Effective spin foams provide the most computationally efficient spin foam models yet and are therefore ideally suited for applications, e.g., to quantum cosmology. Here, we provide the first effective spin foam computations of a finite time evolution step in a Lorentzian quantum de Sitter universe. We will consider a setup that computes the no-boundary wave function and a setup describing the transition between two finite scale factors. A key property of spin foams is that they implement discrete spectra for the areas. We therefore study the effects that are induced by the discrete spectra. To perform these computations, we had to identify a technique to deal with highly oscillating and slowly converging or even diverging sums. Here, we illustrate that high-order Shanks transformation works very well and is a promising tool for the evaluation of Lorentzian (gravitational) path integrals and spin foam sums.

ABSTRACT In this paper, we present an efficient method for evaluating the stability of both linear time-invariant (LTI) systems and continuous-time nonlinear systems. Our approach centres around the modulus of the spectral radius derived from a specific matrix transformation of the original system state matrix in the case of LTI systems, or the Jacobian of the nonlinear system. We enhance computational speed by employing the iterated Shanks transformation on Gelfand’s formula, eliminating the need to rely on the system’s characteristic equation. Notably, our stability criterion avoids the computation of all eigenvalues of the system matrix. Illustrative examples, including a fluidised bed reactor, a continuous stirred tank reactor and a hypothetical 20th-order dynamical system, are provided. Additionally, we conduct a CPU-time analysis, revealing that our method demonstrates several orders of magnitude faster computational speed compared to the classical Routh-Hurwitz stability test for systems ranging from two to 80 dimensions.

Evaluation of Turbulence Depending Drag Coefficient in Plume Rise Model for Fire Smoke Dispersion

Shanks Transformation for the Asymptotic Relation of Heat Transfer Rate and Rayleigh Number for Convection in Mantle

A line search in a gradient-based optimization algorithm solves the problem of determining the optimal learning rate for a given gradient or search direction in a single iteration. For most problems, this is determined by evaluating different candidate learning rates to find the optimum, which can be expensive. Recent work has provided an efficient way to perform a line search with the use of the Shanks transformation of a Born series derived from the Lippman-Schwinger formalism. In this paper we show that the cost for performing such a line search can be further reduced with the use of the method of the Schur complement domain decomposition, which can lead to a 10-fold total speed-up resulting from the reduced number of iterations to convergence and reduced wall-clock time per iteration.

Abstract This paper examines a number of extrapolation and acceleration methods and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general framework that encompasses most of the known acceleration strategies. The paper also considers the Anderson Acceleration (AA) method under a new light and exploits a connection with quasi-Newton methods in order to establish local linear convergence results of a stabilized version of the AA method. The methods are tested on a number of problems, including a few that arise from nonlinear partial differential equations.

Firstly, a new sequence transformation that can be expressed in terms of a ratio of two pfaffians is derived based on a special kernel. It can be regarded as a direct generalization of Aitken’s Δ2 process from the point of view of pfaffians and then the corresponding convergence acceleration algorithm is constructed. Numerical examples with applications of this algorithm are also presented. Secondly, we find a way to generalize the Shanks transformation via pfaffians so that a larger class of new sequence transformations are derived. The corresponding recursive algorithms are also proposed.

