Wiener-Hopf Research Articles

Plane wave diffraction by a semi‐infinite parallel‐plate waveguide with partial material loading: The case of H polarisation

AbstractThe analysis of diffraction by a semi‐infinite parallel‐plate waveguide with partial material loading is rigorously carried out using the Wiener–Hopf technique for the H‐polarised plane wave incidence. In solving the Wiener–Hopf equations, the authors apply the Modified Residue Calculus Technique (MRCT) to achieve highly accurate solutions. The scattered field in real space is explicitly derived by performing the inverse Fourier transform of the solution in the transform domain. Within the waveguide, the scattered field is represented in terms of the waveguide TM modes, while the external field is asymptotically evaluated by applying the saddle‐point method to yield a far field expression. The authors present representative numerical examples of the radar cross section for various physical parameters and discuss the far‐field scattering characteristics in detail. Comparisons with the E‐polarisation are also given.

A Fourth-Order Tensorial Wiener Filter Using the Conjugate Gradient Method

The recently developed iterative Wiener filter using a fourth-order tensorial (FOT) decomposition owns appealing performance in the identification of long length impulse responses. It relies on the nearest Kronecker product representation (with particular intrinsic symmetry features), together with low-rank approximations. Nevertheless, this new iterative filter requires matrix inversion operations when solving the Wiener–Hopf equations associated with the component filters. In this communication, we propose a computationally efficient version that relies on the conjugate gradient (CG) method for solving these sets of equations. The proposed solution involves a specific initialization of the component filters and sequential connections between the CG cycles. Different FOT-based decomposition setups are also analyzed from the point of view of the resulting parameter space. Experimental results obtained in the context of echo cancellation confirm the good behavior of the proposed approach and its superiority in comparison to the conventional Wiener filter and other decomposition-based versions.

The Effect of the Wall Thickness of the Material Loaded Cavity On the RCS Reduction

In this paper, the effect of a parallel plate waveguide's wall thickness on radar cross-section reduction (RCS) is rigorously analyzed for H-polarization by using the Wiener-Hopf Technique, when the waveguide region is loaded with dielectric material and terminated with a perfect electric conductor (PEC) plate. Transfer matrices are incorporated into the analysis to account for the effect of different material layers through continuity relations. The Fourier transforms of the diffracted field and the boundary conditions yield a modified scalar Wiener-Hopf equation of the second kind (MWHE-2). The classical procedure to solve the MWHE-2 is applied and the approximate expression of the diffracted far field is obtained. Numerical results are given by comparing with the results available in the literature for the case of the wall thickness of the cavity not being considered.

Identification of flame transfer functions during transient operation

Predicting thermoacoustic instabilities requires knowledge of the flame response to acoustic perturbations, which is characterized by the Impulse Response Function. Obtaining stability maps across an extensive parameter space is prohibitively costly with existing methods, both experimentally and numerically, as the data records must be obtained independently for each stationary operating condition. To address this problem, a nonparametric identification technique that enables the characterization of instantaneous Flame Transfer Functions (FTFs) under nonstationary operating conditions, based on a series expansion of the time-varying impulse response function (TV-IRF), is proposed. The technique is a direct extension of the Wiener-Hopf equation to nonstationary signals, yielding the linear minimum mean squared error estimator (LMMSEE) of the TV-IRF. This method is applied to data obtained from a model of a premixed, hydrogen-enriched swirl flame. It's hydrogen power fraction is rapidly varied from 12-43% over 100 milliseconds and the flame response to a band-limited, white-noise signal is computed. From the resulting input-output data, the TV-IRF is accurately estimated and the FTF for all hydrogen power fractions is obtained from a single data record. This method paves the way for cost-effective estimation of a map of FTF's from computational or experimental data, significantly reducing expenses in both cases.

