Diagonal Matrix Research Articles

Construction of vector vortex beams on hybrid-order Poincaré sphere through highly scattering media

Vector vortex beams (VVBs) have attracted extensive attention due to their unique properties and their wide applications in fields such as optical manipulation and optical imaging. However, the wavefronts of the vector vortex beams are highly scrambled when they encounter highly scattering media (HSM), such as thick biological tissues, which greatly prevents the applications of VVBs behind HSM. To address this issue, we propose a scheme to construct VVBs of freewill position on the surface of hybrid-order Poincaré sphere (HyOPS) through HSM. With the measurement of two orthogonal scalar transmission matrices, the conjugated wavefronts for constructing orbital angular momentum beams with arbitrary topological charge in right and left circularly polarized states through HSM can be calculated, respectively. When an input wavefront superimposed by the two conjugated wavefronts with an appropriate ratio and phase delay, impinges on the HSM, the desired VVB can be created through HSM. To demonstrate the viability of our scheme, a series of VVBs on different locations of various HyOPSs have been reconstructed through a ZnO scattering layer experimentally. Furthermore, to characterize the polarization distribution of the generated beams, the polarization maps of these beams are derived by measuring the four Stokes parameters, which agree well with the theoretical distributions. This work will promote the applications of VVBs in highly scattering environments.

The distribution of the product of independent variance-gamma random variables

Let X and Y be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the product XY is derived. Some basic distributional properties are also derived, including formulas for the cumulative distribution function and the characteristic function, as well as asymptotic approximations for the density, tail probabilities and the quantile function. As special cases, we deduce some key distributional properties for the product of two independent asymmetric Laplace random variables as well as the product of four jointly correlated zero mean normal random variables with a particular block diagonal covariance matrix. As a by-product of our analysis, we deduce some new reduction formulas for the Meijer G-function.

Differential privacy Kalman filtering for graphical dynamic systems: Performance improvement and privacy calibration

This paper investigates differential privacy Kalman filtering for graphical dynamic systems. An enhanced differential privacy Kalman filter is initially developed by leveraging the topology, a unique characteristic of graphical dynamic systems, to improve the filtering performance. And then upper and lower error bounds of the enhanced filter are established to evaluate the recipient's ability to infer sensitive data. Based on these error bounds, a privacy calibration guideline is provided to balance privacy and usability of sensitive data. A notable feature of the enhanced filter is that its Kalman gain matrix is a diagonal matrix, which facilitates the independent update of information at each node by reducing the coupling between nodes. This feature diminishes the accumulation of filtering errors, thereby improving the data availability for the recipient. Finally, the effectiveness of the proposed approach is verified through a simulation of epidemiological data analysis, and the results show that the enhanced filter performs well in systems with high privacy levels.

An analysis of high order FEM and IGA for explicit dynamics: Mass lumping and immersed boundaries

SummaryWe investigate the behavior of different shape functions for the discretization of hyperbolic problems. In particular, we consider classical Lagrange polynomials and B‐splines. The studies focus on the performance of the these functions as a spatial discretization approach combined with an explicit time marching scheme. In this regard, a major concern is the maximum eigenvalue that imposes restrictions on the critical time step size and suitable lumping techniques that yield a diagonal mass matrix. The accuracy of the discretization methods is assessed in an asymptotic manner in terms of the convergence of eigenvalues and eigenvectors. Further, the global accuracy is investigated in terms of the full spectrum. The results show that B‐spline discretization with a consistent mass matrix are more accurate than those based on Lagrange shape functions, which holds true in the boundary‐fitted as well as in the immersed setting. On the other hand, Lagrange shape functions are more robust with respect to standard lumping techniques, which cannot be directly applied for B‐splines without loss of accuracy. In general, we observe that none of the standard lumping schemes yields optimal results for B‐splines, even in the boundary‐fitted setting. For the immersed setting, also Lagrange shape functions show a drop in accuracy which depends on the position of the boundary that cuts the element. Several remedies are considered in order to overcome these issues, including interpolatory B‐spline bases as well as eigenvalue stabilization methods. While accuracy and stability can be improved using these remedies, we conclude from our study that it is still an open question, how to design a discretization method that achieves large critical time step sizes in combination with a diagonal mass matrix and high accuracy in the immersed setting. We note that these considerations primarily relate to linear structural dynamics applications, such as for example, structural acoustics. In nonlinear problems, such as automotive crash dynamics, other considerations predominate. An example of a one‐dimensional elastic‐plastic bar impacting a rigid wall is illustrative.

