High-order Matrices Research Articles

Convex set-oriented singular value decomposition with bounded uncertainties

Singular value decomposition (SVD) has played a crucial role in mathematics and applied mathematics, particularly in the field of practical engineering technology, which is almost indispensable in these fields. However, incomplete information caused by uncertainties from multiple sources and fluctuations in inherent properties during engineering practice can seriously interfere with obtaining singular values. Based on the convex set uncertainty theory, a novel convex set-oriented singular value decomposition (CSSVD) is proposed to overcome the limitations of using the nominal SVD method. The CSSVD aims to accurately and efficiently calculate the uncertainty bounds of singular values and vectors through the bounds and correlations of initial uncertainty parameters. Traditional probability methods rely on a sufficient sample size, and when the sample size is insufficient, accurate results cannot be obtained. However, the CSSVD method is based on set theory and can still demonstrate excellent performance in scenarios with small samples. This study investigates the detailed derivation steps of the proposed CSSVD method, taking into consideration properties such as normalization, orthogonality, and Monte Carlo simulation (MCS) to verify the accurate bounds of convex sets. The proposed method is finally applied to several numerical examples to demonstrate its superiority. This includes verifying singular values and vectors, handling high-order matrices, matrices with close singular values, and rectangular matrices. An application of image denoising is also used to evaluate the effectiveness of the proposed method.

A stochastic SIRD model with imperfect immunity for the evaluation of epidemics

Efficient assessment of epidemic phenomena has an important role in modern epidemiology, while many novel methods propose reliable estimates for epidemic evolution in a population. In this paper, we focus on the stochastic modeling of a novel epidemiological (susceptible-infected-recovered-deceased) SIRD model with imperfect immunity based on a continuous-time Markov process adapted to the characteristics of the epidemic model. We investigate several novel stochastic properties of the SIRD scheme that are both population- and individual-oriented, such as the extinction and alert time of an epidemic, in parallel with the infection and mortality times of a tagged individual. We provide propositions and detailed recursive algorithms for computing probabilities and moments, giving additional information beyond the mean and variance of the stochastic quantities. Important remarks for representing higher-order matrices and system solving are provided, decreasing significantly the operation time of these algorithms. Extensive sensitivity analysis gives light to the influence of the system's parameters, while enhancing the validity of our methodology. Unlike other analyses that focus mainly on fitting the evolution of various diseases, our goal is to complement these studies by highlighting additional noteworthy features that determine an epidemic's future. The proposed methodology is applied on the monkeypox outbreak of 2022 in India and the first COVID infection wave in Barbados. Finally, public health authorities can use the information provided by these indicators to adjust the duration of lockdowns, accordingly, achieving the best possible balance between health benefits and economic disadvantages. Simultaneously, knowledge of mortality probabilities and timing of infection can support early coordination of hospitals and intensive care units, which could notably reduce the high risk of mortality.

Can CT texture analysis parameters be used as imaging biomarkers for prediction of malignancy in canine splenic tumors?

Splenic hemangiosarcoma has morphological similarities to benign nodular hyperplasia. Computed tomography (CT) texture analysis can analyze the texture of images that the naive human eye cannot detect. Recently, there have been attempts to incorporate CT texture analysis with artificial intelligence in human medicine. This retrospective, analytical design study aimed to assess the feasibility of CT texture analysis in splenic masses and investigate predictive biomarkers of splenic hemangiosarcoma in dogs. Parameters for dogs with hemangiosarcoma and nodular hyperplasia were compared, and an independent parameter that could differentiate between them was selected. Discriminant analysis was performed to assess the ability to discriminate the two splenic masses and compare the relative importance of the parameters. A total of 23 dogs were sampled, including 16 splenic nodular hyperplasia and seven hemangiosarcoma. In each dog, total 38 radiomic parameters were extracted from first-, second-, and higher-order matrices. Thirteen parameters had significant differences between hemangiosarcoma and nodular hyperplasia. Skewness in the first-order matrix and GLRLM_LGRE and GLZLM_ZLNU in the second, higher-order matrix were determined as independent parameters. A discriminant equation consisting of skewness, GLZLM_LGZE, and GLZLM_ZLNU was derived, and the cross-validation verification result showed an accuracy of 95.7%. Skewness was the most influential parameter for the discrimination of the two masses. The study results supported using CT texture analysis to help differentiate hemangiosarcoma from nodular hyperplasia in dogs. This new diagnostic approach can be used for developing future machine learning-based texture analysis tools.

Prognostic value of intratumoral heterogeneity of preoperative 18 F-FDG PET/CT in pancreatic cancer.