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

The transient electromagnetic (TEM) method becomes more urgent than ever for marine exploration due to abundant resource reserves and the increasing undersea engineering construction activities, especially in the offshore exploration of mineral deposits such as Sanshandao gold mine. However, the research and application of TEM method in marine environment are still challenged by many problems. Such contradiction motivates our study on the coincident-loop TEM in seafloor exploration. The TEM response of coincident loops is firstly derived in the integral form, based on the potential functions in Helmholtz equations for a magnetic source locating in the whole-space layered model. The frequency-domain vertical magnetic field is described as the Hankel integral with double first-order Bessel functions of first kind. Secondly, the time-domain induced voltage is obtained by transforming the frequency-domain response through the cosine transform and then taking the derivative of time. To simultaneously solve the Hankel transform and the cosine transform, a novel algorithm is introduced by adapting the fixed-point quadrature and extrapolation via the Shanks transformation. Finally, a typical conductivity model for marine polymetallic deposit is designed to investigate the characteristic of TEM response under various conditions. Numerical results demonstrate that existence of conductive seawater causes the TEM response to increase significantly and decay slower. The air-sea reflected electromagnetic waves lead to a significantly large fake negative response (NR) in shallower seawater with depth less than 300 m. Increase in the height of loops will weaken and delay the anomaly response and shorten the observation time-window. The height of configuration should be no more than 100 m for shallower targets and 50 m for deeper targets, respectively. The observation time-window should cover 10-1 000 ms. Increase in the radius of loops only enhances the TEM response proportionally but hardly improves the relative anomaly. The vertical resolution on the low-resistivity target approximates 20 m for the configuration considered in the study. Decreases in D.C. resistivity and chargeability cause the positive response (PR) to increase significantly and decay more rapidly. Meanwhile, the NR is advanced and enlarged significantly and decays slower compared with the PR. The influence of time constant is not monotony and there exists an optimal value for producing the maximum NR. As the frequency parameter increases, the PR is caused to decay more rapidly without magnitude change and the NR is advanced and decays more rapidly with significant increase in magnitude. The influence of frequency parameter is more pronounced than that of time constant.

In an anisotropic model, traveltime can be determined approximately by numerical solution of the eikonal equation in terms of an anellipticity parameter η, using perturbation theory. However, its accuracy decreases under the effect of strong anisotropy at larger offsets. It becomes invalid for determining normal moveout velocity and anellipticity parameter in seismic processing. We propose a new approach using Levin T-transformation to transform the expanded traveltime in the transversely isotropic medium with vertical axis of symmetry (VTI) into rational form. The objective of this study is to provide a new traveltime approximation that is more accurate at larger offsets. In this study, we derive Levin algorithm and determine the optimal value of Levin parameters, which is a key step in achieving better accuracy. In a numerical experiment, we compare the accuracy between Levin T-transformation and second sequence of Shanks transformation in a homogeneous VTI medium. We also implement both approximations in a velocity analysis and stacking traces using synthetic common midpoint gathers on a multilayer earth model. The proposed method shows a superiority in accuracy to existing methods over a range of offsets with offset-to-depth ratio up to 6 and anellipticity parameter 0–0.5.

In this paper, it is proposed an enhanced sequential fully implicit (ESFI) algorithm with a fixed stress split to approximate robustly poro-elastoplastic solutions related to reservoir geomechanics. The constitutive model considers the total strain effect on porosity/permeability variation and associative plasticity. The sequential fully implicit (SFI) algorithm is a popular solution to approximate solutions of a coupled system. Generally, the SFI consists of an outer loop to solve the coupled system, in which there are two inner iterative loops for each equation to implicitly solve the equations. The SFI algorithm occasionally suffers from slow convergence or even convergence failure. In order to improve the convergence (robustness) associated with SFI, a new nonlinear acceleration technique is proposed employing Shanks transformations in vector-valued variables to enhance the outer loop convergence, with a quasi-Newton method considering the modified Thomas method for the internal loops. In this ESFI algorithm, the fluid flow formulation is defined by Darcy’s law including nonlinear permeability based on Petunin model. The rock deformation includes a linear part being analyzed based on Biot’s theory and a nonlinear part being established using Mohr-Coulomb associative plasticity for geomechanics. Temporal derivatives are approximated by an implicit Euler method, and spatial discretizations are adopted using finite element in two different formulations. For the spatial discretization, two weak statements are obtained: the first one uses a continuous Galerkin for poro-elastoplastic and Darcy’s flow; the second one uses a continuous Galerkin for poro-elastoplastic and a mixed finite element for Darcy’s flow. Several numerical simulations are presented to evaluate the efficiency of ESFI algorithm in reducing the number of iterations. Distinct poromechanical problems in 1D, 2D, and 3D are approximated with linear and nonlinear settings. Where appropriate, the results were verified with analytic solutions and approximated solutions with an explicit Runge-Kutta solver for 2D axisymmetric poro-elastoplastic problems.

