Cholesky Decomposition Research Articles

A modified Levenberg–Marquardt algorithm for low order-value optimization problem

AbstractIn this paper, we consider a modified Levenberg–Marquardt algorithm for Low Order Value Optimization problems(LOVO). In the algorithm, we obtain the search direction by a combination of LM steps and approximate LM steps, and solve the subproblems therein by QR decomposition or cholesky decomposition. We prove the global convergence of the algorithm theoretically and discuss the worst-case complexity of the algorithm. Numerical results show that the algorithm in this paper is superior in terms of number of iterations and computation time compared to both LM-LOVO and GN-LOVO algorithm.

Impacts of Shocks to U.S. Macroeconomic Variables on the Regional Economy of Georgia

This paper develops a way to analyze a regional economy and a large economy within a model. Using structural autoregression with block exogeneity and mutual independence assumptions as well as Cholesky factorization, we quantify the impacts of shocks to U.S. macroeconomic variables on the regional economy of Georgia. We found that the largest impact on Georgia's aggregate variables - consumer prices, real per capita personal income, and total employment - is the federal funds rate. For other shocks, Georgia's personal income is most affected by an increase in commodity prices, while Georgia's consumer prices are most affected by U.S. inflation. At the industry level, we found that the impact of real GDP is the largest and the impact of M2 is the smallest on output and employment in Georgia's industries, although the magnitude of the impact on employment is smaller and more short-lived than on output.

Partitioning the Genomic Components of Behavioral Disinhibition and Substance Use (Disorder) Using Genomic Structural Equation Modeling.

Externalizing behaviors encompass manifestations of risk-taking, self-regulation, aggression, sensation-/reward-seeking, and impulsivity. Externalizing research often includes substance use (SUB), substance use disorder (SUD), and other (non-SUB/SUD) "behavioral disinhibition" (BD) traits. Genome-wide and twin research have pointed to overlapping genetic architecture within and across SUB, SUD, and BD. We created single-factor measurement models-each describing SUB, SUD, or BD traits-based on mutually exclusive sets of European ancestry genome-wide association study (GWAS) statistics exploring externalizing variables. We then assessed the partitioning of genetic covariance among the three facets using correlated factors models and Cholesky decomposition. Even when the residuals for indicators relating to the same substance were correlated across the SUB and SUD factors, the two factors yielded a large correlation (rg=0.803). BD correlated strongly with the SUD (rg=0.774) and SUB (rg=0.778)factors. In our initial decompositions, 33% of total BD variance remained after partialing out SUD and SUB. The majority of covariance between BD and SUB and between BD and SUD was shared across all factors, and, within these models, only a small fraction of the total variation in BD operated via an independent pathway with SUD or SUB outside of the other factor. When only nicotine/tobacco, cannabis, and alcohol were included for the SUB/SUD factors, their correlation increased to rg=0.861; in corresponding decompositions, BD-specific variance decreased to 27%. Further research can better elucidate the properties of BD-specific variation by exploring its genetic/molecular correlates.

Adaptive digital antenna array signal processing algorithm based on LDL decomposition

Two algorithms for inverting correlation matrices of digital antenna array signals are compared: LDL decomposition and the fringing method. The idea of comparing algorithms was to take into account the limitations on the length of the input data mantissa in the process of space-time processing using FPGAs. It has been found that rounding errors in the signal processing process in FPGAs can lead to the destruction of the directivity pattern. This is due to the fact that when implementing various algorithms for matrix inversion, the distribution of errors in the channels becomes interconnected. This means that when implementing adaptive antenna arrays on FPGAs, it is necessary to carefully choose the length of the mantissa and the conversion algorithm. In this regard, an algorithm for signal processing in an adaptive antenna array is proposed, its difference from the known ones is that when implemented on FPGAs, floating-point arithmetic of limited mantissa bit is used. When comparing two computationally efficient matrix inversion algorithms (the LDL method and the fringing method), the LDL method has higher stability. It was found that with an increase in the rank of the correlation matrix, the effect of rounding errors will be stronger. In this regard, it is necessary to determine in advance at what ratio of power sources and internal noise it is planned to use an adaptive processor.

