A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity

The Petviashvili's iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: \ $-Mu+u^p=0$, where $M$ is a positive definite self-adjoint operator and $p={\rm const}$. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.Comment: to appear in J. Comp. Phys.; 35 page

Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems

Two approximation algorithms for solving convex vector optimization problems (CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual problem simultaneously. The first algorithm is an extension of Benson's outer approximation algorithm, and the second one is a dual variant of it. Both algorithms provide an inner as well as an outer approximation of the (upper and lower) images. Only one scalar convex program has to be solved in each iteration. We allow objective and constraint functions that are not necessarily differentiable, allow solid pointed polyhedral ordering cones, and relate the approximations to an appropriate \epsilon-solution concept. Numerical examples are provided

Formalising the Foundations of Discrete Reinforcement Learning in Isabelle/HOL

We present a formalisation of finite Markov decision processes with rewards in the Isabelle theorem prover. We focus on the foundations required for dynamic programming and the use of reinforcement learning agents over such processes. In particular, we derive the Bellman equation from first principles (in both scalar and vector form), derive a vector calculation that produces the expected value of any policy p, and go on to prove the existence of a universally optimal policy where there is a discounting factor less than one. Lastly, we prove that the value iteration and the policy iteration algorithms work in finite time, producing an epsilon-optimal and a fully optimal policy respectively

A scalar imaging velocimetry technique for fully resolved four‐dimensional vector velocity field measurements in turbulent flows

This paper presents an experimental technique for obtaining fully resolved measurements of the vector velocity field u(x,t) throughout a four‐dimensional spatiotemporal region in a turbulent flow. The method uses fully resolved four‐dimensional scalar field imaging measurements in turbulent flows [Phys. Fluids A 3, 1115 (1991)] to extract the underlying velocity field from the exact conserved scalar transport equation. A procedure for accomplishing this is described, and results from a series of test cases are presented. These involve synthetically generated scalar fields as well as actual measured turbulent flow scalar fields advected numerically by various imposed flow fields. The imposed velocity fields are exactly known, allowing a careful validation of the technique and its potential accuracy. Results obtained from a zeroth iteration of the technique are found to be very close to the exact underlying vector velocity field. Further results show that successive iterations bring the velocity field from the zeroth iteration even closer to the exact result. It is also shown that the comparatively dense velocity field information that this technique provides is well suited for accurate extraction of the more dynamically insightful strain rate and vorticity fields ϵ(x,t) and ω(x,t).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69930/2/PFADEB-4-10-2191-1.pd

The Third Order Scalar Induced Gravitational Waves

Since the gravitational waves were detected by LIGO and Virgo, it has been promising that lots of information about the primordial Universe could be learned by further observations on stochastic gravitational waves background. The studies on gravitational waves induced by primordial curvature perturbations are of great interest. The aim of this paper is to investigate the third order induced gravitational waves. Based on the theory of cosmological perturbations, the first order scalar induces the second order scalar, vector and tensor perturbations. At the next iteration, the first order scalar, the second order scalar, vector and tensor perturbations all induce the third order tensor perturbations. We present the energy density spectrum of the third order gravitational waves for a monochromatic primordial power spectrum. The shape of the energy density spectrum of the third order gravitational waves is different from that of the second order scalar induced gravitational waves. And it is found that the third order gravitational waves sourced by the second order scalar perturbations dominate the energy density spectrum.Comment: 33 pages, 4 figure

Phase synchronization in discrete chaotic systems

A simple and instantaneous phase definition is proposed for the study of discrete maps by taking the change of chaotic signal at each iteration time as a vector. With such a definition, an exact phase can be calculated at any iteration time for any scalar signal or two-dimensional vector of interest. As examples, the phase synchronization behavior is discussed for a two-dimensional globally coupled map lattice and a one-way coupled map lattice

On the regularity of minima of non-autonomous functionals

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-energy conditions are satisfied. These are related to the occurrence of the so-called Lavrentiev phenomenon that that non-autonomous functionals might exhibit, and which is a natural obstruction to regularity. In the case of vector valued problems we concentrate on higher gradient integrability of minima. Instead, in the scalar case, we prove local Lipschitz estimates. We also present an approach via a variant of Moser's iteration technique that allows to reduce the analysis of several non-uniformly elliptic problems to that for uniformly elliptic ones.Comment: 32 page