Block Circulant Matrix Research Articles

XOR count and block circulant MDS matrices over finite commutative rings

<p>Block circulant MDS matrices are used in the design of linear diffusion layers for lightweight cryptographic applications. Most of the work on construction of block circulant MDS matrices focused either on finite fields or $ GL(m, \mathbb{F}_2) $. The main objective of this paper is to extend the above study of block circulant MDS matrices to finite commutative rings. Additionally, we examine the behavior of the XOR count distribution under different reducible polynomials of equal degree over $ \mathbb{F}_2 $. We show that the determinant of a block circulant matrix over a ring can be expressed in a simple form. We construct $ 4 \times 4 $ and $ 8 \times 8 $ block circulant matrices over a ring. Furthermore, for non-negative integer $ l $, we identify the conditions under which a ring $ \mathfrak{R}_l = \frac{\mathbb{F}_2[x]}{\langle (f(x))^{2^l} \rangle} $, contains a finite field of order $ 2^m $, where $ f(x) $ is an irreducible polynomial of degree $ m $. To facilitate efficient implementation, we analyze XOR distributions within specific rings, such as $ R_1 = \frac{\mathbb{F}_2[x]}{\langle (1+x^2+x^6) \rangle} $ and $ R_2 = \frac{\mathbb{F}_2[x]}{\langle (1+x^4+x^6) \rangle} $. Our calculations reveal distinct XOR distributions when utilizing two reducible polynomials of equal degree, with XOR count distributions 776 and 764, respectively. However, when using irreducible polynomials of the same degree, the XOR count distributions remain the same. This difference is advantageous for applications in lightweight cryptography.</p>

AN ORDER-P TENSOR MULTIPLICATION WITH CIRCULANT STRUCTURE

Research on mathematical operations involving multidimensional arrays or tensors has increased along with the growing applications involving multidimensional data analysis. The -product of order- tensor is one of tensor multiplications. The -product is defined using two operations that transform the multiplication of two tensors into the multiplication of two block matrices, then the result is a block matrix which is further transformed back into a tensor. The composition of both operations used in the definition of -product can transform a tensor into a block circulant matrix. This research discusses the -product of tensors based on their circulant structure. First, we present a theorem of the -product of tensors involving circulant matrices. Second, we use the definition of identity, transpose, and inverse tensors under -product operation and investigate their relationship with circulant matrices. Third, we manifest the computation of the -product involving circulant matrices. The results of the discussion show that the -product of tensors fundamentally involves circulant matrix multiplication, which means that the operation at its core relies on multiplying circulant matrices. This implies the -product operation of tensors having properties analogous to standard matrix multiplication. Furthermore, since the -product of tensors fundamentally involves circulant matrix multiplication, its computation can be simplified by diagonalizing the circulant matrix first using the discrete Fourier transform matrix. Finally, based on the obtained results, an algorithm is constructed in MATLAB to calculate the -product.

Real block-circulant matrices and DCT-DST algorithm for transformer neural network

In the encoding and decoding process of transformer neural networks, a weight matrix-vector multiplication occurs in each multihead attention and feed forward sublayer. Assigning the appropriate weight matrix and algorithm can improve transformer performance, especially for machine translation tasks. In this study, we investigate the use of the real block-circulant matrices and an alternative to the commonly used fast Fourier transform (FFT) algorithm, namely, the discrete cosine transform–discrete sine transform (DCT-DST) algorithm, to be implemented in a transformer. We explore three transformer models that combine the use of real block-circulant matrices with different algorithms. We start from generating two orthogonal matrices, U and Q. The matrix U is spanned by the combination of the reals and imaginary parts of eigenvectors of the real block-circulant matrix, whereas Q is defined such that the matrix multiplication QU can be represented in the shape of a DCT-DST matrix. The final step is defining the Schur form of the real block-circulant matrix. We find that the matrix-vector multiplication using the DCT-DST algorithm can be defined by assigning the Kronecker product between the DCT-DST matrix and an orthogonal matrix in the same order as the dimension of the circulant matrix that spanned the real block circulant. According to the experiment's findings, the dense-real block circulant DCT-DST model with largest matrix dimension was able to reduce the number of model parameters up to 41%. The same model of 128 matrix dimension gained 26.47 of BLEU score, higher compared to the other two models on the same matrix dimensions.