Three-dimensional variational assimilation with a multivariate background error covariance for the Model for Prediction Across Scales – Atmosphere with the Joint Effort for Data assimilation Integration (JEDI-MPAS 2.0.0-beta)

Abstract. This paper describes the three-dimensional variational (3D-Var) data assimilation (DA) system for the Model for Prediction Across Scales – Atmosphere with the Joint Effort for Data assimilation Integration (JEDI-MPAS). Its core element is a multivariate background error covariance implemented through multiple linear variable changes, including a wind variable change from stream function and velocity potential to zonal- and meridional-wind components, a vertical linear regression representing wind–mass balance, and multiplication by a diagonal matrix of error standard deviations. The univariate spatial correlations for the “unbalanced” variables utilize the Background error on Unstructured Mesh Package (BUMP), which is one of the generic components in the JEDI framework. The variable changes and univariate correlations are modeled directly on the native MPAS unstructured mesh. BUMP provides utilities to diagnose parameters of the covariance model, such as correlation lengths, from an ensemble of forecast differences, though some manual adjustment of the parameters is necessary because of mismatches between the univariate correlation function assumed by BUMP and the correlation structure in the sample of forecast differences. The resulting multivariate covariances, as revealed by single-observation tests, are qualitatively similar to those found in previous global 3D-Var systems. Month-long cycling DA experiments using a global quasi-uniform 60 km mesh demonstrate that 3D-Var, as expected, performs somewhat worse than a pure ensemble-based covariance, while a hybrid covariance, which combines that used in 3D-Var with the ensemble covariance, significantly outperforms both 3D-Var and the pure ensemble covariance. Due to its simple workflow and minimal computational requirements, the JEDI-MPAS 3D-Var system can be useful for the research community.

Cryptographic algorithm for multi-path distribution of entangled states of orbital angular momentum based on Fibonacci values

Drawing inspiration from the Fibonacci sequence and its complementary Lucas sequence, this paper introduces an innovative encryption and decryption algorithm tailored for multi-path quantum key distribution. The algorithm capitalizes on the high-quality orbital angular momentum entangled states, harnessing the mathematical elegance of Fibonacci numbers to construct block diagonal matrices. These matrices serve as the foundation for the simultaneous execution of key distribution across multiple communication paths in a structured block distribution format. The encryption process is facilitated through a combination of linear mappings, employing specific transition matrices to manage the cryptographic flow. The security underpinning of this method is firmly rooted in the Heisenberg Uncertainty Principle, a fundamental tenet of quantum mechanics, which ensures the confidentiality and integrity of the quantum communication channel. This approach paves the way for a novel encryption paradigm, fortifying the security framework of quantum communication networks.

Sequential optimal selections of single-qubit gates in parameterized quantum circuits

In variational quantum algorithms, it is important to balance conflicting requirements of expressibility and trainability of a parameterized quantum circuit (PQC). However, appropriate PQC designs are not necessarily trivial. Here, we propose an algorithm for optimizing the PQC structure, where single-qubit gates are sequentially replaced by the optimal ones via diagonalization of a matrix whose elements are evaluated on slightly modified circuits. This replacement leads to a better approximation of target states with limited circuit depth. Furthermore, we clarify the existence of a barren plateau in the sequential optimization in terms of the spectrum concentration of the matrix, which defines the cost landscape with respect to changes in the target gate. Then, we rigorously show the concentration is no faster than polynomials in the number of qubits when an n-qubit PQC depth is O(log⁡n) using local observables. Finally, numerical experiments are provided to show the convergence of our method which is faster than classical optimizers on both simulators and a real device. Our results provide evidences for sequential optimizers as better alternatives to optimize PQCs on near-term quantum devices.

