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An exact parallel algorithm of dynamic mode decomposition (DMD) with Hankel matrices for large-scale flow data is proposed. The proposed algorithm enables the DMD and the Hankel DMD for large-scale data obtained by high-fidelity flow simulations, such as large-eddy simulations or direct numerical simulations using more than a billion grid points, on parallel computations without any approximations. The proposed algorithm completes the computations of the DMD by utilizing block matrices of X T X ∈ R k × k (where X ∈ R n × k is a large data matrix obtained by high-fidelity simulations, the number of snapshot data is n ≳ 10 9 , and the number of snapshots is k ≲ O ( 10 3 ) ) without any approximations: for example, the singular value decomposition of X is replaced by the eigenvalue decomposition of X T X . Then, the computation of X T X is parallelized by utilizing the domain decomposition often used in flow simulations, which reduces the memory consumption for each parallel process and wall-clock time in the DMD by a factor approximately equal to the number of parallel processes. The parallel computation with communication is performed only for X T X , allowing for high parallel efficiency under massively parallel computations. Furthermore, the proposed exact parallel algorithm is extended to the Hankel DMD without any additional parallel computations, realizing the Hankel DMD of large-scale data collected by over a billion grid points with comparable cost and memory to the DMD without Hankel matrices. Moreover, this study shows that the Hankel DMD, which has been employed to enrich information and augment rank, is advantageous for large-scale high-dimensional data collected by high-fidelity simulations in data reconstruction and predictions of future states (while prior studies have reported such advantages for low-dimensional data). Several numerical experiments using large-scale data, including laminar and turbulent flows around a cylinder and transonic buffeting flow around a full aircraft configuration, demonstrate that (i) the proposed exact parallel algorithm reproduces the existing non-parallelized Hankel DMD, (ii) the Hankel DMD for large-scale data consisting of over a billion grid points is feasible by using the proposed exact parallel algorithm with high parallel efficiency on more than 6 thousand CPU cores, and (iii) the Hankel DMD has advantages for high-dimensional data such as n ≳ 10 9 . Graphical abstract

Data completion techniques offer numerous advantages in various fields. However, completing large datasets that must satisfy specific criteria can be challenging, necessitating the use of approximative completion methods. The primary objective of this research paper is to showcase the substantial influence of the initial point selection on the completed matrix, emphasizing the distinctiveness of solutions corresponding to each initial point. However, in the case of using the same initial point, a unique solution is attainable although the employed completion methods may exhibit variations. Each method utilized for the completion of Hankel positive semidefinite matrices strives to achieve a unique solution, assuming feasibility. Nevertheless, the path to reaching this solution can differ among methods due to variations in the number of iterations and accuracy measures required for convergence. The paper will extensively explore theoretical aspects, algorithmic advancements, and empirical findings related to completing Hankel matrices through semidefinite programming (SDP) and combining SDP with second‐order cone optimization (SOCP). The study will present numerical results to support the analyses conducted.

Considering the holomorphic functions in a polycylindrical domain which are rational in some of the variables for the arbitrary fixed values of other variables, we prove that the functions are representable as a ratio of polynomials in these variables whose coefficients are holomorphic in the remaining variables. We use the Kronecker method for establishing some criterion for the rationality of a holomorphic function of one complex variable in a neighborhood of zero by using the properties of Hankel matrices.

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

In an earlier publication of these authors, a unified approach was proposed to the construction of matrix pairs that solve the -commutation problem for Toeplitz and Hankel matrices. Here, this approach is applied for deriving new classes of solutions.

A unified approach is proposed to the construction of matrix pairs that solve the ‑commutation problem for Toeplitz and Hankel matrices. For a certain particular case, a family of solutions is derived.

Annotation In their preceding publication, the authors proposed a unified approach to the construction of matrix pairs that solve the -commutation problem for Toeplitz and Hankel matrices. Here, this approach is applied to the derivation of two classes of solutions that were earlier found by V.N. Chugunov from entirely different considerations.

Spread spectrum communication is a common communication method in underwater communication. Based on the space-time processor received by the array, it can filter the signals arriving along each path separately. Combined with the diversity of space-time clusters, it can effectively improve the communication system’s reliability. The core problem of the space-time processor is the direction of arrival (DOA) and signal source number estimation. Based on the good self-coherence of the spread spectrum sequence, this paper proposes a multiple signal classification algorithm (MUSIC) for accurate DOA estimation. However, since the MUSIC algorithm uses the received signal’s covariance matrix for DOA estimation, the number of sources needs to be predicted in advance. Under a low signal-to-noise ratio (SNR), the signal eigenvalues and the noise eigenvalues of the covariance matrix differ slightly, which makes signal source number estimation difficult. To address this issue, a singular value decomposition method using the delay structure information of the array element is proposed to estimate the number of sources of the spreading sequence under a low SNR. The method proposed in this paper can well estimate the DOA of the signal under a low SNR. Meanwhile, there is no need to convert the signal to the individual sub-bands, which effectively reduces the calculation overhead. At the same time, the Hankel matrix is used to solve the problem that the MUSIC algorithm cannot accurately estimate the number of signal sources under the condition of low SNR. Compared with the conventional algorithm, the Hankel matrix can more accurately estimate the number of signal sources in the case of low SNR. Through simulation experiments, the effectiveness of our DOA estimation algorithm is validated under a low SNR.

This paper presents a modified minimal realization technique to reduce single input single output (SISO) systems from higher-order SISO systems. The reduction process is based on restructuring the Hankel matrix, which consists of two major elements, i.e., Time Moments and Markov parameters. The system transformation is executed to reduce the order of the system by maintaining the desired system properties. The modified Hankel Matrix is proposed to obtain an expected reduce order model, i.e., kth order reduced model by selecting k×k order square matrix and using Silverman’s algorithm. This paper presents a simple solution of model order reduction with the advantages of minimizing the steady-state error, fast convergence of steady-state behavior, better approximation in terms of rise time, peak time, and settling time at higher frequencies. Three different cases have been taken from the literature to validate the proposed technique with the comparisons of performance in terms of a quality check through performance indices and response matching between original and reduced-order models.

When using ultrasound to detect the thickness of protective coatings on assembled steel structures, the coatings are extremely thin, which can cause echo signals to overlap and impair the detection accuracy. Therefore, the study of the separation of the superimposed signals is essential for the precise measurement of the thickness of thinner coatings. A method for signal time domain feature extraction based on an adaptive feature dictionary and K–SVD is investigated. First, the wavelet transform, which is sensitive to singular signal values, is used to identify the extreme values of the signal and use them as the new signal to be processed. Then, the feature signal extracted by wavelet transform is transformed into Hankel matrix form, and the initial feature dictionary is constructed by period segmentation and random extraction. The optimized feature dictionary is subsequently obtained by enhancing the K–SVD algorithm. Finally, the time domain signal is reconstructed using the optimized feature dictionary. Simulations and experiments demonstrate that the method is more accurate in separating mixed signals and extracting signal time domain feature information than the conventional wavelet transform and Gabor dictionary-based MP algorithm, and that it is more advantageous in detecting the thickness of protective coatings.