Analytical and numerical study of anti-plane elastic wave scattering in a structured quadrant subjected to a boundary point load

We verify the recently developed method by Kisil (Kisil 2024 SIAM J. Appl. Math. 84 , 464–476 (doi: 10.1137/23M1562445 )) through its application in the study of the anti-plane dynamic response of an elastic square-cell lattice quadrant having a free lateral boundary that is subjected to a sinusoidal point load. The vertical boundary of the quadrant is assumed to be either fixed or free. The approach adopted utilizes the discrete Fourier transform in both principal lattice directions to reduce the governing equations to a functional equation linking both the bulk and boundary response of the system. For both problems, this functional equation is shown to reduce to scalar Wiener–Hopf equation whose solution can be exploited to determine the full response of the lattice system. The analytical solutions based on the Wiener–Hopf method are also implemented in numerical investigations that demonstrate the lattice response to various positions and frequencies of the load. In particular, this study includes dynamic regimes induced by the load that possess strong anisotropy and shows how the associated waveforms interact with the boundaries of the quadrant.

Valley-polarized edge plasmons in graphene p–n junctions with pseudomagnetic fields

Abstract Owing to the inherent characteristics of collective excitations in graphene, electrical control of edge plasmons is highly desirable for nanoplasmonic applications. This study investigates valley-polarized edge pseudomagneplasmons in a graphene p–n junction subjected to a strain-induced pseudomagnetic field. A four-component hydrodynamic model is employed and solved via the Wiener–Hopf method, revealing the coexistence of three plasmon modes, including counterpropagating acoustic edge modes, gapless topological edge states, and zero modes. The valley polarization, as determined from the numerically exact solution, is stronger than that predicted by the approximate models. Notably, the confinement of edge plasmons at the graphene p–n junction significantly exceeds that at the graphene/vacuum interface, possibly because of the electron–hole attraction. Furthermore, gate-controlled subwavelength confinement is successfully achieved by applying an appropriate gate voltage, thereby highlighting a unique and promising attribute of edge pseudomagnetoplasmons in graphene p–n junctions.

A Double Legendre Polynomial Order N Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solution, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

Markov additive friendships

The Wiener–Hopf factorisation of a Lévy or Markov additive process describes the way that it attains new extrema in terms of a pair of so-called ladder height processes. Vigon’s theory of friendship for Lévy processes addresses the inverse problem: when does a process exist which has certain prescribed ladder height processes? We give a complete answer to this problem for Markov additive processes, provide simpler sufficient conditions for constructing processes using friendship, and address in part the question of the uniqueness of the Wiener–Hopf factorisation for Markov additive processes.

The uniqueness of the Wiener–Hopf factorisation of Lévy processes and random walks

AbstractWe prove that the spatial Wiener–Hopf factorisation of a Lévy process or random walk without killing is unique.

Riemann-Hilbert problems, Toeplitz operators and ergosurfaces

The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix M depending on the Weyl coordinates ρ and v, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the (ρ, v) plane on which some elements of M(ρ, v) tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.

A simplified Wiener–Hopf factorization method for pricing double barrier options under Lévy processes

A simplified Wiener–Hopf factorization method for pricing double barrier options under Lévy processes

Recent Developments in General Quasi Variational Inequalities

In this paper, we present a number of new and known numerical techniques for solving general quasi variational inequalities, introduced by Noor [34] in 1988, using various techniques including projection, Wiener-Hopf equations, auxiliary principle, dynamical systems coupled with finite difference approach and sensitivity analysis. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis is also investigated. Some special cases are discussed as applications of the main results. Several open problems are suggested for future research.