A recursive trapezoid-based algorithm designed to compute the strength- and stiffness-related geometric properties of beams with polygonal cross-sections

A new recursive trapezoid-based algorithm is proposed for calculating the strength- and stiffness-related geometric characteristics of polygonal beam sections. By taking as input the coordinate vertices of the polygon, the approach may effectively be applied to determine the cross-section area and the position of its centroid along with the axial area moments of inertia and product of inertia relative to the cross-sections’ centroidal axes. As the method combines the theory of planar moments with the well-known trapezoid numerical method, it provides a straightforward tabular procedure that can be easily implemented using any spreadsheet software. In addition, aiming to determine the principal area moments of inertia, a computation matrix formalism based on the theory of orthogonal diagonalization of symmetric matrices is put forward as an alternative to the classical method of vanishing the first derivative and solving for the principal directions. Some benchmark examples are worked out to prove the relevance and accuracy of the proposed approach for several beam sections ranging from the most common triangular and quadrilateral ones to very complex open and closed polygonal profiles. The solution presented here appears to be virtually foolproof compared to the conventional one, as it does not necessitate the decomposition of complex polygonal cross-section area into elementary surfaces of different shapes, then use the parallel axes theorem, and so forth; rather, it requires only insertion of the coordinates of the vertices of the polygon, turn by turn in the clockwise order, relative to an arbitrary cartesian coordinate system, and no further steps.

Accurate complex Jacobi rotations

This note shows how to compute, to high relative accuracy under mild assumptions, complex Jacobi rotations for diagonalization of Hermitian matrices of order two, using the correctly rounded functions cr_hypot and cr_rsqrt, proposed for standardization in the C programming language as recommended by the IEEE-754 floating-point standard. The rounding to nearest (ties to even) and the non-stop arithmetic are assumed. The numerical examples compare the observed with theoretical bounds on the relative errors in the rotations’ elements, and show that the maximal observed departure of the rotations’ determinants from unity is smaller than that of the transformations computed by LAPACK.

An algebraic algorithm for the diagonalization of a biquaternion matrix in the biquaternionic mechanics

An algebraic algorithm for the diagonalization of a biquaternion matrix in the biquaternionic mechanics

Ensemble variational Bayesian approximation for the inversion and uncertainty quantification of Darcy flows in heterogeneous porous media with random parameters

The inversion and uncertainty quantification of physical parameters are important for many science and engineering problems. For nonlinear parameter inversion problems of flows in hydrocarbon reservoirs, estimating the true posterior probability distribution function for the random parameter is difficult when the dimension of the parameter is relatively high. Inversion methods such as gradient-based or heuristic optimization methods searching for the maximum a posterior are generally point-estimate, while uncertainty quantification using MCMC is computationally expensive. In the current study, a new ensemble variational Bayesian approximation method is developed to estimate the posterior probability density using an ensemble of realizations regarded as a trial sample distribution. The objective is to optimize the trial sample distribution such that the evidence lower bound is maximized indicating that the distance between the trial and true posterior distribution is minimized. A subspace is built using principle component analysis based on finite samples to obtain a diagonal positive-definite covariance matrix, and ensemble-based data assimilation is used to optimize the trial distribution efficiently. Heterogeneous 2D and 3D test cases of transient single- and two-phase Darcy flows are presented for validation.

Coprime Factor Model Reduction of Multidimensional Systems

In this work, the right coprime factor for balancing and truncating unstable systems is extended to parameter-varying multidimensional systems employing recent works on coprime factor model reduction of one-dimensional uncertain systems. Since the balanced truncation method cannot be applied directly to unstable systems, state feedback gains should be computed and incorporated in order to stabilize the given system and to be able to apply the balanced truncation technique via defining the so-called stable coprime factor, as coprime factorization overcomes the stability condition required for model reduction. Parameter-dependent state feedback and parameter-dependent Gramians are considered in this work yielding less conservative techniques. In addition, the Gramians are defined as block diagonal matrices, which are partitioned according to the structure of the multidimensional systems. The application to a simulation example demonstrates the applicability and validity of the proposed reduction approach leading to small error bounds between the full and the reduced models.

