Saddle-point Integration Research Articles

On the Deficiency of the First-Order Ott–Clemmow Saddle Point Method as Applied to the Sommerfeld Half-Space Problem

The accuracy of the first-order multiplicative (Ott–Clemmow) and additive (van der Waerden) modified saddle point integration methods for the Sommerfeld problem of a vertical Hertzian dipole over a lossy half-space is investigated using two theoretically equivalent formulations with either a positive or a negative image term extracted. It is demonstrated that whereas the additive method leads to the same asymptotic field representation irrespective of the sign of the image, the multiplicative variant yields distinct results in these two cases, both of which differ from the unique result of the additive method. It is further found that the positive-image Ott–Clemmow method yields the well-known Norton formula, but the negative-image variant is inaccurate and predicts a zero field in the on-surface transmitter–receiver configuration. It is also demonstrated that the first-order multiplicative method yields an incomplete representation of the first order in the inverse distance.

On the Complete Radiation Pattern of a Vertical Hertzian Dipole Above a Low-Loss Ground

The complete radiation field pattern of a vertical Hertzian dipole antenna on or above a lossless or low-loss dielectric half-space is studied using a rigorous Sommerfeld formalism. The reflected fields in the air above the interface and the subsurface fields transmitted into the dielectric are computed by numerical integration of the Sommerfeld integrals. Furthermore, to facilitate the physical interpretation of these results, a detailed asymptotic saddle-point integration method analysis is presented, which includes terms that vary in magnitude with the second power of the inverse distance from the dipole. It is shown that the second-order field constituents are dominant at the interface, where the first-order geometrical fields vanish. These second-order terms comprise an evanescent wave propagating along the interface in the upper half-space and a lateral wave, also known as the head wave, which propagates in the subsurface along the direction of the critical angle. The two waves only exist between two cones whose half-angles are equal to the critical angle, and their interference with the geometrical-optics fields determines the radiation pattern for elevation angles near the horizon. The far zone surface fields on either side of the interface comprise two second order waves that propagate along the interface, one with the phase velocity in the air, and the other with the phase velocity in the dielectric. Away from the interface, the leading field components vary with the first power of the inverse distance, which explains the sharp dip in the field pattern at the interface-a phenomenon known as the interface pattern extinction. Another distinctive phenomenon, observed in the subsurface field pattern, is the rippling that occurs in the angular range between the critical angle cone and the interface. The asymptotic analysis has shown that this pattern scalloping results from the interference of the lateral wave with the geometrical-optics spherical wave.

On the Sommerfeld half-space problem: appraisal of approximate solutions with extensions to plasmonics

The classical Sommerfeld problem of a vertical electric dipole radiating over an imperfectly conducting half-space is revisited. Two variants of the modified saddle-point integration method are employed to derive new, closed-form, second-order approximations of the far-zone electromagnetic fields. The new formulas are applied to compute radiation patterns and surface fields for sample problems including both ordinary and plasmonic media, and their accuracy is assessed by comparison with the results of rigorous numerical evaluation of the Sommerfeld integrals.

Strong-field ionization via a high-order Coulomb-corrected strong-field approximation

Signatures of the Coulomb corrections in the photoelectron momentum distribution during laser-induced ionization of atoms or ions in tunneling and multiphoton regimes are investigated analytically in the case of an one-dimensional problem. High-order Coulomb corrected strong-field approximation is applied, where the exact continuum state in the S-matrix is approximated by the eikonal Coulomb-Volkov state including the second-order corrections to the eikonal. Although, without high-order corrections our theory coincides with the known analytical R-matrix (ARM) theory, we propose a simplified procedure for the matrix element derivation. Rather than matching the eikonal Coulomb-Volkov wave function with the bound state as in the ARM-theory to remove the Coulomb singularity, we calculate the matrix element via the saddle-point integration method as by time as well as by coordinate, and in this way avoiding the Coulomb singularity. The momentum shift in the photoelectron momentum distribution with respect to the ARM-theory due to high-order corrections is analyzed for tunneling and multiphoton regimes. The relation of the quantum corrections to the tunneling delay time is discussed

