Linear Problem Research Articles

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.

The pseudospectrum and transient of Kaluza–Klein black holes in Einstein–Gauss–Bonnet gravity

Abstract The spectrum and dynamical instability, as well as the transient effect of the tensor perturbation for the so-called Maeda–Dadhich black hole, a type of Kaluza–Klein black hole, in Einstein–Gauss–Bonnet gravity have been investigated in framework of pseudospectrum. We cast the problem of solving quasinormal modes (QNMs) in AdS-like spacetime as the linear evolution problem of the non-normal operator in null slicing by using ingoing Eddington–Finkelstein coordinates. In terms of spectrum instability, based on the generalized eigenvalue problem, the QNM spectrum and ε-pseudospectrum has been studied, while the open structure of ε-pseudospectrum caused by the non-normality of operator indicates the spectrum instability. In terms of dynamical instability, we introduce the concept of the distance to dynamical instability, which plays a crucial role in bridging the spectrum instability and the dynamical instability. We calculate such distance, named the complex stability radius, as parameters vary. Finally, we show the behavior of the energy norm of the evolution operator, which can be roughly reflected by the three kinds of abscissas in context of pseudospectrum, and find the transient growth of the energy norm of the evolution operator.

Thermodynamic linear algebra

AbstractLinear algebra is central to many algorithms in engineering, science, and machine learning; hence, accelerating it would have tremendous economic impact. Quantum computing has been proposed for this purpose, although the resource requirements are far beyond current technological capabilities. We consider an alternative physics-based computing paradigm based on classical thermodynamics, to provide a near-term approach to accelerating linear algebra. At first sight, thermodynamics and linear algebra seem to be unrelated fields. Here, we connect solving linear algebra problems to sampling from the thermodynamic equilibrium distribution of a system of coupled harmonic oscillators. We present simple thermodynamic algorithms for solving linear systems of equations, computing matrix inverses, and computing matrix determinants. Under reasonable assumptions, we rigorously establish asymptotic speedups for our algorithms, relative to digital methods, that scale linearly in matrix dimension. Our algorithms exploit thermodynamic principles like ergodicity, entropy, and equilibration, highlighting the deep connection between these two seemingly distinct fields, and opening up algebraic applications for thermodynamic computers.

Equilibrium Control in Uncertain Linear Quadratic Differential Games with V-Jumps and State Delays: A Case Study on Carbon Emission Reduction

Uncertainty, time delays, and jumps often coexist in dynamic game problems due to the complexity of the environment. To address such issues, we can utilize uncertain delay differential equations with jumps to depict the dynamic changes in differential game problems that involve uncertain noise, delays, and jumps. In this paper, we first examine a linear quadratic differential game optimistic value problem within an uncertain environment characterized by jumps and delays. By applying the Z(x,y) transform, we convert the infinite-dimensional problem into a finite-dimensional one. We then demonstrate that the condition for the existence of a Nash equilibrium strategy is equivalent to the existence of solutions to two cross-coupled matrix Riccati equations. Furthermore, we explore the saddle point equilibrium strategy of the linear quadratic differential game optimistic value model and derive the corresponding saddle point equilibrium solution. Finally, we apply our results to solve a carbon emission reduction game problem.

Some rapidly mixing hit-and-run samplers for latent counts in linear inverse problems

Some rapidly mixing hit-and-run samplers for latent counts in linear inverse problems

Transfer learning for high-dimensional linear regression via the elastic net

In this paper, the high-dimensional linear regression problem is explored via the Elastic Net under the transfer learning framework. Within this framework, potentially related source datasets are leveraged to enhance estimation or prediction beyond what can be achieved solely with the target data. When transferable sources are known, an oracle transfer learning algorithm is proposed based on the Elastic Net. Additionally, the ℓ1/ℓ2 estimation error bounds for the corresponding estimator are established. When the transferable sources are unknown, a novel procedure for detecting transferable sources via the Elastic Net is also proposed, with its selection consistency demonstrated under regular conditions. This method transforms the source detection problem into a variable selection problem in high-dimensional space and always gets results that are consistent with the true outcomes. The performance of these methods is further demonstrated through a variety of numerical examples. Finally, our approach is applied to analyze several real datasets for illustrative purposes.