Diagonalizably realizable implies universally realizable

A spectrum $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix $A$. The spectrum $\Lambda$ is diagonalizably realizable ($\mathcal{DR}$) if the realizing matrix $A$ is diagonalizable, and $\Lambda$ is universally realizable ($\mathcal{UR}$) if it is realizable for each possible Jordan canonical form allowed by $\Lambda.$ In 1981, Minc proved that if $\Lambda$ is the spectrum of a diagonalizable positive matrix, then $\Lambda$ is universally realizable. One of the main open questions about the problem of universal realizability of spectra iswhether $\mathcal{DR}$ implies $\mathcal{UR}$. Here, we prove a surprisingly simple result, which shows how diagonalizably realizable implies universally realizable.

On Goethals and Seidel Array

Objectives: In this article, we aim to find a series of Hadamard matrices by suitable selection of the special class of matrices given in the Goethals and Seidel array and study the pattern obtained. Methods: In the presented work, the search technique of Hadamard matrices has been done by selecting special class of (0,1) negacyclic matrices instead of the back diagonal identity matrix given in Geothals and Seidel arrays and the possible existence of negacyclic matrices for the corresponding four matrices have been explored in each case. Findings: Corresponding to the special class of (0,1) negacyclic matrices, no sets of four negacyclic matrices have been obtained in the Goethal Seidel array, for even orders. For odd orders, except in the case when all four matrices are equal and the case when there is a pair of equal matrices, many outputs have been obtained for the remaining cases, the search domain being upto 11,9 and 7 for the case of two different, three different and four different matrices respectively, in the Goethal Seidel array. Novelty: The selection of special class of negacyclic matrices instead of the back diagonal identity matrix and finding the corresponding four negacyclic matrices in Goethals and Seidel arrays for constructing Hadamard matrices provides a new approach to the construction of Hadamard matrices. Keywords: Hadamard matrix, Circulant matrix, Williamson matrices, Orthogonal array, Goethals and Seidel array

A Relation between Krylov and Nielsen Complexity.

Krylov complexity and Nielsen complexity are successful approaches to quantifying quantum evolution complexity that have been actively pursued without much contact between the two lines of research. The two quantities are motivated by quantum chaos and quantum computation, respectively, while the relevant mathematics is as different as matrix diagonalization algorithms and geodesic flows on curved manifolds. We demonstrate that, despite these differences, there is a relation between the two quantities. Namely, the time average of Krylov complexity of state evolution can be expressed as a trace of a certain matrix, which also controls an upper bound on Nielsen complexity with a specific custom-tailored penalty schedule adapted to the Krylov basis.

Computing a compact local Smith–McMillan form

We define a compact local Smith–McMillan form of a rational matrix R ( λ ) as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of R ( λ ) . We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of R ( λ ) around an arbitrary complex point λ 0 , allows us to compute a compact local Smith-McMillan form of that rational matrix R ( λ ) at the point λ 0 , provided we keep track of the transformation matrices used in the rank search. It also allows us to recover the root polynomials of a polynomial matrix and root vectors of a rational matrix, at an expansion point λ 0 . Numerical tests illustrate the promising performance of the resulting algorithm.

