Optimal Linear Glauber Model

Abstract

The Kawasaki model is not exactly solvable as any choice of the exchange rate (w jj ′ ) which satisfies the detailed balance condition is highly nonlinear. In this work we address the issue of writing w jj ′ in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least. In the continuum limit, our approach leads to a Cahn-Hilliard equation of conservative dynamics. The work presented in this paper will help us anticipate how the conservative dynamics of an arbitrary Ising system depends on the temperature and the coupling constants. In particular, for two and three dimensional systems, the critical temperatures estimated in our work are in good agreement with the actual values. We also calculate the dynamic and some of the critical exponents of the model.

Sequential Model-Based A-Optimal Design of Experiments When the Fisher Information Matrix Is Noninvertible

“…As indicated in the studies above, one approach for dealing with a noninvertible FIM is to make simplifying assumptions about parameters requiring estimation. An alternative approach, considered for the first time in the current study, is to use a pseudoinverse (PI) to deal with singular or ill-conditioned FIM s. The Moore-Penrose PI is popular in the mathematical modeling and parameter estimation literature because it has attractive mathematical properties and results in unique parameter estimates. − Nevertheless, the resulting parameter estimates may be unreliable . Other approaches for dealing with singular/ill-conditioned FIM s during parameter estimation include Tikhonov regularization approaches such as ridge regression and LASSO estimation , and Bayesian methods, which are beyond the scope of the current study but will be considered in future MBDoE studies.…”

The Moore–Penrose inverse: a hundred years on a frontline of physics research

“…An everlasting interest in the least squares method is well reflected by a immutable stream of articles, in which the method is utilized and most often plays a relevant role. Regarding recent research within broadly understood physics, exemplary applications of the method can be encountered in the following articles: Chou et al (2018) proposing a framework for privacy preserving compressive analysis (which exploits formulae for the Moore-Penrose inverse of columnwise partitioned matrices), Gaylord and Kilby (2004) specifying a procedure of measuring optical transmittance of photonic crystals, Huang et al (2006) introducing the extreme learning machine algorithm, Le Bigot et al (2008) presenting high-precision energy level calculations in atomic hydrogen and deuterium, Ordones et al (2019) deriving frequency transfer function formalism for phase-shifting algorithms, Sahoo and Ganguly (2015) optimizing the linear Glauber model to analyse kinetic properties of an arbitrary Ising system, Stanimirović et al (2013) introducing a computational method of the digital image restoration, Wang et al (1993) identifying sources of neuronal activity within the brain from measurements of the extracranial magnetic field, Wang and Zhang (2012) deriving an online linear discriminant analysis algorithm (which exploits formulae for the Moore-Penrose inverse of modified matrices), and White et al (2014) elaborating a method for computing the initial post-buckling response of variable-stiffness cylindrical panels (even though the least squares method was not explicitly mentioned in the paper, we conclude it was exploited from remarks on pp. 141 and 143 stating that the systems of equations solved were overdetermined).…”