In gradient-based optimization of photonic devices, within the overall design parameter space, one iteratively performs a line search in a one-dimensional subspace as spanned by the search direction. While the search direction can be efficiently determined with the adjoint variable method, there has not been an efficient algorithm that determines the optimal learning rate that controls the distance one moves along the search direction. Here we introduce an efficient algorithm of determining the optimal learning rate, using the Shanks transformation in the Lippmann-Schwinger formalism. Our approach can determine very accurately the optimal learning rates at each epoch, with only a modest increase of computational cost. We show that this approach can significantly improve the figure of merits of the final structure, as compared to conventional methods for estimating the learning rate.

Improved asymptotic predictions for the effective slip over a corrugated topography

In this paper, we want to exemplify the use of extrapolation methods (namely, Shanks transformations, the recursive algorithms for their implementation, and the freely available corresponding MATLAB software) in the solution of nonlinear Fredholm integral equations of the second kind. Extrapolations methods are well known in some domains of numerical analysis and applied mathematics, but, unfortunately, they are not frequently used in other domains. Thus, after presenting the most simple iterative method for the solution of Fredholm equations, we will show how the sequence it produces can be accelerated (under some assumptions) and also how the underlying system of nonlinear equations generated by it can be solved quite efficiently by a restarting method. Numerical examples and comparisons with other methods demonstrate the usefulness of these procedures.

The rate of heat conduction (or mass transfer by diffusion) from a cylindrical or a spherical particle confined between two walls is determined as a function of the position and the radius of the particle. It is shown that the appropriate Green's function can be determined using the method of images even when the resulting series is divergent with the help of Shanks transformation. Asymptotic expansions for small particle radius compared to the distance between the walls are combined with the expressions for the case in which the gap between the particle and one of the walls is small compared to the particle radius to provide formulas that are surprisingly accurate for estimating the rate of heat transfer for the entire range of parameters that include the radius and the position of the particle. Results are also presented for the thermal dipole induced by a spherical or a cylindrical particle placed between two walls with unequal temperatures and these are used to predict the effective thermal conductivity of thin composite films containing spherical or cylindrical particles.

Convergence acceleration in scattering series and seismic waveform inversion using nonlinear Shanks transformation

This paper presents a general framework for Shanks transformations of sequences of elements in a vector space. It is shown that Minimal Polynomial Extrapolation (MPE), Modified Minimal Polynomial E...

Minimal area surfaces in ending on a given curve at the boundary are dual to planar Wilson loops in SYM. In previous work it was shown that the problem of finding such surfaces can be recast as the one of finding an appropriate parameterization of the boundary contour that corresponds to conformal gauge. Dekel was able to find such reparameterization in a perturbative expansion around a circular contour. In this work we show that for more general contours such reparameterization can be found using a numerical procedure that does not rely on a perturbative expansion. This provides further checks and applications of the integrability method. An interesting property of the method is that it uses as data the Schwarzian derivative of the contour and therefore it has manifest global conformal invariance. Finally, we apply Shanks transformation to extend the near circular expansion to larger deformations. The results are in agreement with the new method.

We explore in detail how analytic continuation of divergent perturbation series by generalized hypergeometric functions is achieved in practice. Using the example of strong-coupling perturbation series provided by the two-dimensional Bose–Hubbard model, we compare hypergeometric continuation to Shanks and Padé techniques, and demonstrate that the former yields a powerful, efficient and reliable alternative for computing the phase diagram of the Mott insulator-to-superfluid transition. In contrast to Shanks transformations and Padé approximations, hypergeometric continuation also allows us to determine the exponents which characterize the divergence of correlation functions at the transition points. Therefore, hypergeometric continuation constitutes a promising tool for the study of quantum phase transitions.