Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

A closed-form approximation for pricing temperature-based weather derivatives

This paper develops analytical distributions of temperature indices on which temperature derivatives are written. If the deviations of daily temperatures from their expected values are modelled as an Ornstein-Uhlenbeck process with timevarying variance, then the distributions of the temperature index on which the derivative is written is the sum of truncated, correlated Gaussian deviates. The key result of this paper is to provide an analytical approximation to the distribution of this sum, thus allowing the accurate computation of payoffs without the need for any simulation. A data set comprising average daily temperature spanning over a hundred years for four Australian cities is used to demonstrate the efficacy of this approach for estimating the payoffs to temperature derivatives. It is demonstrated that expected payoffs computed directly from historical records are a particularly poor approach to the problem when there are trends in underlying average daily temperature. It is shown that the proposed analytical approach is superior to historical pricing.

Diffusivity of adatoms on plasma-exposed surfaces determined from the ionization energy approximation and ionic polarizability

Microscopic surface diffusivity theory based on atomic ionization energy concept is developed to explain the variations of the atomic and displacement polarizations with respect to the surface diffusion activation energy of adatoms in the process of self-assembly of quantum dots on plasma-exposed surfaces. These polarizations are derived classically, while the atomic polarization is quantized to obtain the microscopic atomic polarizability. The surface diffusivity equation is derived as a function of the ionization energy. The results of this work can be used to fine-tune the delivery rates of different adatoms onto nanostructure growth surfaces and optimize the low-temperature plasma based nanoscale synthesis processes.

Approximation of Bayesian predictive p-values with regression ABC

In the Bayesian framework a standard approach to model criticism is to compare some function of the observed data to a reference predictive distribution. The result of the comparison can be summarized in the form of a p-value, and it's well known that computation of some kinds of Bayesian predictive p-values can be challenging. The use of regression adjustment approximate Bayesian computation (ABC) methods is explored for this task. Two problems are considered. The first is the calibration of posterior predictive p-values so that they are uniformly distributed under some reference distribution for the data. Computation is difficult because the calibration process requires repeated approximation of the posterior for different data sets under the reference distribution. The second problem considered is approximation of distributions of prior predictive p-values for the purpose of choosing weakly informative priors in the case where the model checking statistic is expensive to compute. Here the computation is difficult because of the need to repeatedly sample from a prior predictive distribution for different values of a prior hyperparameter. In both these problems we argue that high accuracy in the computations is not required, which makes fast approximations such as regression adjustment ABC very useful. We illustrate our methods with several samples.

Transdimensional sequential Monte Carlo using variational Bayes — SMCVB

A new transdimensional Sequential Monte Carlo (SMC) algorithm called SM- CVB is proposed. In an SMC approach, a weighted sample of particles is generated from a sequence of probability distributions which ‘converge’ to the target distribution of interest, in this case a Bayesian posterior distri- bution. The approach is based on the use of variational Bayes to propose new particles at each iteration of the SMCVB algorithm in order to target the posterior more efficiently. The variational-Bayes-generated proposals are not limited to a fixed dimension. This means that the weighted particle sets that arise can have varying dimensions thereby allowing us the option to also estimate an appropriate dimension for the model. This novel algorithm is outlined within the context of finite mixture model estimation. This pro- vides a less computationally demanding alternative to using reversible jump Markov chain Monte Carlo kernels within an SMC approach. We illustrate these ideas in a simulated data analysis and in applications.

Analytical approximation solutions for 3-D optical waveguides: Review

The rectangular dielectric waveguide is the most commonly used structure in integrated optics, especially in semi-conductor diode lasers. Demands for new applications such as high-speed data backplanes in integrated electronics, waveguide filters, optical multiplexers and optical switches are driving technology toward better materials and processing techniques for planar waveguide structures. The infinite slab and circular waveguides that we know are not practical for use on a substrate because the slab waveguide has no lateral confinement and the circular fiber is not compatible with the planar processing technology being used to make planar structures. The rectangular waveguide is the natural structure. In this review, we have discussed several analytical methods for analyzing the mode structure of rectangular structures, beginning with a wave analysis based on the pioneering work of Marcatili. We study three basic techniques with examples to compare their performance levels. These are the analytical approach developed by Marcatili, the perturbation techniques, which improve on the analytical solutions and the effective index method with examples.