On the construction of an optimal network of observation points when solving inverse lin-ear problems of gravimetry and magnetometry

Unique solvability of systems of linear algebraic equations is studied to which many in-verse problems of geophysics are reduced as a result of discretization after applying the method of integral equations or integral representations. Examples of singular and nonsingular systems of vari-ous dimensions that arise when processing magnetometric and gravimetric data from experimental observations are discussed. Conclusions are drawn about methods for constructing an optimal net-work of experimental observation points.

Development of mathematical models of heat conductivity for modern electronic devices with elements containing foreign inclusions

This paper considers the heat conduction process for an isotropic medium containing a foreign half-through inclusion and heated by a locally concentrated heat flow. Linear and non-linear mathematical models for determining the temperature field have been built to establish the temperature regimes for the effective operation of electronic devices. The coefficient of thermal conductivity of a non-uniform structure is represented as a whole, using asymmetric unit functions, which automatically provides the conditions of ideal thermal contact on the surfaces of materials. This results in solving one heat conduction equation with discontinuous and singular coefficients. A linearizing function was introduced to linearize the nonlinear boundary value problem. Analytical-numerical solutions of linear and nonlinear boundary-value problems have been obtained in a closed form. A linear temperature dependence of the coefficient of thermal conductivity of structural materials was chosen for a heat-sensitive medium. As a result, an analytical-numerical solution was derived, which determines the temperature distribution in this medium. On this basis, a numerical experiment was performed, the results of which are graphically displayed and confirm the adequacy of the constructed mathematical models to a real physical process. The materials of the plate and inclusion are silicon and silver. The results for these materials based on the linear and non-linear model differ by 7 %. Their slight difference is explained by the fact that the values of the temperature coefficient of thermal conductivity are small. The models built make it possible to analyze the given environments in terms of their thermal resistance. As a result, it becomes possible to improve it, and protect structures from overheating, which could lead to the failure of individual nodes and their elements and the entire electronic device

On the randomized multiple row-action methods for solving linear least-squares problems

On the randomized multiple row-action methods for solving linear least-squares problems

Cauchy discretization method for solve fractional linear optimal control problems

This paper presents a numerical scheme designed for solving fractional linear optimal control problems (FLOCPs) governed by Caputo fractional derivatives (CFDs) of order 0<α≤1. The method transforms the original fractional control problem into an equivalent linear programming problem (LPP), enabling a more straightforward solution process. To evaluate the performance and accuracy of this approach, several numerical experiments were conducted using MATLAB. These experiments highlight the method’s computational efficiency and its simplicity in terms of implementation. The proposed approach offers accurate results, particularly in scenarios where traditional methods may fall short due to the complexities introduced by fractional derivatives. A detailed numerical example is presented to further illustrate the method’s effectiveness in addressing FLOCPs. The outcomes suggest that this approach is not only practical but also provides valuable insights for researchers dealing with fractional calculus in optimal control theory. This contribution is expected to be beneficial for applications in engineering and the physical sciences, where fractional-order systems play a significant role.

Parameter-adaptive variational autoencoder for linear/nonlinear blind source separation