Efficient Implementation for SBL-Based Coherent Distributed mmWave Radar Imaging

In a distributed frequency-modulated continuous waveform (FMCW) radar system, the echo data collected are not continuous in the azimuth direction, so the imaging effect of the traditional range-Doppler (RD) algorithm is poor. Sparse Bayesian learning (SBL) is an optimization algorithm based on Bayesian theory that has been successfully applied to high-resolution radar imaging because of its strong robustness and high accuracy. However, SBL is highly computationally complex. Fortunately, with FMCW radar echo data, most of the time-consuming SBL operations involve a Toeplitz-block Toeplitz (TBT) matrix. In this article, based on this advantage, we propose a fast SBL algorithm that can be used to obtain high-angular-resolution images, in which the inverse of the TBT matrix can be transposed as the sum of the products of the block lower triangular Toeplitz matrix and the block circulant matrix by using a new decomposition method, and some of the matrix multiplications can be quickly computed using the fast Fourier transform (FFT), decreasing the computation time by several orders of magnitude. Finally, simulations and experiments were used to ensure the effectiveness of the proposed algorithm.

Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields

The Circulant Embedding Method (CEM) is a well known technique to generate stationary Gaussian Random Fields (GRF). The main idea is to embed the covariance matrix in a larger nested block circulant matrix, whose factorization can be rapidly computed thanks to the fast Fourier transform (FFT) algorithm. The CEM requires the extended matrix to be at least positive semidefinite which is proven to be the case if the enclosing domain is sufficiently large, as proven by Theorem 2.3 in [9] for cubic domains. In this paper, we generalize this theorem to the case of rectangular parallelepipeds. Then we propose a new initialization stage of the CEM algorithm that makes it possible to quickly jump to a domain size close to the one needed for the CEM algorithm to work. These domain size estimates are based on fitting functions. Examples of fitting functions are given for the Matérn family of covariances. These functions are inspired by our numerical simulations and by the theoretical work from [9]. The parameters estimation of the fitting functions is done numerically. Several numerical tests are performed to show the efficiency of the proposed algorithms, for both isotropic and anisotropic Matérn covariances.

An Arnoldi-based preconditioner for iterated Tikhonov regularization

Many problems in science and engineering give rise to linear systems of equations that are commonly referred to as large-scale linear discrete ill-posed problems. These problems arise, for instance, from the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or be very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced by applying Tikhonov regularization. We describe a novel “approximate Tikhonov regularization method” based on constructing a low-rank approximation of the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The iterative method so defined is transpose-free. Our work is inspired by a scheme by Donatelli and Hanke, whose approximate Tikhonov regularization method seeks to approximate a severely ill-conditioned block-Toeplitz matrix with Toeplitz-blocks by a block-circulant matrix with circulant-blocks. Computed examples illustrate the performance of our proposed iterative regularization method.

Line attractors of coupled ring neural networks with block circulant weight matrix

ABSTRACTThe continuous attractors of neural networks with symmetric connection weights have been studied widely. However, most of the matrices are asymmetric. In this paper, the model is coupled by some subnetworks. The connection weight matrix of each subnetwork is a circulant matrix, so the whole connection matrix is a block circulant matrix which is asymmetric. The conditions that guarantee this model with block circulant connection weight matrix to possess continuous attractor is obtained. Furthermore, the continuous attractor is a one-dimensional line attractor.

