Reflection Equation Research Articles

Engineering isospectrality in multidimensional photonic systems

Abstract Selective manipulation of energy levels plays an essential role in realizing multichannel wave devices. One of the representative examples is to utilize the concept of quasi-isospectrality: a family of wave systems with an almost identical spectrum except for a part of energy levels. Most approaches toward quasi-isospectrality have employed analytical methods based on symmetry or tridiagonalization, such as supersymmetry, Householder, or Lanczos transformations. Although such analytical approaches provide deterministic and stable designs based on operator factorizations, the mathematical strictness in the factorizations, at the same time, hinders isospectral engineering in a given multidimension. Here we develop the semi-analytical method for engineering isospectrality in multidimensional photonic systems. The method provides the systematic perturbation for the target energy level shifts by decomposing the allowed form of system changes into the perturbation basis. We demonstrate the isospectrality of lower-, higher-, and random-order states while imposing the designed shifts on the other states. The stability analysis shows that the accuracy of the method is determined by the ranges of isospectral state numbers and perturbation strength. The systematic, free-form, and multidimensional natures of the proposed method show great potential for the platform-transparent design of multichannel devices.

Reducing number of gates in quantum random walk search algorithm via modification of coin operators

This paper examines a way to simplify the circuit of quantum random walk search algorithm, when the traversing coin is constructed by both generalized Householder reflection and an additional phase multiplier. If an appropriate relation between corresponding parameters is realized, our algorithm becomes more robust to deviations in the phases. In this modification marking coin is not needed, and all advantages from above mentioned optimization to the stability, are preserved. It is shown explicitly how to construct such walk coin in order to obtain more robust quantum algorithm.

Improved HLLL Lattice Basis Reduction Algorithm to Solve GNSS Integer Ambiguity

Recently, lattice theory has been widely used for integer ambiguity resolution in the Global Navigation Satellite System (GNSS). When using lattice theory to deal with integer ambiguity, we need to reduce the correlation between lattice bases to ensure the efficiency of the solution. Lattice reduction is divided into scale reduction and basis vector exchange. The scale reduction has no direct impact on the subsequent search efficiency, while the basis vector exchange directly impacts the search efficiency. Hence, Lenstra-Lenstra-Lovász (LLL) is applied in the ambiguity resolution to improve the efficiency. And based on Householder transformation, the HLLL improved method is also used. Moreover, to improve the calculation speed further, a Pivoting Householder LLL (PHLLL) method based on Householder orthogonal transformation and rotation sorting is proposed here. The idea of PHLLL method is as follows: First, a sort matrix is introduced into the lattice basis reduction process to sort the original matrix. Then, the sorted matrix is used for Householder transformation. After transformation, it needs to be sorted again, until the diagonal elements in the matrix meet the ascending order. In addition, when using the Householder image operator for orthogonalization, the old column norm is modified to obtain a new norm, reducing the number of column norm calculations. Compared with the LLL reduction algorithm and HLLL reduction algorithm, the experimental results show that the PHLLL algorithm has higher reduction efficiency and effectiveness. The theoretical superiority of the algorithm is proved.

Color image watermarking based on a fast structure-preserving algorithm of quaternion singular value decomposition

This paper presents a new color image watermarking algorithm based on a fast structure-preserving algorithm of quaternion singular value decomposition (QSVD). We first propose F structure-preserving algorithm of QSVD by using quaternion Householder transformation, then apply it to color image watermarking, finally compare it with other algorithms and get better invisibility, security and almost robustness.

