative parameter for the potential use of this material in ultrafast photonic functionalities, is determined.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

Sign in / Sign up

Export Citation Format

Share Document