An efficient AIM for the scattering computation of the finite sparse array

Based on the traditional adaptive integral method(AIM), a fast method called array AIM is proposed to accelerate the scattering calculation of the finite periodic array and the sparse array. On one hand, this method could eliminate the idle grids through the utilization of 5-level block-Toeplitz matrix. Furthermore, the procedure of near correction is eliminated by applying the zeros shielding technique. On the other hand, the block Jacobi preconditioning technique is used to improve the iterative convergence, and the technique of wave path difference compensation is applied to accelerate the post-processing. The numerical results show that the proposed method not only possesses good accuracy, but also has much less cost both in time and memory, in comparison with the traditional AIM. Moreover, this method could be applied to solve the scattering problems for the finite periodic array, as well as the sparse array.

Block circulant matrices and the spectra of multivariate stationary sequences

Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the first n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a financial time series.

A Modified Adaptive Integral Method for Analysis of Large-scale Finite Periodic Array

A fast algorithm based on AIM is proposed to analyze the scattering problem of the large-scale finite array. In this method, by filling zeros into the local transformation matrix, the near and far fields are isolated thoroughly to eliminate the near correction process. In the far part, a 5-level block-toeplitz matrix is employed to avoid saving the idle grids without adding artificial interfaces. In the near part, only one local cube is required to compute the local translation matrix and near impedance matrix, which can be shared by all elements. Furthermore, the block Jacobi preconditioning technique is applied to improve the convergence, and the principle of pattern multiplication is used to accelerate the calculation of the scattering pattern. Numerical results show that the proposed method can reduce not only the CPU time in filling and solving matrix but also the whole memory requirement dramatically for the large-scale finite array with large spacings.

Nehari’s Theorem for Convex Domain Hankel and Toeplitz Operators in Several Variables

We prove Nehari’s theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley–Wiener space, reads as follows. Let $\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\Xi $, consider the Hankel operator $$\Gamma_f (g)(x)=\int_{\Xi} f(x+y) g(y) \, dy, \quad x \in\Xi.$$ Then $\Gamma _f$ extends to a bounded operator on $L^2(\Xi )$ if and only if there is a bounded function $b$ on ${{\mathbb{R}}}^d$ whose Fourier transform coincides with $f$ on $2\Xi $. This special case has an immediate application in matrix extension theory: every finite multilevel block Toeplitz matrix can be boundedly extended to an infinite multilevel block Toeplitz matrix. In particular, block Toeplitz operators with blocks that are themselves Toeplitz can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.

A HYPONORMAL TOEPLITZ COMPLETION PROBLEM

In this paper we consider the following ‘Toeplitz completion’ problem: Complete the unspecified analytic Toeplitz entries of the partial block Toeplitz matrix $ \begin{linenomath} A:=\begin{bmatrix} T_{\overline\psi_1}& ?\\[4pt] \T_{\overline\psi_2} \end{bmatrix} \end{linenomath} $ to make A hyponormal, where ψi∈H∞ is a non-constant rational function for i=1,2.

A fast reduced-rank interpolation method for prestack seismic volumes that depend on four spatial dimensions

Rank reduction strategies can be employed to attenuate noise and for prestack seismic data regularization. We present a fast version of Cadzow reduced-rank reconstruction method. Cadzow reconstruction is implemented by embedding 4D spatial data into a level-four block Toeplitz matrix. Rank reduction of this matrix via the Lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise. The computational cost of the Lanczos bidiagonalization is dominated by the cost of multiplying a level-four block Toeplitz matrix by a vector. This is efficiently implemented via the 4D fast Fourier transform. The proposed algorithm significantly decreases the computational cost of rank-reduction methods for multidimensional seismic data denoising and reconstruction. Synthetic and field prestack data examples are used to examine the effectiveness of the proposed method.