A Review on Zernike Coefficient-Solving Algorithms (CSAs) Used for Integrated Optomechanical Analysis (IOA)

An integrated optomechanical analysis (IOA) can predict the response of an optomechanical system to temperature, gravity, vibrations, and other local loadings; thus, the normal operation of instruments under special conditions is guaranteed. Zernike polynomials are the most popular for fitting the IOA-derived mechanical deformation data. By solving the Zernike coefficients of all deformed optical surfaces, the relationship between aberrations and deformations can be further revealed. The process of IOA is summarized in this article. The principles of four primary Zernike coefficient-solving algorithms (CSAs) were expounded, and the corresponding applications are reviewed in detail, including the least squares method, the Gram–Schmidt orthogonalized method, the Householder transformation, and singular value decomposition (SVD). Artificial neural networks (ANNs) trained for solving a similar overdetermined set of equations are also discussed; an innovative Zernike CSA based on a one-dimensional convolutional neural network (1D-CNN) was proposed, emphasizing its potential for Zernike CSA. The feasibility of the neural network method was verified by conducting experiments on the primary mirror of the front reflection system of a space camera. This review can provide references for the precise optimization of IOA.

Fracture detection, fluid identification and brittleness evaluation in the gas-bearing reservoir with tilted transverse isotropy via azimuthal Fourier coefficients

Seismic characterization of the sweet spot plays a crucial role in the exploration and exploitation of oil and gas reservoirs. Therefore, we focused on fluid discrimination, natural fracture detection and brittleness evaluation in the reservoir permeated by an inclined set of parallel fractures using pre-stacked seismic inversion method. To this end, we proved by numerical experiments that the suggested fluid indicator is available for identifying the gas-bearing reservoir due to its high sensitivity to the gaseous hydrocarbon and extremely weak dependences on the porosity and the fracture density of the gas-bearing formation. To reduce the cumulative errors induced by indirect calculations, we derived the reflection coefficient of the TTI (tilted transverse isotropy) rock in terms of Young's modulus, Poisson's ratio, the fluid indicator and the fracture parameters and then rewrote it in the form of the Fourier series. By comparing with the published results, the derived reflectivity equation was proven rational and of satisfactory accuracy. Furthermore, we proposed a hierarchical inversion method incorporating the discrete Fourier transformation (DFT) and the Bayesian inversion strategy, which consists of three steps. Firstly, the DFT method is utilized to decouple anisotropic seismic components from the isotropic seismic component. In the last two steps, the Bayesian inversion approach is taken to acquire the stable and reliable estimations of the fluid and brittleness indicators and the fracture parameters from the second-order and zeroth-order seismic components, respectively. The synthetic example proved that we can realize rational and reliable predictions of the unknown parameters using the hierarchical method even for the case with moderate noise. The field case demonstrated that the proposed method is feasible and stable, and the suggested sweet spot indicators worked well for gaseous hydrocarbon discrimination, natural fracture detection and brittleness evaluation.

Efficient parallel reduction of bandwidth for symmetric matrices

Bandwidth reduction can be a first step in the computation of eigenvalues and eigenvectors for a wide-banded complex Hermitian (or real symmetric) matrix. We present algorithms for this reduction and the corresponding back-transformation of the eigenvectors. These algorithms rely on blocked Householder transformations, thus enabling level 3 BLAS performance, and they feature two levels of parallelism. The efficiency of our approach is demonstrated with numerical experiments.

A flexible block classical Gram–Schmidt skeleton with reorthogonalization

AbstractWe investigate a variant of the reorthogonalized block classical Gram–Schmidt method for computing the QR factorization of a full column rank matrix. Our aim is to bound the loss of orthogonality even when the first local QR algorithm is only conditionally stable. In particular, this allows the use of modified Gram–Schmidt instead of Householder transformations as the first local QR algorithm. Numerical experiments confirm the stable behavior of the new variant. We also examine the use of non‐QR local factorization and show by example that the resulting variants, although less stable, may also be applied to ill‐conditioned problems.