Simulation of multivariate ergodic stochastic processes using adaptive spectral sampling and non-uniform fast Fourier transform

The simulation of multivariate ergodic stochastic processes is critical for structural dynamic analysis and reliability evaluation. Although the traditional spectral representation method (SRM) has a wide application in many areas, it is highly inefficient in simulating stochastic processes with many simulation points or long durations due to the significant computational cost associated with matrix factorizations concerning frequency. To address the encountered challenge, this paper presents an efficient approach for simulating ergodic stochastic processes with limited frequencies. Central to this approach is a fusion of the adaptive spectral sampling and the non-uniform fast Fourier transform (NUFFT) techniques. The adaptive spectral sampling of the envelope spectrum enables the determination of limited non-equispaced frequencies, which are randomly sampled according to a uniform distribution. Thus, the Cholesky decomposition is only required at limited specific frequencies, which dramatically reduces the computational cost of matrix factorizations. Since the randomly sampled frequencies are not equispaced, utilizing FFT to accelerate the summation of trigonometric functions becomes impractical. Then, the NUFFT that adapts the non-equispaced sampling points is employed instead to expedite this process with the non-uniform increment approximated through reduced interpolation. By taking the wind field simulation of a long-span suspension bridge as an example, a parametric analysis is conducted to investigate the effect of random frequencies on the simulation error of the developed approach and the convergence of spectra. Finally, the developed approach is further validated by focusing on the spectra and probabilistic density functions of the simulated wind samples, and the simulation performance is compared with that of the traditional approach. The analytical results demonstrate the efficiency and accuracy of the developed approach in simulating ergodic stochastic processes.

Developmental trajectories of child and adolescent emotional problems: associations with early adult alcohol use behaviors.

Whether emotional problems during childhood and adolescence are longitudinally associated with adult alcohol use behaviors is unclear. This study examined associations between developmental trajectories of emotional problems and early adult alcohol use behaviors, while considering co-occurring conduct problems, developmental change/timing, sex differences, and potential confounds. Participants were from the Twins Early Development Study (analytic N = 19,908 individuals). Emotional and conduct problems were measured by parent reports at child ages 4, 7, and 9 years and via self-reports at ages 9, 11, and 16 years on the Strengths and Difficulties Questionnaire. Alcohol use behaviors (alcohol consumption and alcohol-related problems) were self-reported by the twins on the Alcohol Use Disorders Identification Test at age 22 years. Piecewise latent growth curve models described nonlinear developmental trajectories of emotional and conduct problems from ages 4 to 16. At age 22, alcohol use was regressed on emotional and conduct problems' intercepts and slopes from piecewise latent growth curve model and sex differences in regression coefficients were tested. Using twin modeling, Cholesky decompositions and direct path models were compared to test whether significant phenotypic associations were best explained by direct phenotypic influences or correlated genetic and environmental influences. Emotional problems had different associations with alcohol-related problems versus alcohol consumption. After accounting for direct influences from conduct problems, emotional problems were not associated with alcohol-related problems, while emotional problems at age 9 were negatively associated with alcohol consumption in males. Overall, findings did not support emotional problems as prospective risk factors for severe alcohol use above and beyond risks associated with conduct problems. Sex- and age-specific links between emotional problems and alcohol consumption in early adulthood may be worthy of further exploration, particularly as twin analyses improved our confidence that such links may be underpinned by causal mechanisms.

Exploring age-related differences in metacognitive self-regulation: the influence of motivational factors in secondary school students.

