Technical Note—On Nested Partitions Method for Global Optimization

2020 ◽  
Author(s):  
Tao Wu
Keyword(s):  
Global Optimization ◽  
Finite Time ◽  
Technical Note ◽  
Time Behavior ◽  
Nested Partitions

Finite-Time Behavior of Nested Partitions Method for Global Optimization

Well-posedness and blow-up properties for the generalized Hartree equation

2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Hartree Equation ◽  
Energy Threshold ◽  
Local Theory ◽  
Positive Energy ◽  
Small Data ◽  
Well Posedness ◽  
Time Behavior ◽  
Mass Energy

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

Nested Partitions Method for Global Optimization

2000 ◽  
Vol 48 (3) ◽  
pp. 390-407 ◽  
Author(s):  
Leyuan Shi ◽  
Sigurdur Ólafsson
Keyword(s):  
Global Optimization ◽  
Nested Partitions

A Note on the Finite Time Behavior of Simulated Annealing

2000 ◽  
Vol 25 (3) ◽  
pp. 476-484 ◽  
Author(s):  
Andreas Nolte ◽  
Rainer Schrader
Keyword(s):  
Simulated Annealing ◽  
Finite Time ◽  
Time Behavior

Convergence and finite-time behavior of simulated annealing

1986 ◽  
Vol 18 (03) ◽  
pp. 747-771 ◽  
Author(s):  
Debasis Mitra ◽  
Fabio Romeo ◽  
Alberto Sangiovanni-Vincentelli
Keyword(s):  
Markov Chain ◽  
Simulated Annealing ◽  
Finite Time ◽  
Randomized Algorithm ◽  
Global Optimum ◽  
Time Behavior ◽  
Annealing Schedule ◽  
Complete Problems ◽  
Inhomogeneous Markov Chain ◽  
Np Complete

Simulated annealing is a randomized algorithm which has been proposed for finding globally optimum least-cost configurations in large NP-complete problems with cost functions which may have many local minima. A theoretical analysis of simulated annealing based on its precise model, a time-inhomogeneous Markov chain, is presented. An annealing schedule is given for which the Markov chain is strongly ergodic and the algorithm converges to a global optimum. The finite-time behavior of simulated annealing is also analyzed and a bound obtained on the departure of the probability distribution of the state at finite time from the optimum. This bound gives an estimate of the rate of convergence and insights into the conditions on the annealing schedule which gives optimum performance.

scholarly journals Finite-time behavior of inner systems

2003 ◽  
Vol 48 (7) ◽  
pp. 1134-1149 ◽  
Author(s):  
J.H.A. Ludlage ◽  
S. Weiland ◽  
A.A. Stoorvogel ◽  
T.A.C.P.M. Backx
Keyword(s):  
Finite Time ◽  
Time Behavior

scholarly journals Technical Note: A simple generalization of the Brutsaert and Nieber analysis

2014 ◽  
Vol 11 (11) ◽  
pp. 12519-12530
Author(s):  
T. L. Chor ◽  
N. L. Dias
Keyword(s):  
Mathematical Formulation ◽  
Early Time ◽  
Technical Note ◽  
Soil Parameters ◽  
Late Time ◽  
Time Behavior ◽  
Water Stream ◽  
Discharge Data ◽  
Simple Generalization ◽  
Predicted Values

Abstract. The Brutsaert and Nieber (1977) analysis is a well known method that can estimate soil parameters given discharge data for some aquifers. It has been used for several cases where the observed late-time behavior of the recession suggests that the water stream that is adjacent to the aquifer has non-zero depth. However, its mathematical formulation is, strictly speaking, not capable of reproducing these real-case scenarios since the early time behavior is based on a solution for which the aquifer stream has zero depth (Polubarinova-Kochina, 1962). We propose a simple generalization for the Brutsaert and Nieber (1977) method that can estimate soil parameters for aquifers discharging into a water stream of finite non-zero depth. The generalization is based on already available solutions by Polubarinova-Kochina (1962), Chor et al. (2013) and Dias et al. (2014) and can be readily implemented with little effort. A sensitivity analysis shows that the modification can have significant impact on the predicted values of the drainable porosity.

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