An improved seismic fluid identification method incorporating squirt flow and frequency-dependent fluid-solid inversion

Joint inversion of seismic amplitude and frequency information for fluid discrimination works as a popular approach for reservoir fluid identification. However, ignorance of the factor caused by fluid flow and the viscosity of oil and gas in current frequency-dependent inversion approaches leads to the inaccuracy of oil and gas prediction. Therefore, based on squirt flow and viscoelastic theory, a new frequency-dependent viscoelastic squirt-flow fluid factor is established, which is compared with other conventional fluid factors to verify its advantages in fluid detection. In addition, a new linearized reflectivity equation related to frequency-dependent viscoelastic squirt-flow fluid factor is derived. Finally, to verify the feasibility of the new fluid factor in predicting oil/gas, a frequency-dependent elastic impedance inversion method is introduced. Synthetic data and field data examples find that frequency-dependent viscoelastic squirt-flow fluid factor can eliminate the interference of “false bright spots” in conventional fluid factors inversion and has a higher vertical resolution, which demonstrated the effectiveness and stability of our method in fluid prediction.

ТРАНСФОРМАЦІЯ МЕХАНІЗМУ ДЕРЖАВНОГО РЕГУЛЮВАННЯ ЗАОЩАДЖЕНЬ ДОМОГОСПОДАРСТВ В УМОВАХ НЕСТАЦІОНАРНОЇ ЕКОНОМІКИ

The mechanism of transformation of households’ funds into credit and investment resources through a system of financial intermediaries in the open and market economy, which has a certain margin of financial stability, develops quite efficiently and steadily. However, in the countries that are characterized by economic instability, insufficiently developed financial market, certain fluctuations and unpredictable phenomena can lead to the emergence of crisis and destructive processes of this mechanism functioning, which leads to the complicated relations between economic entities and financial institutions, decrease in their confidence, deterioration of the financial status of these institutions and possible even loss of the own funds of the clients. The purpose of the article is to substantiate theoretical provisions of the transformation of the mechanism of state regulation of households’ savings in the non-stationary economy. In the article, the essence of the mechanism of state regulation of households’ savings is considered in detail, that is carried out based on the results of the content analysis of such definitions as “mechanism of state regulation of economy”, “system of state regulation of households’ savings”. As a result, it is proposed to consider the mechanism of state regulation of households’ savings as a complex of interrelated components that together allow the impact of state authorities of the economic behavior of households to form their own, temporarily free financial resources that are not used for current consumption, but are stored for future use. Within the article, the structure of this mechanism, features of changes in the economic behavior of households in an unstable economic environment are also considered in details. It is established that state authorities play a key role in ensuring stable formation and preservation of households’ savings in the non-stationary economy. Certain areas of their operation in such conditions are considered to ensure future implementation of households’ savings and attract them to the national economy system.

On the choice of sign defining Householder transformations

On the choice of sign defining Householder transformations

A Linearized Alternating Direction Method of Multipliers Algorithm for Prestack Seismic Inversion in VTI Media Using the Quadratic PP-Reflectivity Approximation

Periodic horizontal thin interlayers, or formations developing horizontally layered fractures in a homogeneous background, are commonly studied as a transverse isotropy with a vertical axis of symmetry (VTI) media. Anisotropy needs to be considered in predicting such special reservoirs, and can be inferred from the amplitude variation with offset (AVO) by the exact or linear reflectivity equations. However, the linear approximation is usually derived based on the assumptions of weak anisotropy and weak contrast, the accuracy suffers when the anisotropy or the contrast of two adjacent media increases. In addition, due to the high nonlinearity of the Graebner equation, the stability and efficiency of the exact inversion are limited. To address these issues, we propose a robust pre-stack seismic inversion for VTI media using quadratic PP-reflectivity approximation. Firstly, by deriving quadratic approximations for phase velocity and polarization, we analyze the advantage of the second-order terms at a higher-contrast interface. Then, based on perturbation theory, we derive a quadratic PP-reflectivity approximation from the exact equation. It consists of first- and second-order terms with isotropic and anisotropic components, and the accuracy is significantly improved even with enhanced anisotropy. Finally, we introduce the Hadamard product operator to linearly characterize the objective function, and develop an improved Alternating Direction Method of Multipliers (ADMM) algorithm to handle the nonlinearity of the quadratic equations. The feasibility and stability of the proposed method are tested on two datasets, that is, the logging curves used as numerical experiments and a field dataset from a shale reservoir.