Parallel Lattice Structure for Dual Windows Computation in Multiwindow Gabor Transform

The multiwindow discrete Gabor transform (M-DGT) is a useful time-frequency analysis tool for non-stationary signal processing. Given an arbitrary Gabor frame, a parallel lattice structure of block time-recursive algorithm for fast and efficient computation of dual/analysis Gabor windows for M-DGT is presented. By using a multiple window dual Gabor frame, the dual Gabor windows can be expressed by synthesis and analysis windows with a block-circulant matrix. Then, a parallel lattice structure of block time-recursive can be derived to solve the dual Gabor windows by the block-circulant matrix computed by the fast discrete Fourier transform (FFT). When compared to three existing methods, the proposed algorithm can reduce computational complexity and save computational time. Experimental results indicate that the proposed algorithm is valid to compute the dual windows of the M-DGT, which make the algorithm attractive for fast time-frequency signal analysis and processing.

An Ultra-weak Discontinuous Galerkin Method for Schrödinger Equation in One Dimension

In this paper, we develop an ultra-weak discontinuous Galerkin method to solve the one-dimensional nonlinear Schrodinger equation. Stability conditions and error estimates are derived for the scheme with a general class of numerical fluxes. The error estimates are based on detailed analysis of the projection operator associated with each individual flux choice. Depending on the parameters, we find out that in some cases, the projection can be defined element-wise, facilitating analysis. In most cases, the projection is global, and its analysis depends on the resulting $$2\times 2$$ block-circulant matrix structures. For a large class of parameter choices, optimal a priori $$L^2$$ error estimates can be obtained. Numerical examples are provided verifying theoretical results.

Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

In a previous paper (Graham et al. in J Comput Phys 230:3668–3694, 2011), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but no convergence theory was provided. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.

Acoustic Scattering for Rotational and Translational Symmetric Structures in Nonuniform Potential Flow

An efficient approach is proposed to predict acoustic scattering with nonuniform potential flow effects for structures with rotational and translational symmetries. The convected wave equation is transformed to Helmholtz and Laplace equations using a time transformation. The boundary-element method is used to formulate scattering by rotationally symmetric structures as two separate block circulant matrix equations and, similarly, as two separate block Toeplitz matrix equations for structures with translational symmetry. Discrete Fourier transform is employed to solve the block circulant systems. The block Toeplitz systems are solved using the generalized minimal residual method along with the discrete Fourier transform. Solving the convected wave equation using structured matrices significantly reduces computational time and storage requirements. To demonstrate the application of the formulation, two exterior acoustic case studies are considered. The first case study examines acoustic scattering from a sphere submerged in potential flow under monopole source excitation. Directivity plots obtained using the proposed technique are compared with analytical results. The second case study examines flow-induced noise generated by a rigid cylinder immersed in low-Mach-number flow, with the effect of mean flow on the scattered acoustic field taken into account using nonuniform potential flow. The fluctuating flowfield is obtained using an incompressible computational fluid dynamics solver. Acoustic sources based on Lighthill’s analogy are extracted from the flowfield data using a high-order reconstruction scheme. Results from the hybrid computational fluid dynamics– boundary-element method technique are presented for turbulent flow past the cylinder, with Reynolds number based on cylinder diameter of and Mach number . The aeroacoustic results are compared with data from literature.

Acoustic scattering for 3D multi-directional periodic structures using the boundary element method.

An efficient boundary element formulation is proposed to solve three-dimensional exterior acoustic scattering problems with multi-directional periodicity. The multi-directional periodic acoustic problem is represented as a multilevel block Toeplitz matrix. By exploiting the Toeplitz structure, the computational time and storage requirements to construct and to solve the linear system of equations arising from the boundary element formulation are significantly reduced. The generalized minimal residual method is implemented to solve the linear system of equations. To efficiently calculate the matrix-vector product in the iterative algorithm, the original matrix is embedded into a multilevel block circulant matrix. A multi-dimensional discrete Fourier transform is then employed to accelerate the matrix-vector product. The proposed approach is applicable to a periodic acoustic problem for any arbitrary shape of the structure in both full space and half space. Two case studies involving sonic crystal barriers are presented. In the first case study, a sonic crystal barrier comprising rigid cylindrical scatterers is modeled. To demonstrate the effectiveness of the proposed technique, periodicity in one, two, or three directions is examined. In the second case study, the acoustic performance of a sonic crystal barrier with locally resonant C-shaped scatterers is studied.