Metacognitive self-regulation is a crucial factor that promotes students' learning and achievement. However, the evidence regarding age differences in metacognitive skills is rather mixed, with some evidence pointing toward further refinement and development and other evidence suggesting declining levels. Academic motivation, an important antecedent of metacognitive self-regulation, has also been reported to decline steeply in adolescence. Hence, this raises the question whether there are any age-related differences in academic motivation and metacognitive self-regulation of adolescents and whether age differences in academic motivation drive decreases in metacognitive self-regulation. A large sample size of 1,027 Greek adolescents (ages 12-16, M age = 13.95, SD = 0.78) was utilized in the present study. Multigroup measurement invariance analyses were deployed to compare the latent means of motivational factors (self-efficacy, task value, mastery, and performance goals) and metacognitive self-regulation across age groups. Cholesky decomposition was applied to test the independent contribution of motivational factors to and the indirect effects of age on metacognitive self-regulation. Invariance analyses revealed scalar invariance for metacognitive self-regulation, language self-efficacy, mastery and performance goal orientations and partially scalar invariance for task value. Older adolescents scored lower on metacognitive self-regulation, mastery and performance goals, and self-efficacy. Older students scored lower on metacognitive self-regulation via indirect effects through Cholesky decomposed motivational factors. Self-efficacy, mastery and performance goals, and task value are similarly understood across adolescents in different age groups. Decreased mastery and performance goals and task value can lead to reduced metacognitive self-regulation in adolescents. The implications of the findings underscore the key role of making students more engaged with lessons' content in order to promote greater academic motivation and prevent decreases in metacognitive self-regulation.

Least angle regression, relaxed lasso, and elastic net for algebraic multigrid of systems of elliptic partial differential equations

AbstractIn a sequence of papers, the author examined several statistical affinity measures for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (Cnodes) in algebraic multigrid (AMG) for systems of elliptic partial differential equations (PDEs). These measures were applied to a set of relaxed vectors that exposes the problematic error components. Once the CDOFs are determined using any one of these measures, the interpolation operator is constructed in a bootstrap AMG (BAMG) procedure. However, in a recent paper of Kahl and Rottmann, the statistical least angle regression (LARS) method was utilized in the coarsening procedure and shown to be promising in the CDOF selection. This method is generally used in the statistics community to select the most relevant variables in constructing a parsimonious model for a very complicated and high‐dimensional model or data set (i.e., variable selection for a “reduced” model). As pointed out by Kahl and Rottmann, the LARS procedure has the ability to detect group relations between variables, which can be more useful than binary relations that are derived from strength‐of‐connection, or affinity measures, between pairs of variables. Moreover, by using an updated Cholesky factorization approach in the regression computation, the LARS procedure can be performed efficiently even when the original set of variables is large; and due to the LARS formulation itself (i.e., its ‐norm constraint), sparse interpolation operators can be generated. In this article, we extend the LARS coarsening approach to systems of PDEs. Furthermore, we incorporate some modifications to the LARS approach based on the so‐called elastic net and relaxed lasso methods, which are well known and thoroughly analyzed in the statistics community for ameliorating several major issues with LARS as a variable selection procedure. We note that the original LARS coarsening approach may have addressed some of these issues in similar or other ways but due to the limited details provided there, it is difficult to determine the extent of their similarities. Incorporating these modifications (or effecting them in similar ways) leads to improved robustness in the LARS coarsening procedure, and numerical experiments indicate that the changes lead to faster convergence in the multigrid method. Moreover, the relaxed lasso modification permits an indirect BAMG (iBAMG) extension to the interpolation operator. This iBAMG extension applied in an intra‐ or inter‐variable interpolation setting (i.e., nodal‐based coarsening), as well as in variable‐based coarsening, which will not preserve the nodal structure of a finest‐level discretization on the lower levels of the multilevel hierarchy, will be examined. For the variable‐based coarsening, because of the parsimonious feature of LARS, the performance is reasonably good when applied to systems of PDEs albeit at a substantial additional cost over a nodal‐based procedure.