Adaptive Polynomial Harmonic Distortion Compensation in Current and Voltage Transformers Through Iteratively Updated QR Factorization

Measuring current and voltage harmonics has paramount importance for improving the power quality of distribution grids. However, the achieved accuracy strongly depends on the adopted instrument transformer. The present paper proposes an adaptive technique that enables an effective compensation of both the filtering behavior and the harmonic distortion introduced by current and voltage transformers, namely the strongest nonlinear effect at low-order harmonics. The approach is based on a flexible, linear in the parameters polynomial modeling of harmonic distortion in the frequency domain. Model complexity can be different from one harmonic to the other, and it is selected through an automatic iterative process to suit the nonlinear behavior at each specific harmonic order, while avoiding overfitting. In particular, the number of parameters is increased by progressively updating the QR factorization of the regressor matrix trough Householder reflections, until a convergence condition is reached. Experimental tests performed on an inductive VT and CT highlight the effectiveness of the approach.

Христианизация античной архитектуры: крупные метрополии и далматинские провинции

The article examines the peculiarities of the processes of Christianization of architecture in the major metropolises (Philippi, Thessalonica, Athens, Corinth) of the Roman Empire, as well as in the Dalmatian provinces (Jader, Salona, Pola, small settlements and islands) on the border between them. The specifics of the religious situation in these regions and the missionary activity of Christian preachers led to the transformation or adaptation of pagan cult structures, household or residential buildings. At the same time, pagan temples were very rarely destroyed intentionally, but as a rule they became abandoned and gradually destroyed naturally, providing building material for Christian churches. The use of such building material of previous structures could be symbolic, and in some cases architectural details changed their function from constructive to liturgical. Both in the capitals and in the provinces, there was a certain tolerance when different religious buildings functioned at the same time. Christian conversions could occur while preserving the special functions of the revered place. With the advent of Christianity in the Dalmatian provinces of the Roman and then the Byzantine Empire, very conservative ancient traditions were preserved there for a long time, which allowed these lands to organically survive the process of Christianization of ancient architecture.

Estimation of Parameters in Regression Analysis Based on QR Decomposition of Rectangular Matrices by Householder Reflections

An approach to eliminate multicollinearity problems in regression analysis using QR decomposition of rectangular matrices by Householder reflection has been proposed. The reliability of this computational procedure has been proved. Povzetek: Predložen je pristup uklanjanju problema multikolinearnosti u regresijskoj analizi korištenjem QR dekompozicije pravokutnih matrica Householderovom refleksijom. Dokazana je pouzdanost ovog proračunskog postupka.

Stokes Phenomenon and Reflection Equations

In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik–Zamolodchikov equation associated to the symmetric pair \((\mathfrak {gl}_n,\mathfrak {so}_n)\), and prove that its two types of Stokes matrices satisfy the Yang–Baxter and reflection equations respectively. We briefly discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists and quantum symmetric pairs.

Quantum embedding of multi-orbital fragments using the block-Householder transformation.

Recently, some of the authors introduced the use of the Householder transformation as a simple and intuitive method for embedding local molecular fragments [see Sekaran et al., Phys. Rev. B 104, 035121 (2021) and Sekaran et al., Computation 10, 45 (2022)]. In this work, we present an extension of this approach to the more general case of multi-orbital fragments using the block version of the Householder transformation applied to the one-body reduced density matrix, unlocking the applicability to general quantum chemistry/condensed matter physics Hamiltonians. A step-by-step construction of the block Householder transformation is presented. Both physical and numerical areas of interest of the approach are highlighted. The specific mean-field (noninteracting) case is thoroughly detailed as it is shown that the embedding of a given N spin-orbital fragment leads to the generation of two separated sub-systems: (1) a 2N spin-orbitals "fragment+bath" cluster that exactly contains N electrons and (2) a remaining cluster's "environment" described by so-called core electrons. We illustrate the use of this transformation in different cases of embedding scheme for practical applications. We particularly focus on the extension of the previously introduced Local Potential Functional Embedding Theory and Householder-transformed Density Matrix Functional Embedding Theory to the case of multi-orbital fragments. These calculations are realized on different types of systems, such as model Hamiltonians (Hubbard rings) and ab initio molecular systems (hydrogen rings).

