746 research outputs found
New Step-Size Criterion for the Steepest Descent based on Geometric Numerical Integration
This paper deals with unconstrained optimization problems based on numerical
analysis of ordinary differential equations (ODEs). Although it has been known
for a long time that there is a relation between optimization methods and
discretization of ODEs, research in this direction has recently been gaining
attention. In recent studies, the dissipation laws of ODEs have often played an
important role. By contrast, in the context of numerical analysis, a technique
called geometric numerical integration, which explores discretization to
maintain geometrical properties such as the dissipation law, is actively
studied. However, in research investigating the relationship between
optimization and ODEs, techniques of geometric numerical integration have not
been sufficiently investigated. In this paper, we show that a recent geometric
numerical integration technique for gradient flow reads a new step-size
criterion for the steepest descent method. Consequently, owing to the discrete
dissipation law, convergence rates can be proved in a form similar to the
discussion in ODEs. Although the proposed method is a variant of the existing
steepest descent method, it is suggested that various analyses of the
optimization methods via ODEs can be performed in the same way after
discretization using geometric numerical integration
Crystal Structure and Pseudopolymorphism of Bisdemethoxycurcumin-Alcohol Solvates
 Crystal structures of pseudopolymorphic (1E,6E)-1,7-bis (4-hydroxyphenyl) -1,6 heptadiene-3,5-dione (bisdemethoxycurcumin, BDMC) methanol solvate BDMC・CH3OH (1) and 2-propanol solvate BDMC・(CH3) 2CHOH (2) were determined
A new unified framework for designing convex optimization methods with prescribed theoretical convergence estimates: A numerical analysis approach
We propose a new unified framework for describing and designing
gradient-based convex optimization methods from a numerical analysis
perspective. There the key is the new concept of weak discrete gradients (weak
DGs), which is a generalization of DGs standard in numerical analysis. Via weak
DG, we consider abstract optimization methods, and prove unified convergence
rate estimates that hold independent of the choice of weak DGs except for some
constants in the final estimate. With some choices of weak DGs, we can
reproduce many popular existing methods, such as the steepest descent and
Nesterov's accelerated gradient method, and also some recent variants from
numerical analysis community. By considering new weak DGs, we can easily
explore new theoretically-guaranteed optimization methods; we show some
examples. We believe this work is the first attempt to fully integrate research
branches in optimization and numerical analysis areas, so far independently
developed
Space laser interferometers can determine the thermal history of the early Universe
It is shown that space-based gravitational wave detectors such as DECIGO
and/or Big Bang Observer (BBO) will provide us with invaluable information on
the cosmic thermal history after inflation and they will be able to determine
the reheat temperature provided that it lies in the range preferred by
the cosmological gravitino problem, GeV. Therefore it is
strongly desired that they will be put into practice as soon as possible.Comment: 5 page
Screening for Narrow Angles in the Japanese Population Using Scanning Peripheral Anterior Chamber Depth Analyzer
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