The purpose of current study was to investigate the value of textural features used fluorine-18-fluorodeoxyglucose positron emission tomography/computed tomography (18 F-FDG PET/CT) in predicting recurrence-free survival (RFS) and overall survival (OS) in patients with pancreatic cancer. Seventy two patients with pancreatic cancer who underwent 18F-FDG PET/CT prior to curative surgical treatment were enrolled. Conventional parameters, such as maximum standardize uptake value (SUVmax), metabolic tumor volume (MTV) and total lesion glycolysis (TLG) were measured. Twenty-two textural features were extracted from PET images derived from first-order and 4 higher-order matrices: Grey level co-occurrence matrix (GLCM), neighborhood grey-level different matrix (NGLDM), grey-level run length matrix (GLRLM), and grey-level zone length matrix (GLZLM). Independent predictive factors for survival were determined using Cox proportional hazards regression models. The SUVmax did not have prognostic values for RFS and OS. Median values of TLG and intratumoral heterogeneity parameter (kurtosis) were 63.95, and 3.15. High TLG and high kurtosis patients group showed shorter OS. Cox proportional hazards regression analysis revealed differentiation of the tumor, kurtosis and TLG were the significant predictive factors on OS. Besides, kurtosis presented no correlation with the conventional PET parameters, such as SUVmax, MTV and TLG. This study suggests that intratumoral heterogeneity and volumetric parameter induced by 18F-FDG PET/CT could be significant prognostic surrogate markers in patients with pancreatic cancers.

TOWARDS OVERCOMING THE CURSE OF DIMENSIONALITY IN PREDICTIVE MODELLING AND UNCERTAINTY QUANTIFICATION

This invited presentation summarizes new methodologies developed by the author for performing high-order sensitivity analysis, uncertainty quantification and predictive modeling. The presentation commences by summarizing the newly developed 3rd-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for linear systems, which overcomes the “curse of dimensionality” for sensitivity analysis and uncertainty quantification of a large variety of model responses of interest in reactor physics systems. The use of the exact expressions of the 2nd-, and 3rd-order sensitivities computed using the 3rd-ASAM is subsequently illustrated by presenting 3rd-order formulas for the first three cumulants of the response distribution, for quantifying response uncertainties (covariance, skewness) stemming from model parameter uncertainties. The use of the 1st-, 2nd-, and 3rd-order sensitivities together with the formulas for the first three cumulants of the response distribution are subsequently used in the newly developed 2nd/3rd-BERRU-PM (“Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”), which aims at overcoming the curse of dimensionality in predictive modeling. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing a subjective user-defined “cost functional quantifying the discrepancies between measurements and computations.” By utilizing the 1st-, 2nd- and 3rd-order response sensitivities to combine experimental and computational information in the joint phase-space of responses and model parameters, the 2nd/3rd-BERRU-PM generalizes the current data adjustment/assimilation methodologies. Even though all of the 2nd- and 3rd-order are comprised in the mathematical framework of the 2nd/3rd-BERRU-PM formalism, the computations underlying the 2nd/3rd-BERRU-PM require the inversion of a single matrix of dimensions equal to the number of considered responses, thus overcoming the curse of dimensionality which would affect the inversion of hessian and higher-order matrices in the parameter space.

TOWARDS OVERCOMING THE CURSE OF DIMENSIONALITY IN PREDICTIVE MODELLING AND UNCERTAINTY QUANTIFICATION

This invited presentation summarizes new methodologies developed by the author for performing high-order sensitivity analysis, uncertainty quantification and predictive modeling. The presentation commences by summarizing the newly developed 3rd-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for linear systems, which overcomes the “curse of dimensionality” for sensitivity analysis and uncertainty quantification of a large variety of model responses of interest in reactor physics systems. The use of the exact expressions of the 2nd-, and 3rd-order sensitivities computed using the 3rd-ASAM is subsequently illustrated by presenting 3rd-order formulas for the first three cumulants of the response distribution, for quantifying response uncertainties (covariance, skewness) stemming from model parameter uncertainties. The use of the 1st-, 2nd-, and 3rd-order sensitivities together with the formulas for the first three cumulants of the response distribution are subsequently used in the newly developed 2nd/3rd-BERRU-PM (“Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”), which aims at overcoming the curse of dimensionality in predictive modeling. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing a subjective user-defined “cost functional quantifying the discrepancies between measurements and computations.” By utilizing the 1st-, 2nd- and 3rd-order response sensitivities to combine experimental and computational information in the joint phase-space of responses and model parameters, the 2nd/3rd-BERRU-PM generalizes the current data adjustment/assimilation methodologies. Even though all of the 2nd- and 3rd-order are comprised in the mathematical framework of the 2nd/3rd-BERRU-PM formalism, the computations underlying the 2nd/3rd-BERRU-PM require the inversion of a single matrix of dimensions equal to the number of considered responses, thus overcoming the curse of dimensionality which would affect the inversion of hessian and higher-order matrices in the parameter space.