Stochastic modelling of material variability in structural dynamics: A threefold comparison of Monte Carlo, polynomial chaos, and random sampling techniques

This article investigates the influence of random elastic modulus on beam eigenfrequencies using multiple simulation techniques: Monte Carlo simulations (employing Cholesky decomposition (MCS-CD) and Kosambi-Karhunen-Loève expansion (MCS-KKL)), Polynomial Chaos expansion (PCE), and a proposed Random Sampling method (RSM). Anomalies in Monte Carlo simulations, where normally distributed elastic modulus led to negative values and imaginary eigenfrequencies, were effectively addressed by adopting a log-normal distribution. Comparative analyses focused on covariance variation of the first three eigenfrequencies with correlation length and standard deviation of the random field, highlighting nuanced differences between normal and log-normal distributions. PCE exhibited distinct responses, showcasing variations in covariance with different distributions. The study culminates in eigenfrequency estimation using the proposed RSM, wherein the beam is discretised into n elements with randomly assigned elastic moduli. The mean and variance of eigenfrequencies are compared with existing methods, which represent an alternative method for achieving similar outcomes.These comparative studies provide a comprehensive understanding of how different statistical treatments and simulation methods impact the reliability and accuracy of eigenfrequency predictions in beams with random elastic properties, thus contributing valuable insights for structural analysis and design under uncertainty.

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ \epsilon -approximate inverse Cholesky factors of the kernel matrices with complexity O ( N log d ⁡ ( N / ϵ ) ) O(N\log ^d(N/\epsilon )) in space and O ( N log 2 d ⁡ ( N / ϵ ) ) O(N\log ^{2d}(N/\epsilon )) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

A mesh-free framework for high-order simulations of viscoelastic flows in complex geometries

The accurate and stable simulation of viscoelastic flows remains a significant computational challenge, exacerbated for flows in non-trivial and practical geometries. Here we present a new high-order meshless approach with variable resolution for the solution of viscoelastic flows across a range of Weissenberg numbers. Based on the Local Anisotropic Basis Function Method (LABFM) of King et al. (2020), highly accurate viscoelastic flow solutions are found using Oldroyd B and PPT models for a range of two dimensional problems — including Kolmogorov flow, planar Poiseulle flow, and flow in a representative porous media geometry. Convergence rates up to 9th order are shown. Three treatments for the conformation tensor evolution are investigated for use in this new high-order meshless context (direct integration, Cholesky decomposition, and log-conformation), with log-conformation providing consistently stable solutions across test cases, and direct integration yielding better accuracy for simpler unidirectional flows. The final test considers symmetry breaking in the porous media flow at moderate Weissenberg number, as a precursor to a future study of fully 3D high-fidelity simulations of elastic flow instabilities in complex geometries. The results herein demonstrate the potential of a viscoelastic flow solver that is both high-order (for accuracy) and meshless (for straightforward discretisation of non-trivial geometries including variable resolution). In the near-term, extension of this approach to three dimensional solutions promises to yield important insights into a range of viscoelastic flow problems, and especially the fundamental challenge of understanding elastic instabilities in practical settings.

Comparison of Matrix Decomposition in Null Space-Based LDA Method

Problems with small sample sizes and high dimensionality are common in pattern recognition. Almost all machine learning algorithms degrade in high-dimensional data, so that singularities in the scatter matrices, the main problem of the Linear Discriminant Analysis (LDA) technique, might result. A null space-based LDA (NLDA) has been conceived to address the singularity issue. NLDA aims to maximize the distance between classes in the null space of the within-class scatter matrix. In the first research, the NLDA method was performed by computing the eigenvalue decomposition and singular value decomposition (SVD). This research led to several new implementations of the NLDA method that use other matrix decompositions. The new implementations include NLDA using Cholesky decomposition and NLDA using QR decomposition. This paper compares the performance of three NLDA methods that use different matrix decompositions, namely, SVD, Cholesky decomposition, and QR decomposition. Two sets of data were used in the experiments that used three different NLDA algorithms. To determine the performance of the NLDA methods, the classification accuracy of the three methods was measured using the confusion matrix. The results show that the NLDA method using SVD has the best performance when compared to the other two methods, achieving a precision of 77.8% accuracy for the Colon dataset and a precision of 98.8% accuracy for the TKI-resistance dataset.

