Applied Mathematics Research Express 

About: Applied Mathematics Research Express is an academic journal. The journal publishes majorly in the area(s): Nonlinear system & Bounded function. It has an ISSN identifier of 1687-1197. Over the lifetime, 133 publications have been published receiving 2756 citations.

A traffic model for velocity data assimilation

Abstract: This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data

Rational Construction of Stochastic Numerical Methods for Molecular Sampling

Abstract: In this article, we focus on the sampling of the configurational Gibbs-Boltzmann distribution, that is, the calculation of averages of functions of the position coordinates of a molecular N -body system modelled at constant temperature. We show how a formal series expansion of the invariant measure of a Langevin dynamics numerical method can be obtained in a straightforward way using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property (4th order accuracy where only 2nd order would be expected) of one method in the high friction limit; this method, moreover, can be reduced to a simple modification of the Euler-Maruyama method for Brownian dynamics involving a non-Markovian (coloured noise) random process. In the Brownian dynamics case, 2nd order accuracy of the invariant density is achieved. All methods considered are efficient for molecular applications (requiring one force evaluation per timestep) and of a simple form. In fully resolved (long run) molecular dynamics simulations, for our favoured method, we observe up to two orders of magnitude improvement in configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared to common alternative methods. keywords: molecular dynamics; sampling; Langevin dynamics; Brownian dynamics; stochastic dynamics; diffusion equation. ∗School of Mathematics and Maxwell Institute of Mathematical Sciences, James Clerk Maxwell Building, Kings Buildings, University of Edinburgh, Edinburgh, EH9 3JZ, UK

Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation

Abstract: driving path x(t) is nondifferentiable, has recently been developed by Lyons. I develop an alternative approach to this theory, using (modified) Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. I also give some other examples showing that the main results are reasonably sharp.

Comparative Analysis of a Novel M-TOPSIS Method and TOPSIS

Abstract: In this study, we introduce a novel modified synthetic evaluation method (M-TOPSIS) based on the concept of original TOPSIS and calculate the distance between the alternatives and ‘optimized ideal reference point’ in the D D-plane. It could avoid rank reversals and solve the problem on evaluation failure that often occurs in original TOPSIS, so we believe that the mechanism of M-TOPSIS is more reasonable. Furthermore, the MTOPSIS method is simple in both concept and calculation procedures.

On Blow-up Solutions to the 3D Cubic Nonlinear Schrödinger Equation

Abstract: For the 3d cubic nonlinear Schrodinger (NLS) equation, which has critical (scaling) norms L 3 and u H 1/2 , we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time blow-up. For the rest of the paper, we focus on the study of finite-time radial blow-up solutions, and prove a result on the concentration of the L 3 norm at the origin. Two disparate possibilities emerge, one which coincides with solutions typically observed in numer- ical experiments that consist of a specific bump profile with maximum at the origin and focus toward the origin at rate ∼ (T − t) 1/2 , where T > 0 is the blow-up time. For the other possibility, we propose the existence of "contracting sphere blow-up solutions", i.e. those that concentrate on a sphere of radius ∼ (T −t) 1/3 , but focus towards this sphere at a faster rate ∼ (T − t) 2/3 . These conjectured solutions are analyzed through heuristic arguments and shown (at this level of precision) to be consistent with all conservation laws of the equation.