Analytical modeling and simulations of dynamic mode-III fracture in pre-stressed dry sandy elastic continuum

A mathematical model has been developed to investigate the SIF (stress intensity factor) and COD (crack opening displacement) for the propagation of mode-III fracture influenced by propagating SH-wave in a pre-stressed dry sandy elastic continuum. The Wiener-Hopf mathematical method along with the Fourier integral transform (FIT) technique have been successively implemented to obtain the solution of the boundary value problem associated with the mathematical model. Furthermore, the closed form solutions of SIF and COD for non-static and static conditions of mode-III fracture have been obtained, and are found to be in well agreement with the existing results present in the literature. The sound effects of various affecting parameters viz. horizontal and vertical (compressive/tensile) pre-stresses, crack depth, and crack velocity on SIF and COD have been explored. In order to delineate the influences of these aforementioned parameters on SIF and COD graphically, numerical simulation has been accomplished.

Diffraction by a set of collinear cracks on a square lattice: An iterative Wiener–Hopf method

The diffraction of a time-harmonic plane wave on collinear finite defects in a square lattice is studied. This problem is reduced to a matrix Wiener–Hopf equation. This work adapts the recently developed iterative Wiener–Hopf method to this situation. The method was motivated by wave scattering in continuous media but it is shown here that it can also be employed in a discrete lattice setting. The numerical results are validated against a different method using discrete Green’s functions. Unlike the latter approach, the complexity of the present algorithm is shown to be virtually independent of the length of the cracks.

Interaction between a uniform current and a submerged cylinder in a marginal ice zone

The interaction between a uniform current with a circular cylinder submerged in a fluid covered by a semi-infinite ice sheet is considered analytically. The ice sheet is modelled as an elastic thin plate, and the fluid flow is described by the linearised velocity potential theory. The Green function or the velocity potential due to a source is first obtained. As the water surface is divided into two semi-infinite parts with different boundary conditions, the Wiener–Hopf method (WHM) offers significant advantages over alternative approaches and is consequently adopted. To do that, the distribution of the roots of the dispersion equation for fluid fully covered by an ice sheet in the complex plane is first analysed systematically, which does not seem to have been done before. The variations of these roots with the Froude number are investigated, especially their effects or factorisation and decomposition required in the WHM. The result is verified by comparing with that obtained from the matched eigenfunction expansion method. Through differentiating the Green function with respect to the source position, the potentials due to multipoles are obtained, which are employed to construct the velocity potential for the circular cylinder. Extensive results are provided for hydrodynamic forces on the cylinder and wave profiles, and some unique features are discussed. In particular, it is found that the forces can be highly oscillatory with the Froude number when the body is below the ice sheet, whereas such an oscillation does not exist when the body is below the free surface.

Identifiability and estimation of possibly non-invertible SVARMA Models: The normalised canonical WHF parametrisation

This article focuses on the parametrisation, identifiability, and (quasi-) maximum likelihood (QML) estimation of possibly non-invertible structural vector autoregressive moving average (SVARMA) models. SVAR models are routinely adopted due to their well-known implementation strategy. However, for various economic and statistical reasons, multivariate SVARMA settings are often more suitable. These settings introduce complexity in the analysis, primarily due to the presence of the moving average (MA) polynomial. We propose a novel representation of the MA polynomial matrix using the Wiener–Hopf factorization (WHF). A significant advantage of the WHF is its ability to handle possible non-invertibility and thus models with informational asymmetry between economic agents and outside observers. Since solutions of Dynamic Stochastic General Equilibrium (DSGE) models often involve this informational asymmetry, SVARMA models in WHF parametrisation can be considered data-driven alternatives to DSGE models and used for their evaluation. Furthermore, we provide low-level conditions for the asymptotic normality of the (Q)ML estimator and analytic expressions for the score and information matrix. As application, we estimate the Blanchard and Quah model, and compare our results and implied impulse response function with the ones in the SVAR model by Blanchard and Quah and a non-invertible SVARMA model by Gouriéroux and co-authors. Importantly, we have implemented this novel method in a well-documented R-package, making it readily accessible for researchers and practitioners.