Optimal hydrothermal load flow: Formulation and a successive approximation solution for fixed head systems

This paper deals with the optimal load flow problem in a fixed-head hydrothermal electric power system. Equality constraints on the volume of water available for active power generation at the hydro plants as well as inequality constraints on the reactive power generation at the voltage controlled buses are imposed. Conditions for optimal load flow are derived and a successive approximation algorithm for solving the optimal generation schedule is developed. Computer implementation of the algorithm is discussed, and the results obtained from the computer solution of test systems are presented.

Bias reduction using stochastic approximation

The paper studies stochastic approximation as a technique for bias reduction. The proposed method does not require approximating the bias explicitly, nor does it rely on having independent identically distributed (i.i.d.) data. The method always removes the leading bias term, under very mild conditions, as long as auxiliary samples from distributions with given parameters are available. Expectation and variance of the bias-corrected estimate are given. Examples in sequential clinical trials (non-i.i.d. case), curved exponential models (i.i.d. case) and length-biased sampling (where the estimates are inconsistent) are used to illustrate the applications of the proposed method and its small sample properties.

A Fuzzy Approximation Scheme for Sequential Learning in Pattern Recognition

An adaptive learning scheme, based on a fuzzy approximation to the gradient descent method for training a pattern classifier using unlabeled samples, is described. The objective function defined for the fuzzy ISODATA clustering procedure is used as the loss function for computing the gradient. Learning is based on simultaneous fuzzy decisionmaking and estimation. It uses conditional fuzzy measures on unlabeled samples. An exponential membership function is assumed for each class, and the parameters constituting these membership functions are estimated, using the gradient, in a recursive fashion. The induced possibility of occurrence of each class is useful for estimation and is computed using 1) the membership of the new sample in that class and 2) the previously computed average possibility of occurrence of the same class. An inductive entropy measure is defined in terms of induced possibility distribution to measure the extent of learning. The method is illustrated with relevant examples.

Theoretical description of time-resolved pump/probe photoemission in TaS2: a single-band DFT+DMFT(NRG) study within the quasiequilibrium approximation

In this work, we theoretically examine recent pump/probe photoemission experiments on the strongly correlated charge-density-wave insulator TaS2.We describe the general nonequilibrium many-body formulation of time-resolved photoemission in the sudden approximation, and then solve the problem using dynamical mean-field theory with the numerical renormalization group and a bare density of states calculated from density functional theory including the charge-density-wave distortion of the ion cores and spin-orbit coupling. We find a number of interesting results: (i) the bare band structure actually has more dispersion in the perpendicular direction than in the two-dimensional planes; (ii) the DMFT approach can produce upper and lower Hubbard bands that resemble those in the experiment, but the upper bands will overlap in energy with other higher energy bands; (iii) the effect of the finite width of the probe pulse is minimal on the shape of the photoemission spectra; and (iv) the quasiequilibrium approximation does not fully describe the behavior in this system.

Approximation algorithm for maximum independent set in planar traingle-free graphs

The maximum independent set problem is NP-complete even when restricted to planar graphs, cubic planar graphs or triangle free graphs. The problem of finding an absolute approximation still remains NP-complete. Various polynomial time approximation algorithms, that guarantee a fixed worst case ratio between the independent set size obtained to the maximum independent set size, in planar graphs have been proposed. We present in this paper a simple and efficient, O(|V|) algorithm that guarantees a ratio 1/2, for planar triangle free graphs. The algorithm differs completely from other approaches, in that, it collects groups of independent vertices at a time. Certain bounds we obtain in this paper relate to some interesting questions in the theory of extremal graphs.

Coherent potential approximation, averaged T-matrix approximation and Lloyd's model

It is shown how the single-site coherent potential approximation and the averaged T-matrix approximation become exact in the calculation of the averaged single-particle Green function of the electron in the Anderson model when the site energy is distributed randomly with lorentzian distribution. Using these approximations, Lloyd's exact result is reproduced.

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x1,…, xm) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.

An appendix to the paper "approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic"

The theory of Varley and Cumberbatch [l] giving the intensity of discontinuities in the normal derivatives of the dependent variables at a wave front can be deduced from the more general results of Prasad which give the complete history of a disturbance not only at the wave front but also within a short distance behind the wave front. In what follows we omit the index M in Eq. (2.25) of Prasad [2].