Wavelet-based treatment for nonuniqueness problem of acoustic scattering using integral equations (original) (raw)

On the use of discrete wavelet transform for solving integral equations of acoustic scattering

Acoustical Physics, 2010

In this paper, the discrete wavelet transform (DWT) is used to solve the dense system of equations which arises from integral equation of acoustic scattering. The DWT using appropriate wavelet family for acquiring larger sparsification of the system matrix is used to obtain a sparse approximation to the trans formed matrix that is used in place of the original matrix in an iterative solver. Alternatively DWT is also used to design sparse preconditioners for an iterative method. Also, DWT-based preconditioners are constructed to accelerate iterative Krylov subspace methods. Convergence rates and number of operations are discussed for each case.

Neural network model for solving integral equation of acoustic scattering using wavelet basis

Communications in Numerical Methods in Engineering, 2006

In this work, neural networks (NNs) are trained in order to obtain a fast and efficient solution of the integral equation of acoustic scattering. Wavelets sparsification methods are utilized to reduce the NN size and complexity. The non-uniqueness problem which arises in solving this integral equation at characteristic frequencies can, also, be solved using such network without any additional computational load. Experimental results show excellent agreement between the NN-based solution and the analytical solution of a spherical scatterer.

An efficient method for solving the nonuniqueness problem in acoustic scattering

Communications in Numerical Methods in Engineering, 2006

The problem of acoustic wave scattering by closed objects via second kind integral equations, is considered. Based on, combined Helmholtz integral equation formulation (CHIEF) method, an e cient method for choosing and utilizing interior ÿeld relations is suggested for solving the nonuniqueness problem at the characteristic frequencies. The implementation of the algorithm fully utilizes previous computation and thus signiÿcantly reduces the CPU time compared to the usual least-squares treatment. The method is tested for acoustic wave scattering by both acoustically hard and soft spheres. Accurate results compared to the known exact solutions are obtained.

Application of wavelet transforms for solving integral equations that arise in rough-surface scattering

2001

We consider the problem of scattering a plane wave from a periodic rough surface. The scattered field is evaluated once the field on the boundary is calculated. The latter is the solution of an integral equation. In fact, different integral equation formulations are available in both coordinate and spectral space. We solve these equations using standard numerical techniques and compare the results to corresponding solutions of the equations using wavelet transform methods for sparsification of the impedance matrix. Using an energy check, the methods are shown to be highly accurate. We limit the discussion in this paper to the Dirichlet problem (scalar) or TE-polarized case for a one-dimensional surface. The boundary unknown is thus the normal derivative of the total (scalar) field or equivalently the surface current. We illustrate two conclusions. First, sparsification (using thresholded wavelet transforms) can significantly reduce accuracy. Second, the wavelet transform did not speed up the overall solution. For our examples the solution time was considerably increased when thresholded wavelet transforms were used.

Wavelet methods for the solution of wave-body problems

Fredholm integral equations of the second kind that arise in wave analysis of floating bodies are solved using a wavelet method. The two-dimensional linear wave-body problem for arrays of rectangular cylinders floating in the free surface of an otherwise unbounded fluid is considered. Both spline wavelets and the Daubechies wavelets with adaption to an interval are used as basis functions. An a priori compression strategy taking into account the singularities of the kernel of the integral equation, which arise at the corners of the geometry, is developed. The algorithm is O(n), where n is the number of unknowns. Computations of the hydrodynamic properties of the cylinders using the compression strategy are performed. The strategy is found to work well. A very high compression rate is obtained, still keeping a high accuracy of the computations. The accuracy of the potential close to the corners (singular points) is examined in a special case where an analytical solution is available.

Wavelet-based integral representation for solutions of the wave equation

Journal of Physics A: Mathematical and Theoretical, 2009

An integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built. The solutions from a wide class can be used as building blocks for the representation. Considerations are based on mathematical techniques of continuous wavelet analysis. The formulas obtained are justified from the point of view of distribution theory. A comparison of the results with those by G. Kaiser is carried out. Methods of obtaining physical wavelets are discussed.

The inverse problem of an impenetrable sound-hard body in acoustic scattering

Journal of Physics: Conference Series, 2008

We study the inverse problem of recovering the scatterer shape from the far-field pattern(FFP) of the scattered wave in presence of noise. This problem is ill-posed and is usually addressed via regularization. Instead, a direct approach to denoise the FFP using wavelet technique is proposed by us. We are interested in methods that deal with the scatterer of the general shape which may be described by a finite number of parameters. To study the effectiveness of the technique we concentrate on simple bodies such as ellipses, where the analytic solution to the forward scattering problem is known. The shape parameters are found based on a least-square error estimator. Two cases with the FFP corrupted by Gaussian noise and/or computational error from a finite element method are considered. We also consider the case where only partial data is known in the far field.

Analysis of Scattering Problems by MOM with Intervallic Wavelets and Operators

Applied Computational Electromagnetics Society Journal

In this paper a method for the solu-II. WAVELETS ON THE INTERVAL AND tion of scattering problems is proposed. In partic-OPERATORS ular the EFIE is solved by a collocation point procedure, where the unknown current density is ex-The concepts of scaling functions, wavelets, timepanded by Daubechies wavelets on the interval and scale analysis, multiresolution analysis are here the integration is performed in the wavelet domain by the integral operator, hence without the use of considered known [1]; there are many wavelet any quadrature formula. Comparison with induced bases available in the literature, and we chose the currents calculated by a standard MoM and with Daubechics Wavelets on the interval [2] for their fields calculated by a FEM code are reported.