An extended Gauss-Newton method for full-waveform inversion

Full-waveform inversion (FWI) is a large-scale nonlinear ill-posed problem for which computationally expensive Newton-type methods can become trapped in undesirable local minima, particularly when the initial model lacks a low-wavenumber component and the recorded data lack low-frequency content. A modification to the Gauss-Newton (GN) method is developed to address these issues. The standard GN system for multisource multireceiver FWI is reformulated into an equivalent matrix equation form, with the solution becoming a diagonal matrix rather than a vector as in the standard system. The search direction is transformed from a vector to a matrix by relaxing the diagonality constraint, effectively adding a degree of freedom to the subsurface offset axis. The relaxed system can be explicitly solved with only the inversion of two small matrices that deblur the data residual matrix along the source and receiver dimensions, which simplifies the inversion of the Hessian matrix. When used to solve the extended-source FWI objective function, the extended GN (EGN) method integrates the benefits of model and source extension. The EGN method effectively combines the computational effectiveness of the reduced FWI method with the robustness characteristics of extended formulations and offers a promising solution for addressing the challenges of FWI. It bridges the gap between these extended formulations and the reduced FWI method, enhancing inversion robustness while maintaining computational efficiency. The robustness and stability of the EGN algorithm for waveform inversion are demonstrated numerically.

Research on synthesis method of weakly coupled three-translation parallel mechanism with large bearing capacity

Aiming at the problem of poor bearing capacity in the direction of decoupled DOF of weakly coupled three-translation parallel mechanism, a class of new configurations of large load-bearing capacity weakly coupled three-translation parallel mechanisms with closed-loop unit has been proposed in this paper. Based on the screw theory, configuration evolution method, and graphical approach method, the synthesis method of these kinds of mechanisms has been proposed, and the decoupled criterion of this mechanism and the addition conditions of the kinematic pair diagram are given. The synthesis process has been developed to synthesize the single degree of freedom translational decoupled branch chain and 2T coupled branch chain. To better apply the weakly coupled mechanism to large load occasions, a closed-loop unit construction method has been proposed. Based on the theory of singular value decomposition, a theoretical formula for increasing the load-bearing capacity of a mechanism by constructing a closed-loop unit with redundant drives has been derived for the first time. The closed-loop unit is added to the 2T coupled branch chain and the single DOF decoupled branch chain to construct the large load-bearing capacity single DOF decoupled three-translation parallel mechanism. Kinematic analysis and static analysis have been carried out on the synthesized mechanism. The Jacobian matrix of the synthesized mechanism is a block diagonal matrix, and the bearing capacity is significantly stronger than that of the decoupled mechanism, even higher than that of the coupled parallel mechanism. The synthesized weak coupling three translation parallel mechanism with large load bearing characteristics has a good application prospect in the field of high precision and large load bearing.

Low-complexity GFDM transmission scheme for low earth orbit satellite communications

As a promising multi-carrier modulation scheme, Generalized Frequency Division Multiplexing (GFDM) has certain advantages in low earth orbit satellite communications. However, the cost of GFDM to obtain these advantages is the self-interference caused by non-orthogonality and the increase of computational complexity. This paper proposes two low-complexity transceiver schemes for GFDM based on matrix sparsity. Firstly, the paper summarizes two GFDM modulation structures, referred to as structure types I and II, based on modulation data. Next, the modulation matrix in both structures is divided into sub-matrices, and the modulation sub-matrix is made sparse by leveraging the properties of the Fourier transform, so as to reduce the complexity of the transmitter. Similarly, by performing sparse processing on the received matrix and converting it into a block diagonal matrix form, the complexity of the receiver can be effectively reduced. The computational costs of our proposed schemes are analyzed in detail and compared with the existing solutions that are known to have the lowest complexity. The simulation results demonstrate that the two proposed schemes can provide satisfactory low-complexity performance depending on the time–frequency structure.

Federal Algorithm Design and Numerical Experiments of Ensemble Kalman Filter Data Assimilation Analysis Process

Based on the federated Kalman filter proposed by Carlson with linear systems, we propose a federated computing scheme for the ensemble Kalman filter (EnKF) assimilation analysis process for nonlinear systems, and give an optimal information fusion estimation algorithm weighted by diagonal matrix under the linear minimum variance criterion, that is, the assimilation analysis values of each variable in the global estimation are linear combinations of the assimilation analysis values of the corresponding variables in the local estimation of sub-filters, and the calculation of the combination coefficients is given. The federated algorithm of the EnKF assimilation analysis process for nonlinear systems is verified by the Lorenz (1963) system.