Abstract TP47: Inverse PWI/DWI Mismatch is a Typical Feature of Acute Subcortical Ischemic Stroke

Introduction / Hypothesis: Diffusion/perfusion mismatch (MM) is commonly used to assess tissue at risk in stroke patients. In small subcortical infarcts (SSI) the reverse pattern i.e. “inverse mismatch” (IM) has been shown [1]. We aimed to evaluate the frequency and characteristics of IM in a larger cohort using two different perfusion diffusion analysis approaches. Methods: We enrolled 129 patients with SSI within 24 hours of symptom onset (1000+ study, NCT00715533) comprising 44 infarcts of the thalamus, 9 located juxtacortically, 58 in the anterior choroidal artery, 18 in the lenticulostriate territory. Lesion volumes were assessed on DWI images using AnToNIa software. To evaluate perfusion maps a time-shift insensitive technique using a block-circulant matrix for deconvolution (oSVD) [2] was employed. Perfusion lesions were evaluated on MTT and Tmax (delay >2s and >6s) maps. A perfusion/diffusion ratio of 0.7 was used to define IM and 1.3 to define mismatch (MM). Results: Results are summarized in table1. For all types of data analysis and all time points, IM was the by far most frequent pathophysiological pattern. Obviously, IM was seen most often at Tmax>6s, but it was still clearly observed with looser thresholds (Tmax>2s, MTT). At earlier time of onset IM occurred less often, but was still the most frequent pattern (p=0.0026). No significant association between infarct location and perfusion pattern was observed Conclusion: IM is the most common pathophysiological constellation in SSI. We hypothesize that this is due to a lack of collateral circulation in these “endstream infarcts” and that the diffusion changes in areas without perfusion deficit reflect cytotoxic mechanisms. 1. Doege CA et al. AJNR 2003;24:1355-63 2. Wu O et al. Magn Reson Med 2003;50:164-174

Boundary element solution for periodic acoustic problems

This work shows when using the boundary element method to solve 3D acoustic scattering problems from periodic structures, the coefficient matrix can be represented as a block Toeplitz matrix. By exploiting the Toeplitz structure, the computational time and storage requirements to construct the coefficient matrix are significantly reduced. To solve the linear system of equations, the original matrix is embedded into a larger and more structured matrix called the block circulant matrix. Discrete Fourier transform is then employed in an iterative algorithm to solve the block Toeplitz system. To demonstrate the effectiveness of the formulation for periodic acoustic problems, two exterior acoustic case studies are considered. The first case study examines a continuous structure to predict the noise generated by a sharp-edged flat plate under quadrupole excitation. Directivity plots obtained using the periodic boundary element method technique are compared with numerical results obtained using a conventional boundary element model. The second case study examines a discrete periodic structure to predict the acoustic performance of a sonic crystal noise barrier. Results for the barrier insertion loss are compared with both finite element results and available data in the literature.

Imaging method with low data rate for autonomous star trackers based on compressive sensing

Compressive sensing (CS) theory is popular and has led to the development of new imaging methods in many fields. Specifically, CS theory can reduce the sampling rate of a high-accuracy star tracker and maintain the quality of star images. However, the imaging system of a star tracker contains certain properties that cannot correspond to conventional measurement architectures. Thus, a compressive imaging system that is suitable for the star trackers used in space missions is proposed. First, the CS framework is briefly introduced and the feasibility of typical measurement architectures is analyzed. Second, an imaging method with a low data rate and based on deterministic phase modulation (DPM) is presented to form a weight block circulant matrix (WBCM) of deterministic measurement. The advantage of the proposed imaging method is determined by comparing it with the typical measurement architectures. Third, numerical simulations are conducted to investigate the performance of the proposed method. Results show that the reconstruction quality of the DPM architecture is close to that of random phase modulation architecture. Meanwhile, the WBCM measurement is better than the conventional block circulant matrix measurement. Furthermore, the statistical result significantly ensures accurate star centroid estimation and confirms the feasibility of CS application to star tracker imaging at low data rates.