Dynamic reduction in the calculation of bridge structures

In modern bridge construction, on the one hand, there are increasing trends towards increasing bridge spans, which requires reducing the weight of structures. On the other hand, the use of structural elements made of various composite materials is expanding, which allows to significantly reduce the weight of the bridge structures as a whole. However, the creation of new forms of span structures of bridges requires more detailed calculations in order to optimize such forms, in particular the role of calculating the dynamic impact, because with increasing spans and weight loss, increases design flexibility and sensitivity to dynamic loads. In the present paper, the problems of solving an incomplete algebraic problem of eigenvalues and eigenvectors are considered. To increase the accuracy of the calculation and exclude the use of high-order matrices, a method of sequential reduction of the stiffness and equivalent mass matrices is proposed. The method is based on the construction of partial systems using a static transformation, followed by the solution of its own problem for the partial system. In the process of solving this problem through the eigenvectors of the system, the minor unknowns are reduced to the main ones. Dynamic reduction showed high calculation accuracy..

Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

In the Lagrangian meshless (particle) methods, such as the smoothed particle hydrodynamics (SPH), moving particle semi-implicit (MPS) method and meshless local Petrov-Galerkin method based on Rankine source solution (MLPG_R), the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities (such as the viscous stresses). In some meshless applications, the Laplacians are also needed as stabilisation operators to enhance the pressure calculation. The particles in the Lagrangian methods move following the material velocity, yielding a disordered (random) particle distribution even though they may be distributed uniformly in the initial state. Different schemes have been developed for a direct estimation of second derivatives using finite difference, kernel integrations and weighted/moving least square method. Some of the schemes suffer from a poor convergent rate. Some have a better convergent rate but require inversions of high order matrices, yielding high computational costs. This paper presents a quadric semi-analytical finite-difference interpolation (QSFDI) scheme, which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices, i.e. 3 × 3 for 3D cases, compared with 6 × 6 or 10 × 10 in the schemes with the best convergent rate. Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations. The convergence, accuracy and robustness of the present schemes are compared with the existing schemes. It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures, particularly for estimating the Laplacian of given functions.

A Numerical Method for Computing the Roots of Non-Singular Complex-Valued Matrices

A method for the computation of the n th roots of a general complex-valued r × r non-singular matrix ? is presented. The proposed procedure is based on the Dunford–Taylor integral (also ascribed to Riesz–Fantappiè) and relies, only, on the knowledge of the invariants of the matrix, so circumventing the computation of the relevant eigenvalues. Several worked examples are illustrated to validate the developed algorithm in the case of higher order matrices.

Simulation of an incomplete algebraic problem of eigenvalues and vectors by the method of frequency-dynamic condensation based on FEM in the form of the classical mixed method

Relevance . Dynamic analysis of complex structures using numerical methods leads to the solution of the algebraic problem of eigenvalues and the corresponding eigenvectors of high orders. The solution of this problem for high order matrices is performed using reduction methods. One of the most effective methods is the method of sequential frequency-dynamic condensation, which allows partial consideration of the dynamic properties of the structure in the minor degrees of freedom. This allows for more accurate results compared to static condensation. Frequency-dynamic condensation is traditionally used to reduce frequency equations derived from the finite element method in the form of the displacement method or the force method. Methods. The authors have developed an algorithm for the frequency-dynamic condensation method for the frequency equation obtained on the basis of the FEM in the form of the classical mixed method. That allows to obtain not only the spectrum of the lower vibration frequencies, but also the corresponding vibration modes and the stress-strain state of the structure. Results . This article describes the algorithm and its practical implementation in the problem of dynamic analysis of a rectangular plate. The results of the numerical analysis of the problem are presented. An assessment of the accuracy of the method and recommendations for its use are given.

Semianalytical solution of the random-product problem of matrices and discrete-time random evolution

We consider the product of a large number of matrices chosen randomly (with some correlation) as $A$ or $B$. We introduce an exact functional equation, whose solution gives the limiting distribution and the Lyapunov exponent for the product of large number of matrices. Thus we suggest new semianalytical method to solve the random product problem of correlated matrices. The method can be easily generalized to high order matrices, as we deal with a larger systems of functional equations. We have provided a known numerical algorithm for the solution of the matrix product problem. We have also applied our method for a solution of discrete time random evolution. Our functional equation method provides exact transition point.