Benchmarking mean-field approximations to level densities

We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point approximation to extract the number-projected densities is not a significant source of error compared to other errors inherent to the mean-field theory. We also present an alternative formulation of the saddle-point approximation that makes direct use of an approximate particle-number projection and avoids computing the usual three-dimensional Jacobian of the saddle-point integration. We find that the pairing condensate is less amenable to approximate particle-number projection methods due to the explicit violation of particle-number conservation in the pairing condensate. Nevertheless, the Hartree-Fock-Bogoliubov theory is accurate to less than one unit of entropy for $^{148}$Sm at the neutron threshold energy, which is above the pairing phase transition.

High-order boundary integral equation solution of high frequency wave scattering from obstacles in an unbounded linearly stratified medium

We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve, the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. (Δ+E+x2)u(x1,x2)=0 where E is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 102 nodes, with an effort that is independent of the frequency parameter E. By combining with a high-order Nyström quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop.

Modified saddle-point integral near a singularity for the large deviation function

Long-time-integrated quantities in stochastic processes, in or out of equilibrium, usually exhibit rare but huge fluctuations. Work or heat production is such a quantity, for which the probability distribution function displays an exponential decay characterized by the large deviation function (LDF). The LDF is often deduced from the cumulant generating function through the inverse Fourier transformation. The saddle-point integration method is a powerful technique to obtain the asymptotic results in the Fourier integral, but special care should be taken when the saddle point is located near a singularity of the integrand. In this paper, we present a modified saddle-point method to handle such a difficulty efficiently. We investigate the dissipated and injected heat production in equilibration processes with various initial conditions, for example, where the generating functions contain branch-cut singularities as well as power-law ones. Exploiting the new modified saddle-point integrations, we obtain the leading finite-time corrections for the LDFs, which are confirmed by numerical results.

Quantum statistical correlations in thermal field theories: Boundary effective theory

We show that the one-loop effective action at finite temperature for a scalar field with quartic interaction has the same renormalized expression as at zero temperature if written in terms of a certain classical field ${\ensuremath{\phi}}_{c}$, and if we trade free propagators at zero temperature for their finite-temperature counterparts. The result follows if we write the partition function as an integral over field eigenstates (boundary fields) of the density matrix element in the functional Schr\"odinger field representation, and perform a semiclassical expansion in two steps: first, we integrate around the saddle point for fixed boundary fields, which is the classical field ${\ensuremath{\phi}}_{c}$, a functional of the boundary fields; then, we perform a saddle-point integration over the boundary fields, whose correlations characterize the thermal properties of the system. This procedure provides a dimensionally reduced effective theory for the thermal system. We calculate the two-point correlation as an example.

Chiral perturbation theory in aθvacuum

We consider chiral perturbation theory (ChPT) with a non-zero theta term. Due to the CP violating term, the vacuum of chiral fields is shifted to a non-trivial element on the SU(N_f) group manifold. The CP violation also provides mixing of different CP eigenstates, between scalar and pseudoscalar, or vector and axialvector operators. We investigate upto O(theta^2) effects on the mesonic two point correlators of ChPT to the one-loop order. We also address the effects of fixing topology, by using saddle point integration in the Fourier transform with respect to theta.

Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations

The use of hyperasymptotics (H) and the Weniger transformation (WT) has been proposed, in a joint fashion, for decoding the divergent asymptotic series generated by the steepest descent on a wide class of saddle-point integrals evaluated across Stokes sets [R. Borghi, Phys. Rev. E 78, 026703 (2008)]. In the present sequel, the full development of the hyperasymptotic-Weniger transformation (H-WT) up to the second order in H is derived. Numerical experiments, carried out on several classes of saddle-point integrals, including the swallowtail diffraction catastrophe, show the effectiveness of the second-level H-WT, in particular when the integrals are evaluated beyond the asymptotic realm.

Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals

When the Weniger transformation is systematically used in resumming the asymptotic series arising from the application of the steepest descent method can display dramatic numerical instabilities that prevent it from improving the accuracy achievable via superasymptotics. In the present paper an explanation of such pathologies through the concept of resurgence, introduced by Berry and Howls several years ago within the context of hyperasymptotics, is proposed. In particular, the way the topology of the whole set of the saddles influences the resummation capabilities of the Weniger transformation is here investigated for the integrals defining the lowest-order cuspoid diffraction catastrophes. Eventually, a powerful and easily implementable resummation scheme, based on a joint use of the Weniger transformation and hyperasymptotics, is proposed for taking care of the above pathologies. Such a joint action seems to encompass the main virtues of both approaches, and the preliminary numerical results obtained from its application show that relative errors several orders of magnitude smaller than those achievable via superasymptotics can be achieved with modest implementation efforts.

Diffraction by an imperfect half plane in a bianisotropic medium

A general theory to study the electromagnetic diffraction by imperfect half planes immersed in linear homogeneous bianisotropic media is presented. The problem is formulated in terms of Wiener‐Hopf equations by deriving explicit spectral domain expressions for the characteristic impedances of bianisotropic media, which allow one to exploit their analytical properties. In the simpler case of perfect electric conducting and perfect magnetic conducting half planes, the Wiener‐Hopf equations involve matrices of order 2, which can be factorized in closed form if the constitutive tensors of the bianisotropic material are of special form. Four of these special cases are discussed in detail. In order to deal with the more general problem, a technique to numerically factorize the Wiener‐Hopf matrix kernels is presented. Our numerical approach is discussed on one example, by considering the previously unsolved problem of a perfect electric conducting half plane in a gyrotropic medium. The reported numerical results show that the diffracted field contribution is obtained by use of the saddle point integration method.

On the Probability Density Function of an Optical DPSK Signal in the Presence of Intrachannel Crosstalk and ASE Noise

The application of the saddle point integration method for evaluating the probability density function of a DPSK signal in the presence of intrachannel crosstalk and ASE noise is investigated. Numerical results for a 10 Gbit/s optically pre-amplified direct detection receiver show that the shape of the density has an asymmetric two-pronged structure. Moreover, the reliability of a saddle point approximation is also studied, and shown that this technique estimates very well the tails of the density, but lacks accuracy in the central region.

Theory of neural information processing systems

I INTRODUCTION TO NEURAL NETWORKS 1. General introduction 2. Layered networks 3. Recurrent networks with binary neurons II ADVANCED NEURAL NETWORKS 4. Competitive unsupervised learning processes 5. Bayesian techniques in supervised learning 6. Gaussian processes 7. Support vector machines for binary classification III INFORMATION THEORY AND NEURAL NETWORKS 8. Measuring information 9. Identification of entropy as an information measure 10. Building blocks of Shannon's information theory 11. Information theory and statistical inference 12. Applications to neural networks IV MACROSCOPIC ANALYSIS OF DYNAMICS 13. Network operation: macroscopic dynamics 14. Dynamics of online learning in binary perceptrons 15. Dynamics of online gradient descent learning V EQUILIBRIUM STATISTICAL MECHANICS OF NEURAL NETWORKS 16. Basics of equilibrium statistical mechanics 17. Network operation: equilibrium analysis 18. Gardner theory of task realizability APPENDICES A. Historical and bibliographical notes B. Probability theory in a nutshell C. Conditions for central limit theorem to apply D. Some simple summation identities E. Gaussian integrals and probability distributions F. Matrix identities G. The delta-distribution H. Inequalities based on convexity I. Metrics for parametrized probability distributions J. Saddle-point integration REFERENCES

Joint statistics of combined first- and second-order random processes

Two-term Volterra series are often used to describe non-linear random processes in subject areas as diverse as communication theory and the dynamics of offshore structures. It is well known that the characteristic function, the moments and the cumulants of such processes can be calculated analytically and that the probability density function can be calculated accurately and efficiently. This paper considers the joint statistics between two such random processes by deriving: (i) an exact expression for the joint characteristic function; (ii) an efficient means for calculating the joint moments; (iii) an ‘exact’ numerical means for calculating the joint probability density function (jpdf). For the special case of a combined first- and second-order process and a pure first-order process it is shown that it is possible to derive analytical expressions for the characteristic function and to calculate the jpdf accurately and efficiently using saddle-point integration. In addition to the above, the maximum entropy principle is used to calculate the jpdf.

