On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

Generalized criterion of solvent polarity as a tool for control of chromatographic process

The generalized (reductive) criterion of solvent polarity was obtained by the method of multiparametric optimization of the Snyder index P", Hildebrand parameter _T , permittivity _r , and solvatochromism parameter E _T (30). Possibilities of employing this criterion for estimation of the elution power of normal and reversed mobile phases used in high-performance liquid chromatography were considered.     

A hidden analytic structure of the Rabi model

The Rabi model describes the simplest interaction between a cavity mode with a frequency ω_c${\omega }_c$ and a two-level system with a resonance frequency ω₀${\omega }_0$. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to Δ=ω₀/(2ω_c)=0$\Delta ={\omega }_0/\left(2{\omega }_c\right)=0$, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy ?$?$, which are orthogonal on an equidistant lattice. A non-zero value of Δ$\Delta$ leads to non-classical discrete orthogonal polynomials ?_k(?)${?}_k\left(?\right)$ and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. 1350 calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak’s solution. Any first n$n$ eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ?_N(?)${?}_N\left(?\right)$ of at least the degree N=n+n_t$N=n+n_t$. The value of n_t>0$n_t>0$, which is slowly increasing with n$n$, depends on the required precision. For instance, n_t?26$n_t?26$ for n=1000$n=1000$ and dimensionless interaction constant κ=0.2$\kappa =0.2$, if double precision is required. Given that the sequence of the l$l$th zeros x_nl$x_{nl}$’s of ?_n(?)${?}_n\left(?\right)$’s defines a monotonically decreasing discrete flow with increasing n$n$, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for dimensionless interaction constant κ<1$\kappa <1$, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥1$\kappa \ge 1$.     

An improved automatic time-of-flight picker for medical ultrasound tomography

Objective and motivationTime-of-flight (TOF) tomography used by a clinical ultrasound tomography device can efficiently and reliably produce sound-speed images of the breast for cancer diagnosis. Accurate picking of TOFs of transmitted ultrasound signals is extremely important to ensure high-resolution and high-quality ultrasound sound-speed tomograms. Since manually picking is time-consuming for large datasets, we developed an improved automatic TOF picker based on the Akaike information criterion (AIC), as described in this paper.MethodsWe make use of an approach termed multi-model inference (model averaging), based on the calculated AIC values, to improve the accuracy of TOF picks. By using multi-model inference, our picking method incorporates all the information near the TOF of ultrasound signals. Median filtering and reciprocal pair comparison are also incorporated in our AIC picker to effectively remove outliers.ResultsWe validate our AIC picker using synthetic ultrasound waveforms, and demonstrate that our automatic TOF picker can accurately pick TOFs in the presence of random noise with absolute amplitudes up to 80% of the maximum absolute signal amplitude. We apply the new method to 1160 in vivo breast ultrasound waveforms, and compare the picked TOFs with manual picks and amplitude threshold picks. The mean value and standard deviation between our TOF picker and manual picking are 0.4 μs and 0.29 μs, while for amplitude threshold picker the values are 1.02 μs and 0.9 μs, respectively. Tomograms for in vivo breast data with high signal-to-noise ratio (SNR) (∼25 dB) and low SNR (∼18 dB) clearly demonstrate that our AIC picker is much less sensitive to the SNRs of the data, compared to the amplitude threshold picker.Discussion and conclusionsThe picking routine developed here is aimed at determining reliable quantitative values, necessary for adding diagnostic information to our clinical ultrasound tomography device - CURE. It has been successfully adopted into CURE, and allows us to generate such values reliably. We demonstrate that in vivo sound-speed tomograms with our TOF picks significantly improve the reconstruction accuracy and reduce image artifacts.     

Convergence behavior of a new DSMC algorithm

The convergence rate of a new direct simulation Monte Carlo (DSMC) method, termed “sophisticated DSMC”, is investigated for one-dimensional Fourier flow. An argon-like hard-sphere gas at 273.15 K and 266.644 Pa is confined between two parallel, fully accommodating walls 1 mm apart that have unequal temperatures. The simulations are performed using a one-dimensional implementation of the sophisticated DSMC algorithm. In harmony with previous work, the primary convergence metric studied is the ratio of the DSMC-calculated thermal conductivity to its corresponding infinite-approximation Chapman–Enskog theoretical value. As discretization errors are reduced, the sophisticated DSMC algorithm is shown to approach the theoretical values to high precision. The convergence behavior of sophisticated DSMC is compared to that of original DSMC. The convergence of the new algorithm in a three-dimensional implementation is also characterized. Implementations using transient adaptive sub-cells and virtual sub-cells are compared. The new algorithm is shown to significantly reduce the computational resources required for a DSMC simulation to achieve a particular level of accuracy, thus improving the efficiency of the method by a factor of 2.     

The stability analysis of rolling motion of hypersonic vehicles and its validations

The stability of the rolling motion of near space hypersonic vehicles with rudder control is studied using method of qualitative analysis of nonlinear differential equations, and the stability criteria of the deflected rolling motions are improved. The outcomes can serve as the basis for further study regarding the influence of pitching and lateral motion on the stability of rolling motion. To validate the theoretical results, numerical simulations were done for the rolling motion of two hypersonic vehicles with typical configurations. Also, wind tunnel experiments for four aircraft models with typical configurations have been done. The results show that: 1) there exist two dynamic patterns of the rolling motion under statically stable condition. The first one is point attractor, for which the motion of aircraft returns to the original state. The second is periodic attractor, for which the aircraft rolls periodically. 2) Under statically unstable condition, there exist three dynamic patterns of rolling motion, namely, the point attractor, periodic attractor around deflected state of rolling motion, and double periodic attractors or chaotic attractors.     

Composition-dependent melting behaviour of NaxK55–x core–shell nanoalloys

Molecular dynamics simulations at constant temperature are performed to investigate melting-like transition in Na₁₃K₄₂, Na₁₉K₃₆ and Na₂₆K₂₉ nanoalloys using a second-moment-approximation tight-binding analytic potential to calculate the forces on the constituent atoms. A weighted histogram analysis method is employed to remove non-ergodicity issues due to the complex potential energy surface of these nanoalloys. The heat capacity shows three distinctive steps in melting for Na₁₃K₄₂, while Na₂₆K₂₉ and Na₁₉K₃₆ have two-step and one-step melting transition, respectively. The steepest descent method is used to quench the configurations in a given interval during the simulation and also study the isomerisation processes occurring at different temperatures. Analysing the configuration energies of quenched structures for the entire nanoalloy and the core atoms separately gives more details about the melting mechanism. The Lindemann parameter is also calculated at several temperatures during the simulation which shows a gradual increase for Na₁₃K₄₂ and Na₂₆K₂₉ while a sharp change is observed for Na₁₉K₃₆. These findings are in agreement with the multi-step nature of the phase transition in Na₁₃K₄₂ and Na₂₆K₂₉ and one-step melting of the Na₁₉K₃₆ magic composition.     

A new nonmonotone trust-region method of conic model for solving unconstrained optimization

In this paper, we present a new nonmonotone trust-region method of conic model for solving unconstrained optimization problems. Both the local and global convergence properties are analyzed under reasonable assumptions. Numerical experiments are conducted to compare this method with some existed ones which indicate that the new method is efficient.