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

Mathematical analysis and numerical comparison of energy-conservative schemes for the Zakharov equations

Furihata and Matsuo proposed in 2010 an energy-conserving scheme for the Zakharov equations, as an application of the discrete variational derivative method (DVDM). This scheme is distinguished from conventional methods (in particular the one devised by Glassey in 1992) in that the invariants are consistent with respect to time, but it has not been sufficiently studied both theoretically and numerically. In this study, we theoretically prove the solvability under the loosest possible assumptions. We also prove the convergence of this DVDM scheme by improving the argument by Glassey. Furthermore, we perform intensive numerical experiments for comparing the above two schemes. It is found that the DVDM scheme is superior in terms of accuracy, but since it is fully-implicit, the linearly-implicit Glassey scheme is better for practical efficiency. In addition, we proposed a way to choose a solution for the first step that would allow Glassey's scheme to work more efficiently

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 $T_R$ provided that it lies in the range preferred by the cosmological gravitino problem, $T_R\sim 10^{5-9}$ GeV. Therefore it is strongly desired that they will be put into practice as soon as possible.Comment: 5 page