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann?Hilbert Problem

We consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V 1 (x), V 2 (y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (‘‘windows’’), of lengths equal to the degrees of the potentials V 1 and V 2 , satisfy systems of ODE’s with polynomial coefficients as well as PDE’s (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.

Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering

A novel approach to analytically evaluate the bit error probability in optically preamplified direct-detection systems is presented, which can take into account the effects of pulse shaping, chirping, filtering at the transmitter and the receiver, both pre- and postdetection, chromatic dispersion, and ASE noise. The method is computationally very fast in that the saddle point integration method for solving the resulting line integral of a particular moment generating function is adopted. A closed-form approximation for the bit error probability is also provided, which is within 0.01 dB from the exact numerical results.

Interacting Bose Gas in an Optical Lattice

A grand canonical system of hard-core bosons in an optical lattice is considered. The bosons can occupy randomly N equivalent states at each lattice site. The limit N→∞ is solved exactly in terms of a saddle-point integration, representing a weakly-interacting Bose gas. In the limit N→∞ there is only a condensate if the fugacity of the Bose gas is larger than 1. Corrections in 1/N increase the total density of bosons but suppress the condensate. This indicates a depletion of the condensate due to increasing interaction at finite values of N.

EDGEWORTH APPROXIMATION IN THE AR(1) PROCESS WITH SOME POSSIBLY NONZERO INITIAL VALUE

where c1, c2 ≥ 0 (see Ochi (1983)). The estimator θ1,0 is the least-squares estimator of θ which minimizes ∑T t=2(Xt − θXt−1), and for the model (1.1) with initial value X0 = 0, it is also the maximum likelihood estimator of θ. Furthermore, the estimator θ1,1 is the Yule-Walker estimator, and the estimator θ1/2,1/2 is the Burg estimator (see also Daniels (1956)). The random variable θc1,c2 is a ratio of quadratic forms of correlated normal random variables. The methods for calculating the probability distribution of such ratios are given in Mathai and Provost (1992; Section 4.5). The formulae they present are complicated, and one resorts instead to numerical methods such as Imhof (1961) and Helstrom (1996). Imhof (1961) gave a numerical method for inverting the characteristic function of a linear combination of several noncentral chi-squares random variables, and Helstrom (1996) used the saddlepoint integration for the numerical inversion. The saddlepoint approximation method in statistics was originally introduced by Daniels (1954, 1956). Following Daniels (1956), Phillips (1978) and Kakizawa (1996) derived the saddlepoint density for

Performance analysis of DS/CDMA systems with shadowing and flat fading

A new method is developed for evaluating the error probability ( P e) for direct sequence, code division multiple access (DS/CDMA) wireless systems that includes the effects of shadowing and fading. The method is based on saddle point integration (SPI) of the test statistic's moment generating function (MGF) in the complex plane. The SPI method is applicable to both ideal and wireless channels. For wireless channels, a Padé approximation (PA) of the MGF, which is derived from the moments of the channel's shadowing and fading distributions, allows efficient evaluation of the P e. The SPI method can be used to model independent channels using separate shadowing and fading moments for each individual channel. The relative error between the probability density function (PDF) of the composite variate representing log-normal shadowing and Rayleigh fading and the PDF found from the inverse Laplace transform of the PA is negligible. Results show that log-normal shadowing increases the P e by 100–1000% compared to channels exhibiting fading only.