QR decomposition for the least squares method: theory and practice

QR decomposition is widely used for solving the least squares problem. However, existing materials about it may be too abstract for non-mathematicians, especially STEM students, and/or require serious background in linear algebra. The paper describes theoretical background and examples of GNU Octave compatible MATLAB scripts that give relatively simple but complete explanations about how to use QR decomposition for the least squares problem solution. Only basic knowledge of linear algebra and calculus are required. Both Givens rotations and Householder reflections usage for the linear least squares problem were considered. It was shown that the algorithm based on Givens rotations is even easier to program than explicit formation of the normal equations with subsequent usage of Gaussian elimination.

Azimuthal amplitude difference inversion constrained by azimuth velocity anisotropy

SUMMARYAmplitude variation with incident angle and azimuth (AVAZ) inversion is usually used for estimating subsurface medium properties from azimuthal stacked seismic data for fracture prediction. And weak anisotropy parameters play an important role in the fracture prediction of shale reservoirs. However, the low contribution of the anisotropic parameters to the reflection coefficient and too many condition numbers of forward solver lead to insufficient inversion stability of the anisotropic parameters. Therefore, a new azimuthal-amplitude-difference inversion method with the azimuth-velocity-anisotropic constraints is proposed to improve the stability of inversion and obtain a reasonable estimation of anisotropic parameters. The azimuthal amplitude difference AVAZ forward solver in the anisotropic medium is initially established from a rewritten approximate reflectivity equation and variable substitution. Sensitivity analysis of estimated parameters of the rewritten approximate reflectivity indicates that our approach is sufficiently suitable for seismic inversion of anisotropic parameters. Furthermore, an azimuthal amplitude difference Bayesian AVAZ inversion approach with azimuth-velocity-anisotropic constraint is developed to improve the stability of the inversion. We add Gaussian noise to the synthetic seismic records and assume the Cauchy distributions as a prior constraint on the model parameters. The anisotropic information obtained from the azimuth-velocity-anisotropic inversion is utilized as a constraint for improving the stability of inversion. Synthetic data tests reveal that the anisotropic parameters can be estimated stably even with moderate noise. Field data tests illustrate the feasibility and reliability of the presented inversion method for estimating anisotropic parameters in a shale reservoir with vertical or near-vertical fractures.

Privacy-Preserving and Verifiable Cloud-Aided Disease Diagnosis and Prediction With Hyperplane Decision-Based Classifier

With the vigorous development and gradual maturity of machine learning (ML) technologies, the AI-assisted disease diagnosis and prediction ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {AADP}$ </tex-math></inline-formula> ) system has been extensively studied and can be expected to be intensively deployed in the real world. However, as the scale of ML data increases exponentially, the training and application of ML models impose a great burden on resource-constrained terminals. Designing cloud/edge server-aided <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {AADP}$ </tex-math></inline-formula> protocols is becoming a popular topic. Whereas, the sensitivity of ML data, the intellectual property of ML models, and the uncontrollability of servers bring great security challenges to this promising computing paradigm. In this article, we initialize a new four-party framework for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {AADP}$ </tex-math></inline-formula> system which consists of users, third-party test institution, AI doctor, and cloud/edge server. With this framework, we design two high-efficiency and secure outsourcing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {AADP}$ </tex-math></inline-formula> protocols under two different security models. By comprehensively employing secure hash functions, Householder transformations, and random permutations, we realize the following design objectives: 1) user’s actual identification is invisible to the other parties; 2) user’s feature vector is blinded to the AI doctor and the server; 3) the ML model of the AI doctor is confidential to the server; 4) AI doctor can obtain decent computational savings compared with achieving the diagnosis task by itself; and 5) AI doctor can verify the server’s misbehaviors with a nonnegligible probability under the security model with a fully malicious server. We argue these claims with rigid theoretical proofs and corroborate them with extensive experimental analysis.