AbstractBlind source separation (BSS) serves as an important technique in the field of structural health monitoring (SHM), particularly for solving modal decomposition tasks. This study proposes a novel approach to both linear and nonlinear BSS problems in the Variational Autoencoder (VAE) framework, where the encoding and decoding processes of VAE are interpreted as procedures for inferring sources from observations and remixing these sources, respectively. In this way, the distribution of latent variables inferred by VAE is equivalent to the distribution of sources. We make improvements to the vanilla VAE to augment its effectiveness for BSS. First, we substitute standard normal distributions with trainable Gaussian processes (GP) as priors for latent variables and implement an exponential function as the activation function for adaptive parameters in the GP kernel functions. While the form of the priors is set as GP, the parameters of their kernel functions are not fixed but automatically converge to suitable values during the model training process. Additionally, a hyperparameter is introduced to balance the terms in the loss function. The proposed method is referred to as parameter-adaptive VAE (PAVAE). Then, upon different assumptions of the variances of sources, the proposed PAVAE is branched into two types: homoscedastic PAVAE (Ho-PAVAE) and heteroscedastic PAVAE (He-PAVAE). Through numerical and laboratory experiments, we demonstrate the effectiveness of the proposed method in solving BSS problems and their potential to underpin future research in SHM.

Classifying Abelian Groups Through Acyclic Matchings

AbstractThe inquiry into identifying sets of monomials that can be eliminated from a generic homogeneous polynomial via a linear change of coordinates was initiated by E. K. Wakeford. This linear algebra problem prompted C. K. Fan and J. Losonczy to introduce the notion of acyclic matchings in the additive group $$\mathbb {Z}^n$$ Z n , subsequently extended to abelian groups by the latter author. Alon, Fan, Kleitman, and Losonczy established the acyclic matching property for $$\mathbb {Z}^n$$ Z n . This note aims to classify all abelian groups with respect to the acyclic matching property.

Computing Quadratic Eigenvalues and Solvent by a New Minimization Method and a Split-Linearization Technique

To solve quadratic eigenvalue problems (QEPs), especially the gyroscopic systems, two methods are proposed: an iterative direct detection method (DDM) of the complex eigenvalues of the original QEP, and a split-linearization method (SLM) for finding the solvent matrix, which results to a standard linear eigenvalue problem (LEP) being solved to compute all eigenvalues by the symmetry extension. Reducing the dimension to one-half, the LEP is recast in a simpler QEP involving the square of the solvent. We set up two new merit functions which are minimized to detect the complex eigenvalues from the original QEP and a simpler QEP. For each eigen-parameter the merit function consists of the Euclidean norm of each derived eigen-equation, whose vector variable is solved from a derived inhomogeneous linear system. Then, the golden section search algorithm is employed to minimize the merit functions and locate the complex eigenvalue as a local minimal point. The results are compared with that computed by the cyclic-reduction-based solvent (CRS) method.

Inverse chance-constrained dynamic data envelopment analysis under natural and managerial disposability: Concerning renewable energy efficiency and potentials

The performance of dynamic systems with specific data has been examined in some data envelopment analysis (DEA) studies. However, in many real-life situations, dealing with dynamic systems becomes challenging because of random factors. So, in this paper, a chance-constrained dynamic DEA approach under managerial and natural disposability is first proposed to analyze the dynamic renewable energy efficacy of processes in the presence of random performance measures. Then, the potentials of some performance metrics are addressed for changes of others using the introduced inverse chance-constrained dynamic DEA models. The chance-constrained dynamic DEA techniques and their inverse dynamic stochastic problems are converted into linear problems. The introduced approaches are applied to probe the renewable energy efficiency and potentials of some OECD countries in a span of time. The results show stochastic dynamic DEA and inverse stochastic dynamic DEA provided are practical tools for making the best choices, when there is uncertainty.

Total Positivity and Accurate Computations Related to q-Abel Polynomials

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

A Sixth-Order Cubic B-Spline Approach for Solving Linear Boundary Value Problems: An In-Depth Analysis and Comparative Study

This research presents an efficient and highly accurate cubic B-spline method (CBSM) for solving second-order linear boundary value problems (BVPs). The method achieves sixth-order convergence, supported by rigorous error analysis, ensuring rapid error reduction with mesh refinement. The effectiveness of the CBSM is validated through four numerical examples, showcasing its accuracy, reliability, and computational efficiency, making it well-suited for large-scale problems. A comparative analysis with existing methods confirms the superior performance of the CBSM, positioning it as a practical and powerful tool for solving second-order BVPs.