Memory-usage advantageous block recursive matrix inverse

The inversion of extremely high order matrices has been a challenging task because of the limited processing and memory capacity of conventional computers. In a scenario in which the data does not fit in memory, it is worth to consider exchanging less memory usage for more processing time in order to enable the computation of the inverse which otherwise would be prohibitive. We propose a new algorithm to compute the inverse of block partitioned matrices with a reduced memory footprint. The algorithm works recursively to invert one block of a k × k block matrix M, with k ≥ 2, based on the successive splitting of M. It computes one block of the inverse at a time, in order to limit memory usage during the entire processing. Experimental results show that, despite increasing computational complexity, matrices that otherwise would exceed the memory-usage limit can be inverted using this technique.

In Russian

In this paper heterogeneous subjects are investigated, namely laminated nanoparticles and flat-layered media. Common feature of objects under consideration is presence of surfaces divided into slabs with different physical characteristics. The aim of the work is to demonstrate large possibilities and certain preferences of transfer matrices method for solution different problems of applied physics and generalization of the results taken out by author early. The method is especially convenient for analysis of such systems under action of different external physical fields. The method makes possible to transfer boundary conditions from slab to slab, to avoid the necessity of consideration of high order algebraical systems and matrices inversion problem is reduced to theirs simple multiplication. We presented a general scheme of building transfer matrices based on conversion of partial differential equations to ordinary differential equations with using expansion by a system of orthogonal functions. In addition to known results, the transfer matrices for inhomogeneous differential equations are constructed, i.e. those for physical systems with internal energy sources. Transfer matrix method has been used to solve some applied problems, especially problems of electromagnetic scattering on ellipsoidal nanoparticles (electrostatic approximation), spherical particles (electrostatic and wave approximations), expressions for calculations of electrical energy in slabs are found. Solution of Eigen value problems for multilayer panels is constructed, design formulas for determination of oscillation Eigen frequencies of multilayer panel with using three-dimensional elasticity theory are also obtained. Design formulas and some typical artwork are presented.

Banded, stable, skew-symmetric differentiation matrices of high order

Differentiation matrices play an important role in the space discretisation of partial differential equations. The present work considers grids on a finite interval and treats homogeneous Dirichlet boundary conditions. Differentiation matrices of order 3 and 4 are derived that are banded, stable, and skew symmetric. To achieve these desirable properties, non-equidistant grids have to be considered. The impact of the use of skew-symmetric differentiation matrices is illustrated at examples of advection and diffusion equations.

SU-F-R-13: Decoding 18F-FDG Uptake Heterogeneity for Primary and Lymphoma Tumors by Using Texture Analysis in PET Images

Purpose: To explore 18F-FDG uptake heterogeneity of primary tumor and lymphoma tumor by texture features of PET image and quantify the heterogeneity difference between primary tumor and lymphoma tumor. Methods: 18 patients with primary tumor and lymphoma tumor in lung cancer were enrolled. All patients underwent whole-body 18F-FDG PET/CT scans before treatment. Texture features, based on Gray-level Co-occurrence Matrix, second and high order matrices are extracted from code using MATLAB software to quantify 18F-FDG uptake heterogeneity. The relationships of volume between energy, entropy, correlation, homogeneity and contrast were analyzed. Results: For different cases, tumor heterogeneity was not the same. Texture parameters (contrast, entropy, and correlation) of lymphoma were lower than primary tumor. On the contrast, the texture parameters (energy, homogeneity and inverse different moment) of lymphoma were higher than primary tumor. Significantly, correlations were observed between volume and energy (primary, r=−0.194, p=0.441; lymphoma, r=−0.339, p=0.582), homogeneity (primary, r=−0.146, p=0.382; lymphoma, r=−0.193, p=0.44), inverse difference moment (primary, r=−0.14, p=0.374; lymphoma, r=−0.172, p=0.414) and a positive correlation between volume and entropy (primary, r=0.233, p=0.483; lymphoma, r=0.462, p=0.680), contrast (primary, r=0.159, p=0.399; lymphoma, r=0.341, p=0.584), correlation (primary, r=0.027, p=0.165; lymphoma, r=0.046, p=0.215). For the same patient, energy for primary and lymphoma tumor is equal. The volume of lymphoma is smaller than primary tumor, but the homogeneity were higher than primary tumor. Conclusion: This study showed that there were effective heterogeneity differences between primary and lymphoma tumor by FDG-PET image texture analysis.

