Wiener-Hopf Research Articles

Physical and Spectral Analysis of a Semi-Infinite Grounded Slab Illuminated by Plane Waves

In this article, we study the scattering problem of a truncated grounded slab illuminated by an arbitrarily incident <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E_{\text{z}}$ </tex-math></inline-formula> -polarized plane wave. We present a solution using the novel semianalytical spectral method based on an original extension of the Wiener–Hopf (WH) technique that uses the concept of characteristic Green’s function and the Fredholm factorization technique. The combination of these mathematical tools allows to extend the capabilities of classical WH method to a new set of problems. One of the main benefits of the proposed semianalytical solution is that it allows the computation of field components similar to what is done with closed-form spectral solutions when available. Physical phenomena, such as the reflection, diffraction, and excitation of surface/leaky waves, are reported. Numerical results validate the proposed methodology.

Wiener–Hopf factorization for arithmetic Brownian motion with time-dependent drift and volatility

In this paper we obtain a Wiener–Hopf type factorization for a real-valued arithmetic Brownian motion with time-dependent drift and volatility. To the best of our knowledge, this paper is the very first step towards realizing the objective of deriving Wiener–Hopf type factorizations for (real-valued) time-inhomogeneous Lévy processes. In order to prove our main theorem, we derive some new results regarding time-inhomogeneous noisy Wiener–Hopf factorization. We demonstrate that in the special case of the arithmetic Brownian motion with constant drift and volatility our main result agrees with classical Wiener–Hopf factorization for this particular time-homogenous Lévy process.

Recognizing the aeroacoustic information of noise radiated by an unflanged duct based on convolutional neural networks.

Accurately recognizing the aeroacoustic information of noise propagating into and radiating out of an aero-engine duct is of both fundamental and practical interest. The aeroacoustic information includes (1) the acoustic properties of the noise source, such as the frequency (f) and the circumferential and radial mode numbers (m, n), and (2) the flight conditions, including the ambient flow speed (M0) and the jet flow speed (M1). In this study, a data-driven model is developed to predict the aeroacoustic information of a simplified aero-engine duct noise from the far-field sound pressure level directivity. The model is constructed by the integration of one-dimensional convolutional layers and fully connected layers. The training and validation datasets are calculated from the analytical model for noise radiation from a semi-infinite unflanged duct based on the Wiener-Hopf method. For a single-spinning mode source, a regression model is established for f, M0, and M1 prediction, and a classification model is built up for m and n prediction. Additionally, for a multi-spinning mode source, the regression model is used to predict the coefficient of each mode. Results show that the proposed data-driven model can effectively and robustly predict the acoustic characteristics of noise propagation in and radiation out of an aero-engine bypass duct.

Radiation of sound waves by semi-perforated duct

In this paper, the Wiener-Hopf method is studied to solve the radiation of acoustic waves from a perforated part through an infinite channel. At first the boundary value problem involving a Helmholtz equation with mixed-boundary conditions is formulated into a Wiener Hopf equation by applying the Fourier transform. After that the Wiener-Hopf kernel is split into two functions that are analytic in specified regions of the complexplane and equation is solved analytically. Finally, numerical results showing the effect of the problem’s parameters (channel radius, frequency, etc.) on the radiation phenomenon are graphically displayed.

The Wiener–Hopf Equation with Probability Kernel and Submultiplicative Asymptotics of the Inhomogeneous Term

We consider the inhomogeneous Wiener–Hopf equation whose kernel is a nonarithmetic probability distribution with positive mean. The inhomogeneous term behaves like a submultiplicative function. We establish asymptotic properties of the solution to which the successive approximations converge. These properties depend on the asymptotics of the submultiplicative function.

Solvability of a class of set-valued implicit quasi-variational inequalities: A Wiener–Hopf equation method

In this paper, we consider a class of set-valued implicit quasi-variational inequalities in real Banach spaces and show its equivalence with a class of fixed point equations and a class of Wiener–Hopf equations. Using these equivalences, we propose and analyze some iterative algorithms for this class of variational inequalities. Further, we give existence of solution and discuss the convergence criteria of the sequences generated by the iterative algorithms for this class of set-valued implicit quasi-variational inequalities. The results presented in this paper generalize and unify many known results in the literature.

Running coupling and non-perturbative corrections for O(N) free energy and for disk capacitor

We reconsider the complete solution of the linear TBA equation describing the energy density of finite density states in the O(N) nonlinear sigma models by the Wiener-Hopf method. We keep all perturbative and non-perturbative contributions and introduce a running coupling in terms of which all asymptotic series appearing in the problem can be represented as pure power series without logs. We work out the first non-perturbative contribution in the O(3) case and show that (presumably because of the instanton corrections) resurgence theory fails in this example. Using the relation of the O(3) problem to the coaxial disks capacitor problem we work out the leading non-perturbative terms for the latter and show that (at least to this order) resurgence theory, in particular the median resummation prescription, gives the correct answer. We demonstrate this by comparing the Wiener-Hopf results to the high precision numerical solution of the original integral equation.

ON J-UNITARY MATRIX POLYNOMIALS

An efficient method for construction of J-unitary matrix polynomials is proposed, associated with companion matrix functions the last row of which is a polynomial in 1/t. The method relies on Wiener-Hopf factorization theory and stems from recently developed J-spectral factorization algorithm for certain Hermitian matrix functions.

Design and convergence analysis of some implicit inertial methods for quasi-variational inequalities via the Wiener–Hopf equations

In this work, various implicit methods are proposed solve the quasi-variational inequality problem. It is well known that Tseng's algorithm developed by Thong replaces the high-precision step size. To avoid this shortcoming, several new forward-backward inertial methods are proposed by adopting the technique of the Wiener–Hopf equation. The convergence of the algorithms is investigated under suitable conditions on the operators and parameters involved. The results reported in the paper are substantial improvements with respect to previous works available in the literature. Some computational simulations obtained by using a computational implementation of our techniques show that some of the present algorithms are able to approximate the optimum of standard problems faster than some algorithms already available in the specialized literature.

An explicit Wiener-Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package.

We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.

Design of multichannel adaptive filter by constructing multidimensional Wiener-Hopf equation

In the satellite-to-ground high-speed data transmission link, there are signal self-interference problems of symbols in the co-channel, as well as between orthogonal and polarized channels. A multichannel adaptive filter is designed by constructing a multichannel Wiener-Hopf equation, and the influence of five channel nonideal factors is suppressed to improve the BER performance. Experiments show that this method is effective to suppress the signal self-interference, and the BER floor is optimized from 1E-3 to 1E-7.

High Order Multi-block Boundary-value Integration Methods for Stiff ODEs

In this paper, we present a new family of multi-block boundary value integration methods based on the Enright second derivative type-methods for differential equations. We rigorously show that this class of multi-block methods are generally $A_{k_1,k_2}$-stable for all block number by verifying through employing the Wiener-Hopf factorization of a matrix polynomial to determine the root distribution of the stability polynomial. Further more, the correct implementation procedure is as well determine by Wiener-Hopf factorization. Some numerical results are presented and a comparison is made with some existing methods. The new methods which output multi-block of solutions of the ordinary differential equations on application, and are unlike the conventional linear multistep methods which output a solution at a point or the conventional boundary value methods and multi-block methods which output a block of solutions per step. The second derivative multi-block boundary value integration methods are a new approach at obtaining very large scale integration methods for the numerical solution of differential equations.

Moving semi-infinite crack between dissimilarorthotropic strips

In this paper, we conceive a semi-infinite moving crack between two bonded dissimilar orthotropic strips. Fourier transform has been adopted to reduce the mixed boundary value problem to a standard Wiener–Hopf equation. To determine the stress intensity factor (SIF) and crack opening displacement (COD), the Wiener–Hopf equation is solved in asymptotic sense for stress free boundary. Finally, the variation of SIF and COD is demonstrated graphically with respect to several orthotropic material parameters.

Random matrices associated with general barrier billiards

The paper is devoted to the derivation of random unitary matrices whose spectral statistics is the same as statistics of quantum eigenvalues of certain deterministic two-dimensional barrier billiards. These random matrices are extracted from the exact billiard quantisation condition by applying a random phase approximation for high-excited states. An important ingredient of the method is the calculation of S-matrix for the scattering in the slab with a half-plane inside by the Wiener–Hopf method. It appears that these random matrices have the form similar to the one obtained by the author in (2022 J. Phys. A: Math. Theor. 55 024001) for a particular case of symmetric barrier billiards but with different choices of parameters. The local correlation functions of the resulting random matrices are well approximated by the semi-Poisson distribution which is a characteristic feature of various models with intermediate statistics. Consequently, local spectral statistics of the considered barrier billiards is (i) universal for almost all values of parameters and (ii) well described by the semi-Poisson statistics.

Some Novel Aspects of Quasi Variational Inequalities

Quasi variational inequalities can be viewed as novel generalizations of the variational inequalities and variational principles, the origin of which can be traced back to Euler, Lagrange, Newton and Bernoulli's brothers. It is well known that quasi-variational inequalities are equivalent to the implicit fixed point problems. We consider this alternative equivalent fixed point formulation to suggest some new iterative methods for solving quasi-variational inequalities and related optimization problems using projection methods, Wiener-Hopf equations, dynamical systems, merit function and nonexpansive mappings. Convergence analysis of these methods is investigated under suitable conditions. Our results present a significant improvement of previously known methods for solving quasi variational inequalities and related optimization problems. Since the quasi variational inequalities include variational inequalities and complementarity problems as special cases. Results obtained in this paper continue to hold for these problems. Some special cases are discussed as applications of the main results. The implementation of these algorithms and comparison with other methods need further efforts.

A prediction perspective on the Wiener–Hopf equations for time series

The Wiener–Hopf equations are a Toeplitz system of linear equations that naturally arise in several applications in time series. These include the update and prediction step of the stationary Kalman filter equations and the prediction of bivariate time series. The celebrated Wiener–Hopf technique is usually used for solving these equations and is based on a comparison of coefficients in a Fourier series expansion. However, a statistical interpretation of both the method and solution is opaque. The purpose of this note is to revisit the (discrete) Wiener–Hopf equations and obtain an alternative solution that is more aligned with classical techniques in time series analysis. Specifically, we propose a solution to the Wiener–Hopf equations that combines linear prediction with deconvolution. The Wiener–Hopf solution requires the spectral factorization of the underlying spectral density function. For ease of evaluation it is often assumed that the spectral density is rational. This allows one to obtain a computationally tractable solution. However, this leads to an approximation error when the underlying spectral density is not a rational function. We use the proposed solution with Baxter's inequality to derive an error bound for the rational spectral density approximation.

Synthesis of the Current Controller of the Vector Control System for Asynchronous Traction Drive of Electric Locomotives

This paper deals with the analysis of the operating conditions of traction drives of the electric locomotives with asynchronous traction motors. The process of change of the catenary system voltage was found to have a stochastic character. The method of current controller synthesis based on the Wiener–Hopf equation was proposed to enable efficient performance of the traction drive control system under the condition of the stochastic nature of the catenary system voltage and the presence of interferences, when measuring the stator current values of the tractor motor. Performance simulation of the proposed current controller and the current controller used in the existing vector control systems of the traction drives used in the electric locomotives was implemented. The results of the performance simulation of the proposed current controller were compared with the performance of the current controller in existing vector control systems of the traction drives. The results are applicable to the design of vector control systems of traction drives in electric locomotives and to the study of the influence of performance of electric traction drives in electric locomotives on the quality indicators of the power supplied by the traction power supply system under the actual operating conditions of the locomotive.

Resolvent-based tools for optimal estimation and control via the Wiener–Hopf formalism

The application of control tools to complex flows frequently requires approximations, such as reduced-order models and/or simplified forcing assumptions, where these may be considered low rank or defined in terms of simplified statistics (e.g. white noise). In this work we propose a resolvent-based control methodology with causality imposed via a Wiener–Hopf formalism. Linear optimal causal estimation and control laws are obtained directly from full-rank, globally stable systems with arbitrary disturbance statistics, circumventing many drawbacks of alternative methods. We use efficient, matrix-free methods to construct the matrix Wiener–Hopf problem, and we implement a tailored method to solve the problem numerically. The approach naturally handles forcing terms with space–time colour; it allows inexpensive parametric investigation of sensor/actuator placement in scenarios where disturbances/targets are low rank; it is directly applicable to complex flows disturbed by high-rank forcing; it has lower cost in comparison to standard methods; it can be used in scenarios where an adjoint solver is not available; or it can be based exclusively on experimental data. The method is particularly well suited for the control of amplifier flows, for which optimal control approaches are typically robust. Validation of the approach is performed using the linearized Ginzburg–Landau equation. Flow over a backward-facing step perturbed by high-rank forcing is then considered. Sensor and actuator placement are investigated for this case, and we show that while the flow response downstream of the step is dominated by the Kelvin–Helmholtz mechanism, it has a complex, high-rank receptivity to incoming upstream perturbations, requiring multiple sensors for control.

Influence of Interface Shear Cracks Located Near the Angular Point of the Interface in a Bi-Homogeneous Body on the Stress State Formed in the Vicinity of this Point

Within the framework of the hypothesis of plane deformation, we consider a piecewise homogeneous isotropic elastic body with interface shear cracks near the angular point of the interface. The exact solution of the corresponding problem of the theory of elasticity for a wedge-shaped body is constructed by the Wiener–Hopf method. On the basis of the constructed solution, we analyze the behavior of stresses in the vicinity of the angular point.

Tensor Wiener Filter

In signal processing and data analytics, Wiener filter is a classical powerful tool to transform an input signal to match a desired or target signal by a linear time-invariant (LTI) filter. The input signal of a Wiener filter is one-dimensional while its associated least-squares solution, namely Wiener-Hopf equation, involves a two-dimensional data-array, or correlation matrix. However, the actual match should often be carried out between a multi-dimensional filtered signal-sequence, which is the output of a multi-channel filter characterized as a linear-time-invariant MIMO (multi-input and multi-output) system, and a multi-dimensional desired signal-sequence simultaneously. In the presence of such a multi-channel filter, the solution to the corresponding Wiener filter, which we call MIMO Wiener-Hopf equation now, involves a correlation tensor. Therefore, we call this optimal multi-channel filter Tensor Wiener Filter (TWF). Due to lack of the pertinent mathematical framework of needed tensor operations, TWF has never been investigated so far. Now we would like to make the first-ever attempt to establish a new mathematical framework for TWF, which relies on the inverse of the correlation tensor. We propose the new parallel block-Jacobi tensor-inversion algorithm for this tensor inversion. A typical application of the new TWF approach is illustrated as a multi-channel linear predictor (MCLP) built upon a multi-channel autoregressive (MCAR) filter with multi-dimensional input data. Numerical experiments pertaining to seismic data, optical images, and macroeconomic time-series are conducted in comparison with other existing methods. The memory- and computational-complexities corresponding to our proposed parallel block-Jacobi tensor-inversion algorithm are also studied in this paper.