Designing sparse sensing matrix for compressive sensing to reconstruct high resolution medical images

Compressive sensing theory enables faithful reconstruction of signals, sparse in domain Ψ, at sampling rate lesser than Nyquist criterion, while using sampling or sensing matrix Φ which satisfies restricted isometric property. The role played by sensing matrix Φ and sparsity matrix Ψ is vital in faithful reconstruction. If the sensing matrix is dense then it takes large storage space and leads to high computational cost. In this paper, effort is made to design sparse sensing matrix with least incurred computational cost while maintaining quality of reconstructed image. The design approach followed is based on sparse block circulant matrix (SBCM) with few modifications. The other used sparse sensing matrix consists of 15 ones in each column. The medical images used are acquired from US, MRI and CT modalities. The image quality measurement parameters are used to compare the performance of reconstructed medical images using various sensing matrices. It is observed that, since Gram matrix of dictionary matrix...

PH/PH/1 Bulk Arrival and Bulk Service Queue

This paper studies two stochastic bulk arrivals and bulk services PH/PH/1 queue Models (A) and (B) with k1 and k2 as the number of phases of PH arrival and PH service distributions respectively. The system has infinite storing capacity and the arrival and service sizes are finite valued random variables. Matrix partitioning method is used to study the models. In Model (A) the maximum of the arrival sizes is greater than the maximum of the service sizes and the infinitesimal generator is partitioned as blocks of k1k2 times the maximum of the arrival sizes for analysis. In Model (B) the maximum of the arrival sizes is less than the maximum of the service sizes. The generator is partitioned using blocks of k1k2 times the maximum of the service sizes. Block circulant matrix structure is noticed in the basic system generator. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using matrix geometric methods. Numerical examples are presented for illustration.

Comment on ‘Sparse block circulant matrices for compressed sensing’

In `Sparse block circulant matrices for compressed sensing', in order to apply Lemma 4, every off-diagonal element of Gram matrix for the sparse block circulant matrix was separated into two component sums to make sure the terms in each sum are independent. In this comment, however, the authors show that separating every element into two sums is not sufficient to guarantee the independency of the terms in each sum. The authors also prove that the entries should be split into three parts instead of two to satisfy the requirements of Lemma 4. Finally, the authors modify the deduction and the result.

M/M/1 Bulk Arrival and Bulk Service Queue with Randomly Varying Environment

This paper studies two stochastic bulk arrival and bulk service queueing Models (A) and (B) with k varying environments. The system has infinite storing capacity and the arrival and service sizes are finite valued random variables. Matrix partitioning method is used to study the models. In Model (A) the maximum of the arrival sizes in all environments is greater than the maximum of the service sizes in all environments and the infinitesimal generator is partitioned as blocks of k times the maximum of the arrival sizes for analysis and in Model (B) the maximum of the arrival sizes in all environments is less than the maximum of the service sizes in all environments where the generator is partitioned using blocks of k times the maximum of the service sizes. The basic system generator is seen as a block circulant matrix. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using the rate matrix iterated. Numerical examples are presented for illustration. Bulk arrival and bulk service M/PH/1 and PH/M/1 queues become special cases of the models when the PH generator is identified as the generator of the randomly varying environment.

Fast Algorithms for Solving FLSR-Factor Block Circulant Linear Systems and Inverse Problem ofAX=b

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLSR-factor block circulant (retrocirculant) matrix over fieldF. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a fieldFis nonsingular. Fast algorithms for solving the unique solution of the inverse problem ofAX=bin the class of the level-2 FLS(R,r)-circulant(retrocirculant) matrix of type(m,n)over fieldFare given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.