Magnetoelastic effect on semi‐infinite moving crack in an orthotropic structure influenced by anti‐plane wave

AbstractA two dimensional boundary value problem of magnetoelasticity for a semi‐infinite moving crack situated between an orthotropic half‐space and an orthotropic layer has been considered. Also, it has been considered that the crack is agitated by an anti‐plane shear wave. The contemplated problem has been reduced to a typical Wiener–Hopf equation by employing Fourier integral transform to the convenient boundary condition. The accomplished Wiener–Hopf equation has been solved only for asymptotic case to get the analytical expression of stress intensity factor (SIF) and crack tearing displacement (CTD). The dependence of several parameters including the magnetoelastic coupling factor, material constants, crack propagation velocity and crack depth from the upper surface on the SIF and CTD have been illustrated using variety of graphs.

Wiener Filter Using the Conjugate Gradient Method and a Third-Order Tensor Decomposition

In linear system identification problems, the Wiener filter represents a popular tool and stands as an important benchmark. Nevertheless, it faces significant challenges when identifying long-length impulse responses. In order to address the related shortcomings, the solution presented in this paper is based on a third-order tensor decomposition technique, while the resulting sets of Wiener–Hopf equations are solved with the conjugate gradient (CG) method. Due to the decomposition-based approach, the number of coefficients (i.e., the parameter space of the filter) is greatly reduced, which results in operating with smaller data structures within the algorithm. As a result, improved robustness and accuracy can be achieved, especially in harsh scenarios (e.g., limited/incomplete sets of data and/or noisy conditions). Besides, the CG-based solution avoids matrix inversion operations, together with the related numerical and complexity issues. The simulation results are obtained in a network echo cancellation scenario and support the performance gain. In this context, the proposed iterative Wiener filter outperforms the conventional benchmark and also some previously developed counterparts that use matrix inversion or second-order tensor decompositions.

The strain gradient viscoelasticity full field solutions for Mode-I and Mode-II crack problems

The impacts of size and viscosity become distinctly evident when considering micro- and nano-scale phenomena for advanced materials with micro- and nanostructures. In this study, the mechanical behavior of the advanced material is characterized by using the strain gradient viscoelasticity theory, and a novel solution is presented for the mode-I and mode-II cracks, which is formulated based on the strain gradient viscoelasticity theory, employing the Wiener-Hopf method. Besides, the gradient-dependent viscoelastic crack solutions are directly derived by applying the correspondence principle that aligns strain gradient viscoelasticity with strain gradient elasticity within the Maxwell's standard linear solid model. Relative to the influence of elastic strain gradient effects, the involvement of viscous gradient effects instigates a reinforcement to the stress field around the crack tip, thereby offering a more reasonable representation of advanced materials crack behavior. When the viscosity effect is omitted or as time tends to infinity, the solutions based on strain gradient viscoelasticity theory converges to those on the classical strain gradient elasticity theory.

EFFICIENT EVALUATION OF DOUBLE-BARRIER OPTIONS

In the paper, we develop a very fast and accurate method for pricing double barrier options with continuous monitoring in wide classes of Lévy models; the calculations are in the dual space, and the Wiener–Hopf factorization is used. For wide regions in the parameter space, the precision of the order of [Formula: see text] is achievable in seconds, and of the order of [Formula: see text]–[Formula: see text] — in fractions of a second. The Wiener–Hopf factors and repeated integrals in the pricing formulas are calculated using sinh-deformations of the lines of integration, the corresponding changes of variables and the simplified trapezoid rule. If the Bromwich integral is calculated using the Gaver–Wynn Rho acceleration instead of the sinh-acceleration, the CPU time is typically smaller but the precision is of the order of [Formula: see text]–[Formula: see text], at best. Explicit pricing algorithms and numerical examples are for no-touch options, digitals (equivalently, for the joint distribution function of a Lévy process and its supremum and infimum processes), and call options. Several graphs are produced to explain fundamental difficulties for accurate pricing of barrier options using time discretization and interpolation-based calculations in the state space.