Steady-State Equivalent Circuit of Five-Phase Induction Machines with Different Stator Connections under Open Line Conditions

The steady-state modeling of multiphase induction machines (IMs) with m phases under open-line fault conditions is usually carried out based on symmetrical component theory. Hence, the estimation of the machine characteristic curves usually involves complex mathematical calculations and inversion of high-order matrices, which complicates the analysis. Moreover, the effect of the different subspaces’ impedances on the machine performance cannot be easily recognized using this modeling technique. Instead, this paper presents the first attempt to derive a simple equivalent circuit for a five-phase IM under open-line conditions that includes the effect of all subspaces in a single circuit. The derivation is carried out for both star and pentagon connections. The equivalent circuit can be a simple technique to explain the advantages obtained from a pentagon connection compared to the star connection under open-circuit faults. It can also be used to derive a simple expression for torque gain and torque ripple magnitude for different stator connections based on the circuit parameters. The derived equivalent circuits are verified using a 1-kW five-phase prototype machine.

TU-AB-BRA-06: Texture Feature Reproducibility Between PET/CT and PET/MR Imaging Modalities

Purpose: Radiomics involves the extraction of texture features from different imaging modalities with the purpose of developing models to predict patient treatment outcomes. The purpose of this study is to investigate texture feature reproducibility across [18F]FDG PET/CT and [18F]FDG PET/MR imaging in patients with primary malignancies. Methods: Twenty five prospective patients with solid tumors underwent clinical [18F]FDG PET/CT scan followed by [18F]FDG PET/MR scans. In all patients the lesions were identified using nuclear medicine reports. The images were co-registered and segmented using an in-house auto-segmentation method. Fifty features, based on the intensity histogram, second and high order matrices, were extracted from the segmented regions from both image data sets. One-way random-effects ANOVA model of the intra-class correlation coefficient (ICC) was used to establish texture feature correlations between both data sets. Results: Fifty features were classified based on their ICC values, which were found in the range from 0.1 to 0.86, in three categories: high, intermediate, and low. Ten features extracted from second and high-order matrices showed large ICC ≥ 0.70. Seventeen features presented intermediate 0.5 ≤ ICC ≤ 0.65 and the remaining twenty three presented low ICC ≤ 0.45. Conclusion: Features with large ICC values could be reliable candidates for quantification as they lead to similar results from both imaging modalities. Features with small ICC indicates a lack of correlation. Therefore, the use of these features as a quantitative measure will lead to different assessments of the same lesion depending on the imaging modality from where they are extracted. This study shows the importance of the need for further investigation and standardization of features across multiple imaging modalities.

Compact design of photonic crystal ring resonator 2×2 routers as building blocks for photonic networks on chip

A highly compact 2×2 photonic crystal (PhC) passive wavelength router (λ-router) is proposed. The proposed λ-router exploits two photonic crystal ring resonators (PCRRs) having a 3.2 μm diameter and a broadband PhC waveguide crossing. The router topology, allowing assembly into higher-order matrices capable of connecting multiple transmitters with multiple receivers, can be exploited as a basic building block for the design, through a compositional approach, of more complex photonic integrated networks. The design criteria, derived by applying the finite difference time domain and the plane wave expansion methods, are reported. Moreover, the analysis of a 4×4λ-router configuration, obtained by assembling four 2×2 basic switching elements, is reported to highlight the potentiality of the basic routing element to be assembled into compact higher-order matrices that exhibit small footprints of 24 μm×24 μm and a maximum crosstalk between the ports equal to −20.1 dB.

Two inverse-of-N-free methods for [formula omitted

We develop a new representation for the weighted Moore–Penrose inverse AM,N† of a matrix A. From this novel expression, we develop a Gauss–Jordan elimination method and a QR-based method for the computation of AM,N†. One major feature of our methods is that they do not need to compute the inverse of N. Thus, the new methods are less sensitive than the traditional Q-method on the conditioning of N. The former is suitable for computing the generalized inverses of matrices of low order “by hand” and the latter is efficient and robust for matrices of high order. Preliminary numerical testing indicates that the QR-based method is faster than the Matlab’s pinv function for A†.

Chebyshev Polynomials and Generalized Complex Numbers

The generalized complex numbers can be realized in terms of 2 × 2 or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of matrices and to trigonometric functions, we take the quite natural step to discuss them in the context of the theory of generalized complex numbers.We also briefly discuss the two-variable Chebyshev polynomials and their link with the third-order Hermite polynomials.