Time-efficient filtering of imaging polarimetric data by checking physical realizability of experimental mueller matrices.

Imaging Mueller polarimetry has already proved its potential for biomedicine, remote sensing and metrology. The real-time applications of this modality require both video rate image acquisition and fast data post-processing algorithms. First, one must check the physical realizability of the experimental Mueller matrices in order to filter out non-physical data, ie to test the positive semi-definiteness of the 4 × 4 Hermitian coherency matrix calculated from the elements of corresponding Mueller matrix pixel-wise. For this purpose, we compared the execution time for the calculations of i) eigenvalues, ii) Cholesky decomposition, iii) Sylvester's criterion, and iv) coefficients of the characteristic polynomial (two different approaches) of the Hermitian coherency matrix, all calculated for the experimental Mueller matrix images (600 pixels × 700 pixels) of mouse uterine cervix. The calculations were performed using C ++ and Julia programming languages. Our results showed the superiority of the algorithm iv) based on the simplification via Pauli matrices over other algorithms for our dataset. The sequential implementation of latter algorithm on a single core already satisfies the requirements of real-time polarimetric imaging. This can be further amplified by the proposed parallelization (e.g., we achieve a 5-fold speed up on 6 cores). The source codes of the algorithms and experimental data are available at https://github.com/pogudingleb/mueller_matrices.

Enhancing optimization accuracy in power systems: Investigating correlation effects on objective function values

This study addresses the critical yet often overlooked aspect of incorporating correlations among input stochastic variables in power system planning and scheduling optimization. While existing literature has extensively focused on uncertainty modelling, there remains a gap in fully assessing the consequences of disregarding correlations on objective function values across different power network sizes. To bridge this gap, we utilize Monte Carlo simulation with Cholesky decomposition, alongside Quasi-Monte Carlo sampling and Latin Hypercube Sampling, to effectively model uncertainty and capture correlation coefficients among input variables, including wind, solar photovoltaic, and load power. The most efficient technique is then integrated into our optimization model, which is applied to small, medium, and large power network models. Our proposed optimization model addresses conflicting objectives using a hybrid NSGAII-MOPSO, aiming to simultaneously minimize total operational cost, power loss, and voltage deviation. By implementing this model on selected power networks and comparing outcomes between cases with independent and correlated variables, we rigorously assess discrepancies in objective function values. We visualize and analyze these errors across systems of varying sizes, shedding light on the impact of neglecting variable correlations. Notably, the maximum discrepancies are observed at $3.26/h, $40.66/h, and $2754.04/h for the IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus systems, respectively. Crucially, as the system size increases, so does the magnitude of these differences, underlining the escalating impact of neglecting variable correlations on optimization outcomes. We stress the importance of integrating such considerations into future planning and operational strategies to mitigate errors and enhance decision-making processes.

Performance of Linear Algebra Factorization in Multi-Accelerator Architectures

Hardware and software are required to effectively solve problems in many domains. The idea of creating a hybrid architecture based on graphics processors arose to meet the increasing demands of modern scientific problems. Most of these problems are reduced to solving linear algebra problems. A set of efficient linear solutions has been successfully used to solve important scientific problems for many years. Factorizations play a crucial role in solving linear algebra problems. This work presents implementations of LU, QR and Cholesky factorizations on two graphics processors using the MAGMA 2.6.0 library. Their performances are given for matrices with real and complex numbers in single and double precision.

A method of voltage sag detection based on strong tracking SVD-UKF

Abstract Addressing issues of the traditional strong tracking unscented Kalman filtering (STUKF) algorithm in the process of voltage sag detection, such as inadequate precision, large computational complexity, and unstable iterative procedures, a method of voltage sag detection based on strong tracking SVD-UKF is proposed. Only one unscented transformation (UT) is used per iteration by establishing the linear state equation based on the voltage signal model, which simplifies the calculation procedure. Substituting singular value decomposition (SVD) for Cholesky decomposition averts the challenge of non-positive-definite error covariance matrix, ensuring stable execution of the algorithm. The action position of the fading factor is improved to further enhance detection precision. Through simulation experiments, we validate that the proposed method could effectively detect voltage sag characteristics, including amplitude, phase, and duration time.

