A Two-Stage Step Size Rule on Improved Prediction-Correction Method for Monotone Nonlinear Variational Inequalities

2009 ◽  
Author(s):  
Hu Shao ◽  
Guodong Wang
Keyword(s):  
Variational Inequalities ◽  
Correction Method ◽  
Step Size ◽  
Two Stage

Convergence of Two-Stage Method with Bregman Divergence for Solving Variational Inequalities*

2019 ◽  
Vol 55 (3) ◽  
pp. 359-368 ◽  
Author(s):  
D. A. Nomirovskii ◽  
B. V. Rublyov ◽  
V. V. Semenov
Keyword(s):  
Variational Inequalities ◽  
Bregman Divergence ◽  
Two Stage

Convergence analysis of online learning algorithm with two-stage step size

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Weilin Nie ◽  
Cheng Wang
Keyword(s):  
Online Learning ◽  
Optimization Problems ◽  
Learning Algorithm ◽  
Computational Cost ◽  
Reproducing Kernels ◽  
Step Size ◽  
Two Stage ◽  
Logarithmic Term ◽  
Convergence Performance ◽  
Online Learning Algorithm

Abstract Online learning is a classical algorithm for optimization problems. Due to its low computational cost, it has been widely used in many aspects of machine learning and statistical learning. Its convergence performance depends heavily on the step size. In this paper, a two-stage step size is proposed for the unregularized online learning algorithm, based on reproducing Kernels. Theoretically, we prove that, such an algorithm can achieve a nearly min–max convergence rate, up to some logarithmic term, without any capacity condition.

scholarly journals A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2284
Author(s):  
Endre Kovács ◽  
Ádám Nagy ◽  
Mahmoud Saleh
Keyword(s):  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Second Order ◽  
Stiff Systems ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Two Stage ◽  
Time Step Size ◽  
New Methods

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

scholarly journals Converting Climate Change Gridded Daily Rainfall to Station Daily Rainfall—A Case Study at Zengwen Reservoir

Water ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 1516
Author(s):  
Tse-Yu Teng ◽  
Tzu-Ming Liu ◽  
Yu-Shiang Tung ◽  
Ke-Sheng Cheng
Keyword(s):  
Climate Change ◽  
Water Resources ◽  
Bias Correction ◽  
Climate Change Impact ◽  
Daily Rainfall ◽  
Correction Method ◽  
Two Stage ◽  
Climate Change Impact Assessment ◽  
Daily Data ◽  
Change Impact

With improvements in data quality and technology, the statistical downscaling data of General Circulation Models (GCMs) for climate change impact assessment have been refined from monthly data to daily data, which has greatly promoted the data application level. However, there are differences between GCM downscaling daily data and rainfall station data. If GCM data are directly used for hydrology and water resources assessment, the differences in total amount and rainfall intensity will be revealed and may affect the estimates of the total amount of water resources and water supply capacity. This research proposes a two-stage bias correction method for GCM data and establishes a mechanism for converting grid data to station data. Five GCMs were selected from 33 GCMs, which were ranked by rainfall simulation performance from a baseline period in Taiwan. The watershed of the Zengwen Reservoir in southern Taiwan was selected as the study area for comparison of the three different bias correction methods. The results reveal that the method with the wet-day threshold optimized by objective function with observation rainfall wet days had the best result. Error was greatly reduced in the hydrology model simulation with two-stage bias correction. The results show that the two-stage bias correction method proposed in this study can be used as an advanced method of data pre-processing in climate change impact assessment, which could improve the quality and broaden the extent of GCM daily data. Additionally, GCM ranking can be used by researchers in climate change assessment to understand the suitability of each GCM in Taiwan.

scholarly journals A New Implementable Prediction-Correction Method for Monotone Variational Inequalities with Separable Structure

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Feng Ma ◽  
Mingfang Ni ◽  
Zhanke Yu
Keyword(s):  
Global Convergence ◽  
Variational Inequalities ◽  
Decomposition Method ◽  
Correction Method ◽  
Computational Effort ◽  
Numerical Results ◽  
New Method ◽  
Monotone Variational Inequalities ◽  
Separable Structure

The monotone variational inequalities capture various concrete applications arising in many areas. In this paper, we develop a new prediction-correction method for monotone variational inequalities with separable structure. The new method can be easily implementable, and the main computational effort in each iteration of the method is to evaluate the proximal mappings of the involved operators. At each iteration, the algorithm also allows the involved subvariational inequalities to be solved in parallel. We establish the global convergence of the proposed method. Preliminary numerical results show that the new method can be competitive with Chen's proximal-based decomposition method in Chen and Teboulle (1994).

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