Renormalization group approach to 1D cellular automata with large updating neighborhoods

We study self‐similarity in one‐dimensional probabilistic cellular automata (PCA) by applying a real‐space renormalization technique to PCA with increasingly large updating neighborhoods. By studying the flow about the critical point of the renormalization, we may produce estimates of the spatial scaling properties of critical PCA. We find that agreement between our estimates and experimental values are improved by resolving correlations between larger blocks of spins, although this is not sufficient to converge to experimental values. However, applying the technique to PCA with larger neighborhoods, and, therefore, more renormalization parameters, results in further improvement. Our most refined estimate produces a spatial scaling exponent, found at the critical point of the five‐neighbor PCA, of ν = 1.056 which should be compared to the experimental value of ν = 1.097. © 2014 Wiley Periodicals, Inc. Complexity 21: 206–213, 2015     

Directed-bond percolation subjected to synthetic compressible velocity fluctuations: Renormalization group approach

We study the directed-bond percolation process (sometimes called the Gribov process because it formally resembles Reggeon field theory) in the presence of irrotational velocity fluctuations with long-range correlations. We use the renormalization group method to investigate the phase transition between an active and an absorbing state. All calculations are in the one-loop approximation. We calculate stable fixed points of the renormalization group and their regions of stability in the form of expansions in three parameters (ε, y, η). We consider different regimes corresponding to the Kraichnan rapid-change model and a frozen velocity field.     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.     

Renormalization group approach and short-distance expansion in theory of developed turbulence: Asymptotics of the triplex equal-time correlation function

Asymptotics of the triplex equal-time correlation function for the turbulence developed in incompressible fluids in the region of widely separated wave vector values is investigated using the renormalization group approach and short-distance expansion. The problem of the most essential composite operators contributing to these asymptotics is examined. For this purpose, the critical dimensions of a family of composite quadratic tensor operators in the velocity gradient are found. Considered in the one-loop approximation, the contribution of these operators turns out to be less substantial (although not significantly) than the contribution of the linear term. The derived asymptotics of the triplex correlator coincide in form with that predicted by the EDQNM approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 450–461, December, 1995.     

Renormalization group method applied to the primitive equations

In this article we study the limit, as the Rossby number ε goes to zero, of the primitive equations of the atmosphere and the ocean. From the mathematical viewpoint we study the averaging of a penalization problem displaying oscillations generated by an antisymmetric operator and by the presence of two time scales.     

Renormalization group method and Julia sets

The renormalization group (RG) method has been used successfully in treating a variety of phase change and critical-point problems (Wilson KG, Kogut J. Phys Rev C 1974;12:75; Wilson KG. Rev Mod Phys 1975;773; Wilson KG. Phys Rev B 1971;3174). A relatively simple system is considered at the smallest scale; the problem is then renormalized in order to utilize the same system at next larger scale. The process is repeated at larger and larger scales. In the following we consider a model for the flow of a fluid through a porous-medium. The RG transformations for the flow of a fluid through a porous-medium in two and three dimensions are derived and generalized to the complex plane, and the types of the corresponding Julia sets are found and generated. Also, the RG transformation for Ising model on a square lattice is derived and the corresponding Julia set is found.     

Real pole approximations to the exponential function

Rational approximations to the exponential function with real, not necessarily distinct poles are studied in this paper. The orthogonality relation is established in order to show that the zeros of the collocation polynomial of the corresponding Runge-Kutta method are all real, simple and positive. It is proven, that approximants with the smallest error constant are the Restricted Padé approximants of Nørsett. Some results concerning acceptability properties are given.This work was supported by RSS, Ljubljana while the author was at Division of Mathematical Sciences, Norwegian Institute of Technology, Trondheim.     

A finite element recovery approach to Green’s function approximations with applications to electrostatic potential computation

In this paper, a finite element recovery approach is proposed to improve the accuracy of finite element approximations for Green’s functions in three dimensions. This recovery approach is based on some simple postprocessing. It is proved by both theory and numerics that the recovery approach is very efficient. In particular, the approach is successfully applied to some electrostatic potential computations.     

A new approach to spectral approximation

A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence.     

A piecewise approach to piecewise approximation

Tensor-product B-spline surfaces offer a convenient means for representing a set of bivariate data, especially if many surface evaluations are required. This is because the compact support property of the tensor-product spline allows the spline value to be obtained in a time that is (almost) independent of the number of coefficients used to define the surface. The main calculation is the precomputation involved in fitting the data and this can be impractically large if there are many spline coefficients to be calculated. Since the surface produced may be evaluated locally and efficiently, it would be advantageous to exploit local properties in order to fit the data in a piecewise manner. An algorithm to do this is presented.     

Star-shaped approximation approach for stochastic programming problems with probability function

Star-shaped probability function approximation is suggested. Conditions of log-concavity and differentiability of approximation function are obtained. The method for constructing stochastic estimates of approximation function gradient and stochastic quasi-gradient algorithm for probability function maximization are described in the paper     

Tail approximations to the density function in EVT

Let X ₁, X ₂, ...,X _n be independent identically distributed random variables with common distribution function F, which is in the max domain of attraction of an extreme value distribution, i.e., there exist sequences a _n > 0 and b _n ∈ ℝ such that the limit of exists. Assume the density function f (of F) exists. We obtain an uniformly weighted approximation to the tail density function f, and an uniformly weighted approximation to the tail density function of under some second order condition.Partially supported by a grant of the Swiss National Science Foundation.     

A new approximation theory which unifies spherical and Cohen-Macaulay approximations

This paper gives a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies two famous approximation theorems; one is due to Auslander and Bridger and the other is due to Auslander and Buchweitz. Modules admitting such approximations shall be studied.     

Renormalization group and asymptotics of solutions of nonlinear parabolic equations

We present a general method for studying long-time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusion-type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity. In an accompanying paper, [5], the method is applied to systems of equations where some variables are “slaved,” such as the complex Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc.