Wiener-Hopf Integral Equations Research Articles

A fast realization of preconditioned Conjugate Gradients for Wiener-Hopf integral equations

We present an efficient implementation of the Conjugate Gradients algorithm for Wiener-Hopf integral equations based on finite rank approximations of the integral operator and the corresponding preconditioner. The resulting algorithm is of linear complexity. Numerical experiments with this implementation of the preconditioned Conjugate Gradients algorithm show significant speed-up in the ill-conditioned case. This algorithm acts on ill-conditioned equations as a regularization algorithm.

Construction of preconditioners for Wiener-Hopf equations by operator splitting

In this paper, we propose a new type of preconditioners for solving finite section Wiener-Hopf integral equations ( αI + A τ ) x τ = g by the preconditioned conjugate gradient algorithm. We show that for an integer u > 1, the operator αI + A τ > can be decomposed into a sum of operators αI + P τ ( u, v) for 0 ≤ v < u. Here P τ ( u, v) are gw v circulant matrices. For u - 1, our preconditioners are defined as ( 1 u )∑ v(αI+P τ (u,v)) −1 . Thus the way the preconditioners are constructed is very similar to the approach used in the additive Schwarz method for elliptic problems. As for the convergence rate, we prove that the spectra of the resulting preconditioned operators ( 1 u )∑ v(αI+P τ (u,v)) −1][αI+A τ are clustered around 1 and thus the algorithm converges sufficiently fast. Finally, we discretize the resulting preconditioned equations by rectangular rule. Numerical results show that our methods converges faster than those preconditioned by using circulant integral operators.

On the existence of L1(R) solutions to wiener-Hopf type equations

By means of the Fourier transforms of distributions we find necessary and sufficient conditions for the existence of L1(R) solutions to Wiener-Hopf type integral equations. Thus we establish general criteria for the existence of L1(R) filters operating on the observed signal to best approximate the true signal. The theorems apply to wide sense stationary stochastic processes.

Convergence of product integration rules over (0,∞) for functions with weak singularities at the origin

In this paper we consider integrals of the form\[∫0∞e−xK(x,y)f(x)dx,\int _0^\infty {{e^{ - x}}K(x,y)f(x)dx,}\]withf∈Cp[0,∞)∩Cq(0,∞),q≥p≥0f \in {C^p}[0,\infty ) \cap {C^q}(0,\infty ),q \geq p \geq 0, andxif(p+i)(x)∈C[0,∞),i=1,…,q−p{x^i}{f^{(p + i)}}(x) \in C[0,\infty ),i = 1, \ldots ,q - p, whenq&gt;pq &gt; p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nyström method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples.

Convergence of Product Integration Rules Over (0, ∞) for Functions with Weak Singularities at the Origin

In this paper we consider integrals of the form ∫ 0 ∞ e −x K(x, y) f(x) dx, with f ∈ C p [0, ∞) ∩ C q (0, ∞), q ≥ p ≥ 0, and x i f (p+1) (x) ∈ C[0, ∞), i = 1,..., q − p, when q > p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nystrom method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples

Some boundary value problems for a beltrami equation

Many properties of solutions to Beltrami equations were established by I.N. Vekua [11], B. Bojarski [3], A. Dzhuraev [5], R.P. Gilbert [6], H. Begehr [1] and others. Based on Cauchy-type integrals for example Riemann-Hilbert as well as Riemann boundary value problems were studied. Because the kernel of the Cauchy-type integral in general is not known, only existence and uniqueness results were given. In this paper some special cases for the Beltrami equation are considered. A corresponding Cauchy formula and theorem are given where the Cauchy type integral can be represented explicitly as in the analytic case (q= 0). Riemann-Hilbert boundary value problems are solved explicitly for the half-plane and for an ellipse when q is constant. Besides the Riemann-Schwarz reflection principle is established. Using a Wiener-Hopf integral equation and convolution-type dual integral equations the solutions of some nonlocal problems are given in explicit form. For ellipticity of (1) the given function q must satisfy |q|≠1. Without loss of generality is assumed because if |q|q0>1 one can consider equation (1) for .

Fast Preconditioned Conjugate Gradient Algorithms for Wiener–Hopf Integral Equations

In this paper the authors study circulant approximations of finite sections of a Wiener–Hopf integral equation on the half-line. Such circulant operators are defined by periodic kernel functions. They approximate finite sections of the Wiener–Hopf operator within a sum of a small operator and an operator with fixed finite rank. Constructions are given of two such circulant operators and their use as preconditioners for the Conjugate Gradients algorithm is explained. Numerical examples are included to illustrate the results.

On Asymptotic Behavior at Infinity and the Finite Section Method for Integral Equations on the Half-Line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Operational Quadrature Methods for Wiener-Hopf Integral Equations

We study the numerical solution of Wiener-Hopf integral equations by a class of quadrature methods which lead to discrete Wiener-Hopf equations, with quadrature weights constructed from the Fourier transform of the kernel (or rather, from the Laplace transforms of the kernel halves). As the analytical theory of Wiener-Hopf equations is likewise based on the Fourier transform of the kernel, this approach enables us to obtain results on solvability and stability and error estimates for the discretization. The discrete Wiener-Hopf equations are solved by using an approximate Wiener-Hopf factorization obtained with FFT. Numerical experiments with the Milne equation of radiative transfer are included.

Operational quadrature methods for Wiener-Hopf integral equations

We study the numerical solution of Wiener-Hopf integral equations by a class of quadrature methods which lead to discrete Wiener-Hopf equations, with quadrature weights constructed from the Fourier transform of the kernel (or rather, from the Laplace transforms of the kernel halves). As the analytical theory of Wiener-Hopf equations is likewise based on the Fourier transform of the kernel, this approach enables us to obtain results on solvability and stability and error estimates for the discretization. The discrete Wiener-Hopf equations are solved by using an approximate Wiener-Hopf factorization obtained with FFT. Numerical experiments with the Milne equation of radiative transfer are included.

On the Exponential Convergence of Spline Approximation Methods for Wiener-Hopf Equations

We consider the approximate solution of Wiener-Hopf integral equations by Galerkin, collocation and Nystrom methods based on piecewise polynomials where accuracy is achieved by increasing simultaneously the number of mesh points and the degree of the polynomials. We look for the stability of those methods in the Lq norm, 1≤q≤∞. Provided the exact solution is analytic on the half-axis and decays exponentially at infinity, we prove an exponential rate of convergence with respect to the number of degrees of freedom.

Distributional Solutions of the Wiener-Hopf Integral and Integro-differential Equations

Distributional Solutions of the Wiener-Hopf Integral and Integro-differential Equations

Remarks on the origin of the displacement-rank concept

An account is given of how a relationship was found between some equations of Chandrasekhar for solving the Wiener-Hopf integral equations and the nonlinear Riccati differential equations encountered in linear least-squares estimation problems for systems described in state-space form. A paper by Casti, Kalaba, and Murthy (1972) provided the inspiration for seeking such a relationship and then extending it and applying it in many other fields, using the concept of displacement structure.

Analogues of Split Levinson, Schur, and Lattice Algorithms for Three-Dimensional Random Field Estimation Problems

Fast algorithms for computing the linear least-squares estimate of a three-dimensional random field from noisy observations inside a sphere are proposed. The algorithms can be viewed as three-dimensional analogues of the split Levinson, Schur, and lattice algorithms of linear prediction, since they exploit an (assumed) Toeplitz-plus-Hankel structure of the double Radon transform of the random field covariance. Therefore these algorithms require fewer computations than would the solution of the three-dimensional Wiener-Hopf integral equation. Unlike previous generalized Levinson algorithms, no quarter-plane or asymmetric half-plane support assumptions for the filter are necessary; nor is the three-dimensional filtering problem treated as a multichannel (vector) filtering problem.The algorithms work in three stages. First, the three-dimensional split Schur algorithm computes a potential from the covariance of the random field. This potential is a three-dimensional analogue of the parameter appearing in the ...

Active control of low-frequency random sound in enclosures

This paper evaluates the degree to which active methods can be used to reduce the level of sound-pressure fluctuations in an enclosure excited at low frequencies by a stationary random source. The spectral density of the total acoustic potential energy in the enclosure is used as a cost function for evaluating the global effectiveness of the technique. The enclosure is controlled by a secondary source whose output is constrained to act causally with respect to the primary source. The optimal causal filter relating the secondary source output to the primary source output can be deduced using classical Wiener filter theory. A numerical approach is adopted for the solution of the resulting Wiener–Hopf integral equation. It is shown that the resonances of the enclosure can be successfully suppressed by the action of the active control.

On spline collocation for convolution equations

We consider the numerical solution of Wiener-Hopf integral equations and Mellin convolution equations by collocation methods and their iterated and discrete variants, using piecewise polynomials as basis functions. In the present paper we obtain results on stability and optimal convergence in the Lp norm generalizing those of [4]–[6] and [9].

Convolution quadrature and discretized operational calculus. II

Operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular integrals (power and logarithmic singularities, finite part integrals), multiple timescale convolution, Volterra integral equations, Wiener-Hopf integral equations. Frequency domain conditions, which determine, the stability of such equations, can be carried over to the discretization.

A cauchy integral equation method for analytic solutions of half-space convolution equations

This paper is devoted to a method of Cauchy integral equation for the solution of half-space convolution equations. It was introduced by Frisch and Frisch to solve Wiener-Hopf integral equations with algebraically decreasing kernels, arising in non-coherent transfer with complete frequency redistribution. The standard Wiener-Hopf technique, which requires exponentially decreasing kernels, is not directly applicable to those equations. We show here that coherent transfer may also be treated by the Cauchy integral method, which draws its name from the fact that the first step is to perform an inverse Laplace transformation on the convolution equation, in order to recast it into a singular integral equation of the Cauchy-type. For complete redistribution, this Cauchy integral equation may be solved by the classical reduction technique to a boundary value problem in the complex plane. For monochromatic transfer, the situation is somewhat more complicated, because of the presence of discrete eigenvalues in the spectrum of the transport operator. Dirac distributions localized at the isolated eigenvalues must be introduced before the complex plane method of solution becomes applicable again. The analytic solution readily yields the behavior of the radiation field at infinity. The emergent radiation field is also easily obtained. Various examples chosen among standard problems are used to illustrate the method.

Analytical solution of the velocity-slip and diffuslon-slip problems by a cauchy integral method

Abstract Some one-dimensional boundary value problems of the kinetic theory of gases can be solved analytically in closed form when the Boltzmann collision operator is replaced a simple kinetic model such as the linear BGK (Bhatnagar-Gross-Krook) model. Such are the slip-flow and the diffusion slip problems. Analytical solutions were obtained by Wiener-Hopf or singular eigenfunctions expansions methods in the sixties and seventies. Here, a Cauchy integral method of solution, developed for radiative transfer problems, is applied to the slip-flow problem and to the diffusion slip problem for a binary gas mixture (with A. Latyshev1). In the CI method, which can handle Wiener-Hopf integral equations with exponentially or non-exponentially decreasing kernels, the Wiener-Hopf equation is recast into a singular integral equation of the Cauchy type, by an inverse Laplace transformation. The latter equation is solved by a classical method of reduction to a boundary value problem in the complex plane (i.e. a Riemann-Hilbert problem). The equivalence between our solutions and other other analytical representations is discussed.

Numerical solutions of integral equations on the half line

Numerical approximation schemes of quadrature type are investigated for integral equations of the form $$x(s) - \int\limits_0^\infty {k(s,t)x(t)dt = y(s)} $$ . In this paper, the hypotheses will ensure that the integral operator is compact in an appropriate Banach space. A sequel will deal with Wiener-Hopf integral equations.