Variational approximations for renormalization group transformations

Approximate recursion relations which give upper and lower bounds on the free energy are described. Optimal calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, a particularly simple lower bound approximation which preserves the symmetry of the Hamiltonian (the one-hypercube approximation) is described. The approximation is applied to both the Ising model and the Wilson-Fisher model. At the fixed point a parameter is set variationally and critical indices are calculated. For the Ising model the agreement with the exact results atd = 2 is surprisingly good, 0.1%, and is good atd=3 and evend=4. For the Wilson-Fisher model the recursion relation is reduced to a one-dimensional integral equation which can be solved numerically givingv=0.652 atd=3, or by expansion in agreement with the results of Wilson and Fisher to leading order in . The method is also used to calculate thermodynamic functions for thed = 2 Ising model; excellent agreement with the Onsager solution is found.Supported in part by the National Science Foundation under Grants Nos. MPS73-04886A01 and GH-41512 and by the Brown University Materials Research Laboratory supported by the National Science Foundation. M.C.Y. was supported by a grant from the Scientific and Technical Research Council of Turkey.     

Linearization of the interaction principle: Analytic Jacobians in the “Radiant” model

In this paper we present a new linearization of the Radiant radiative transfer model. Radiant uses discrete ordinates for solving the radiative transfer equation in a multiply-scattering anisotropic medium with solar and thermal sources, but employs the adding method (interaction principle) for the stacking of reflection and transmission matrices in a multilayer atmosphere. For the linearization, we show that the entire radiation field is analytically differentiable with respect to any surface or atmospheric parameter for which we require Jacobians (derivatives of the radiance field). Derivatives of the discrete ordinate solutions are based on existing methods developed for the LIDORT radiative transfer models. Linearization of the interaction principle is completely new and constitutes the major theme of the paper. We discuss the application of the Radiant model and its linearization in the Level 2 algorithm for the retrieval of columns of carbon dioxide as the main target of the Orbiting Carbon Observatory (OCO) mission.     

Analysis of the radiative transfer equation with highly assymetric phase function

This paper considers a scalar radiative transfer problem with high scattering anisotropy. Two computational methods are presented based on decomposition of the diffuse light field into a regular and anisotropic part. The first algorithm (DOMAS) singles out the anisotropic radiance in the forward scattering peak using the Small-Angle Modification of RTE. The second algorithm (DOM2+) separates the single scattering radiance as an anisotropic part, which largely defines the fine detail of the total radiance in the backscattering directions. In both cases, the anisotropic part is represented analytically. With anisotropy subtraction, the regular part of the signal, which requires a numerical solution, is essentially smoothed as a function of angles. Further, the transport equation is obtained for the regular part that contains an additional source function from the anisotropic part of the signal. This equation is solved with the discrete ordinates method. A conducted numerical analysis of this work showed that algorithm DOMAS has a strong advantage as compared to the standard discrete ordinates method for simulation of the radiance transmission, and DOM2+ is the best of the three for the reflection computations. Both algorithms offer at least a factor of three acceleration of convergence of the azimuthal series for highly anisotropic phase functions.     

Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation

New numerical techniques are presented for the solution of a two-dimensional anomalous sub-diffusion equation with time fractional derivative. In these methods, standard central difference approximation is used for the spatial discretization, and, for the time stepping, two new alternating direction implicit (ADI) schemes based on the L₁ approximation and backward Euler method are considered. The two ADI schemes are constructed by adding two different small terms, which are different from standard ADI methods. The solvability, unconditional stability and H¹ norm convergence are proved. Numerical results are presented to support our theoretical analysis and indicate the efficiency of both methods.     

Long range effect of ion irradiation on diffusion

Argon ion irradiation enhanced diffusion between Cu and Ni was found beyond the radiation effect zone in classical theory. The enhanced diffusion effect attenuates along with the distance between the diffusion zone and the radiated surface. An interpretation based on irradiation induced discrete breathers was employed as a possible mechanism.     

Statistical mechanics of a dynamical system based on Conway's game of Life

We study a discrete dynamical system whose evolution is governed by rules similar to those of Conway's game of Life but also include a stochastic element (parametrized by a temperature). Statistical properties that are examined are density as a function of temperature and entropy (suitably defined). A phase transition and a certain thermodynamic constant of the motion are observed.Lady Davis Visiting Scientist at the Technion 1974–75.     

Design and optimal tuning of nonlinear PI compensators

In this paper, linear time-invariant single-input single-output (SISO) systems that are stabilizable by linear proportional and integral (PI) compensators are considered. For such systems, a five-parameter nonlinear PI compensator is proposed. The parameters of the proposed compensator are tuned by solving an optimization problem. The optimization problem always has a solution.Additionally, a general nonlinear PI compensator is proposed and is approximated by easy-to-compute compensators, for instance, a six-parameter nonlinear PI compensator. The parameters of the approximate compensators are tuned to satisfy an optimality condition. The superiority of the proposed nonlinear PI compensators over linear PI compensators is discussed and is demonstrated for two feedback systems.     

On the modified discrete KP equation with self-consistent sources

The modified discrete KP equation is the Bäcklund transformation for the Hirota’s discrete KP equation or the Hirota-Miwa equation. We construct the modified discrete KP equation with self-consistent sources via source generation procedure and clarify the algebraic structure of the resulting coupled modified discrete KP system by presenting its discrete Gram-type determinant solutions. It is also shown that the commutativity between the source generation procedure and Bäcklund transformation is valid for the discrete KP equation. Finally, we demonstrate that the modified discrete KP equation with self-consistent sources yields the modified differential-difference KP equation with self-consistent sources through a continuum limit. The continuum limit of an explicit solution to the modified discrete KP equation with self-consistent sources also gives the explicit solution for the modified differential-difference KP equation with self-consistent sources.     

Preconditioning techniques for iterative solvers in the Discrete Sources Method

Different preconditioning techniques for the iterative method MinRes as solver for the Discrete Sources Method (DSM) are presented. This semi-analytical method is used for light scattering computations by particles in the Mie scattering regime. Its numerical schema includes a linear least-squares problem commonly solved using the QR decomposition method. This could be the subject of numerical difficulties and instabilities for very large particles or particles with extreme geometry. In these cases, we showed that iterative methods with preconditioning techniques can provide a satisfying solution.In our previous paper, we studied four different iterative solvers (RGMRES, BiCGStab, BiCGStab(l), and MinRes) considering the performance and the accuracy of a solution. Here, we study several preconditioning techniques for the MinRes method for a variety of oblate and prolate spheroidal particles of different size and geometrical aspect ratio. Using preconditioning techniques we highly accelerated the iterative process especially for particles with a higher aspect ratio.