Data-Efficient Dimensionality Reduction and Surrogate Modeling of High-Dimensional Stress Fields

Abstract Tensor datatypes representing field variables like stress, displacement, velocity, etc., have increasingly become a common occurrence in data-driven modeling and analysis of simulations. Numerous methods [such as convolutional neural networks (CNNs)] exist to address the meta-modeling of field data from simulations. As the complexity of the simulation increases, so does the cost of acquisition, leading to limited data scenarios. Modeling of tensor datatypes under limited data scenarios remains a hindrance for engineering applications. In this article, we introduce a direct image-to-image modeling framework of convolutional autoencoders enhanced by information bottleneck loss function to tackle the tensor data types with limited data. The information bottleneck method penalizes the nuisance information in the latent space while maximizing relevant information making it robust for limited data scenarios. The entire neural network framework is further combined with robust hyperparameter optimization. We perform numerical studies to compare the predictive performance of the proposed method with a dimensionality reduction-based surrogate modeling framework on a representative linear elastic ellipsoidal void problem with uniaxial loading. The data structure focuses on the low-data regime (fewer than 100 data points) and includes the parameterized geometry of the ellipsoidal void as the input and the predicted stress field as the output. The results of the numerical studies show that the information bottleneck approach yields improved overall accuracy and more precise prediction of the extremes of the stress field. Additionally, an in-depth analysis is carried out to elucidate the information compression behavior of the proposed framework.

On the Numerical Solution of Linear and Non-Linear Dirichlet Boundary Value Problems of Ordinary Differential Equations: The Shooting Method

In this article, a numerical technique called shooting which entails the solution of initial value problems with the single-step fourth-order RungeKutta method together with the iterative root finding secant method is formulated for use on both linear and non-linear boundary value problems of ordinary differential equations with Dirichlet boundary conditions. Two examples are illustrated. One, the solution of the linear case with its analytic counterpart is compared, and two, the non-linear case. Graphical outputs of the solutions from two MATHEMATICA codes are presented.

On a posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems

AbstractA posteriori bounds for the error measured in various norms for a standard second‐order explicit‐in‐time Runge–Kutta discontinuous Galerkin (RKDG) discretization of a one‐dimensional (in space) linear transport problem are derived. The proof is based on a novel space‐time polynomial reconstruction, hinging on high‐order temporal reconstructions for continuous and discontinuous Galerkin time‐stepping methods. Of particular interest is the question of error estimation under dynamic mesh modification. The theoretical findings are tested by numerical experiments.

Optimal EV Charging and PV Siting in Prosumers towards Loss Reduction and Voltage Profile Improvement in Distribution Networks

In this paper, the problem of simultaneous charging of Electrical Vehicles (EVs) in distribution networks (DNs) is examined in order to depict congestion issues, increased power losses, and voltage constraint violations. To this end, this paper proposes an optimal EV charging schedule in order to allocate the charging of EVs in non-overlapping time slots, aiming to avoid overloading conditions that could stress the DN operation. The problem is structured as a linear optimization problem in GAMS, and the linear Distflow is utilized for the power flow analysis required. The proposed approach is compared to the one where EV charging is not optimally scheduled and each EV is expected to start charging upon its arrival at the residential charging spot. Moreover, the analysis is extended to examine the optimal siting of small-sized residential Photovoltaic (PV) systems in order to provide further relief to the DN. A mixed-integer quadratic optimization model was formed to integrate the PV siting into the optimization problem as an additional optimization variable and is compared to a heuristic-based approach for determining the sites for PV installation. The proposed methodology has been applied in a typical low-voltage (LV) DN as a case study, including real power demand data for the residences and technical characteristics for the EVs. The results indicate that both the DN power losses and the voltage profile are further improved in regard to the heuristic-based approach, and the simultaneously scheduled penetration of EVs and PVs could yield up to a 66.3% power loss reduction.