Maximum likelihood estimation for constrained parameters of multinomial distributions - Application to Zipf-Mandelbrot models

A numerical maximum likelihood (ML) estimation procedure is developed for the constrained parameters of multinomial distributions. The main difficulty involved in computing the likelihood function is the precise and fast determination of the multinomial coefficients. For this the coefficients are rewritten into a telescopic product. The presented method is applied to the ML estimation of the Zipf–Mandelbrot (ZM) distribution, which provides a true model in many real-life cases. The examples discussed arise from ecological and medical observations. Based on the estimates, the hypothesis that the data is ZM distributed is tested using a chi-square test. The computer code of the presented procedure is available on request by the author

A reliable and efficient implicit a posteriori error estimation technique for the time harmonic Maxwell equations

We analyze an implicit a posteriori error indicator for the time harmonic Maxwell equations and prove that it is both reliable and locally efficient. For the derivation, we generalize some recent results concerning explicit a posteriori error estimates. In particular, we relax the divergence free constraint for the source term. We also justify the complexity of the obtained estimator

Systematic front distortion and presence of consecutive fronts in a precipitation system

A new simple reaction-diffusion system is presented focusing on pattern formation phenomena as consecutive precipitation fronts and distortion of the precipitation front.The chemical system investigated here is based on the amphoteric property of aluminum hydroxide and exhibits two unique phenomena. Both the existence of consecutive precipitation fronts and distortion are reported for the first time. The precipitation patterns could be controlled by the pH field, and the distortion of the precipitation front can be practical for microtechnological applications of reaction-diffusion systems

Error analysis of a continuous-discontinuous Galerkin finite element method for generalized 2D vorticity dynamics

A detailed a priori error estimate is provided for a continuous-discontinuous Galerkin finite element method suitable for two-dimensional geophysical flows. Special attention is given to derive estimates which require only minimal smoothness in the vorticity field

Micro and macro level stochastic simulation of reaction-diffusion systems

We provide numerical simulations for nonlinear reaction-diusion systems, which arise from a micro and macro level model of pattern formation (Liesegang phenomenon). In both cases we apply a stochastic approach: a discrete stochastic model and concentration perturbation in a deterministic model

An excited state coupled-cluster study on indigo dyes

In the present study, the domain-based pair natural orbital implementation of the similarity-transformed equation of motion method is employed to reproduce the vibrationally resolved absorption spectra of indigo dyes. After an initial investigation of multireference, basis set and implicit solvent effects, our calculated 0–0 transition energies are compared to a benchmark set of experimental absorption band maxima. It is established that the agreement between our method and experimental results is well below the desired 0.1 eV threshold in virtually all cases and that the shift in excitation energies upon chemical substitution is also well reproduced. Finally, the entire spectra of some of the main components of the Tyrian purple dye mixture are reproduced and it is found that our computed spectra match the experimental ones without an empirical shift

Transition of liesegang precipitation systems: simulations with an adaptive grid pde method

The dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of a recent adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments

An IMEX scheme combined with Richardson extrapolation methods for some reaction-diffusion equations

An implicit-explicit (IMEX) method is combined with some so-called Richardson extrapolation (RiEx) methods for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The results are applied to a model for determining the overpotential in a Proton Exchange Membrane (PEM) fuel cell

Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations

We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are $i)$ valid in the pre-asymptotic regime; $ii)$ solely depend on the geometry and the polynomial order; and $iii)$ are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order ($p = 1, 2, 3, 4$) hierarchic $H(\mathrm{curl})$-conforming polynomial basis functions

A perturbative approach to multireference equation-of-motion coupled cluster

We introduce a variant of the multireference equation-of-motion coupled-cluster (MR-EOMCC) method where the amplitudes used for the similarity transformations are estimated from perturbation theory. Consequently, the new variant retains the many-body formalism, a reliance on at most two-body densities, and the state-universal character. As a non-iterative variant, computational costs are reduced, and no convergence difficulties with near-singular amplitudes can arise. Its performance was evaluated on several test sets covering transition metal atoms, small diatomics, and organic molecules against (near-)full CI quality reference data. We further highlight its efficacy on the weakly avoided crossing of LiF and place MR-EOMCC and the new variant into context with linear response theory. The accuracy of the variant was found to be at least on par with expectations for multireference perturbation theories, judging by the NEVPT2 method. The variant can be especially useful in multistate situations where the high accuracy of the iterative MR-EOMCC method is not required