Monitoring of high-dimensional and high-frequency data streams: A nonparametric approach

ABSTRACT High-dimensional and high-frequency data streams (HHDS) have become widely available with the prevalence of data-acquisition technologies. Because HHDS have the characteristics of high dimensionality and temporal correlation, traditional statistical process control (SPC) techniques cannot be used directly to address the problem of monitoring HHDS. Most extant monitoring methods commonly assume that the process observations are independent or can be described by some parametric models and face the challenge posed by high-dimensional data. Little research has been conducted on the robust monitoring of HHDS that can simultaneously accommodate high dimensions and high frequencies. Therefore, in this paper, we propose a novel nonparametric approach for monitoring HHDS, based on data decorrelation, dimension reduction, and SPC. Specifically, process observations are first sequentially decorrelated and standardized using Cholesky decomposition, and then the k -nearest neighbors classification algorithm is applied to transform uncorrelated high-dimensional data into one-dimensional data. Finally, the traditional cumulative-sum procedure is used for online monitoring, based on an empirical log-likelihood ratio test. Numerical studies have shown that our monitoring scheme is sufficiently reliable and efficient for the online monitoring of HHDS. An illustrative example about the Dow Jones 30 industrial stock prices is presented to further validate the proposed approach.

Behavioral Modeling and Digital Predistortion for Power Amplifier Based on the Sparse Smooth Twin Support Vector Regression Method

In this paper, a sparse smooth twin support vector regression (Sparse‐STSVR) model for power amplifier (PA) behavioral modeling is obtained by pruning the kernel matrix based on Cholesky decomposition. Based on the primal smooth twin support vector regression (STSVR) model, the Nystrom approximate matrix of the kernel matrix is found to replace the original kernel matrix, thus simplifying the Newton iterative parameter extraction process of the primal STSVR model and accelerating the convergence of the algorithm. In addition, the new rank approximation kernel matrix has the characteristic of sparse parameters, which further reduces the computational complexity of the feedforward link of the digital predistorter. The 100 MHz 5G New Radio (NR) signal is used for verify the effect of PA modeling and digital predistortion (DPD) experiment. The results show that the proposed method can improve the normalized mean square error (NMSE) by about 2 ~ 3 dB with fewer coefficients compared with the previously proposed machine learning model, and the predistortion linearization effect improves by nearly 3 dB on the adjacent channel power ratio (ACPR), which achieves a good trade‐off between model performance and computational complexity. © 2024 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.

Exchange Rate Pass-Through and Inflation: Policy Insights for Vietnam Based on Comparative Analyses Across Asia

The exchange rate is a crucial macro indicator for any open economy, whereby any change in the rate would influence, to a certain extent, various socioeconomic facets. One of the significant impacts is revealed in the pass-through of exchange rate variations to price inflation, also known as the exchange rate pass-through (ERPT). This paper explores the effect of ERPT on domestic inflation in Vietnam by comparing with other Asian economies. Specifically, pass-through mechanisms are examined for every single price as well as a distribution chain from import price index (IMP), producer price index (PPI) to consumer price index (CPI). The empirical results show that exchange rate shocks positively influence the price changes in Vietnam as well as in other Asian countries. According to Cholesky decomposition, the exchange rate plays a limited role in regulating CPI movements. Besides, world commodity prices are found to have a significant impact on domestic inflation across the sample-countries. Our findings reveal profound implications for monetary policy management and price stability.

Dynamic system with correlated noise

Developing mathematical models for dynamic systems presents challenges in selecting state variables, crucial for minimizing computational operations during filter implementation. The paper addresses active identification of dynamic systems amid noise and measurement errors, using the Fisher information matrix. Additionally, it introduces Cholesky decomposition for generating correlated noise, demonstrating its applicability in practical problem-solving.