Linear Problem Research Articles

Weak dangling block reordering and multi-step block compression for efficiently computing and updating PageRank solutions

The PageRank model is a powerful tool for network analysis, utilized across various disciplines such as web information retrieval, bioinformatics, community detection, and graph neural network. Computing this model requires solving a large-dimensional linear system or eigenvector problem due to the ever-increasing scale of networks. Conventional preconditioners and iterative methods for general linear systems or eigenvector problems often exhibit unsatisfactory performance for such problems, particularly as the damping factor parameter approaches 1, necessitating the development of specialized methods that exploit the specific properties of the PageRank coefficient matrix. Additionally, in practical applications, the optimal settings of the hyperparameters are generally unknown in advance, and networks often evolve over time. Consequently, recomputation of the problem is necessary following minor modifications. In this scenario, highly efficient preconditioners that significantly accelerate the iterative solution at a low memory cost are desirable. In this paper, we present two techniques that leverage the sparsity structures and numerical properties of the PageRank system, as well as a preconditioner based on the computed matrix structure. Experiments demonstrate the positive performance of the proposed methods on realistic PageRank computations.

Sparse regression for discovery of constitutive models from oscillatory shear measurements

We propose sparse regression as an alternative to neural networks for the discovery of parsimonious constitutive models (CMs) from oscillatory shear experiments. Symmetry and frame invariance are strictly imposed by using tensor basis functions to isolate and describe unknown nonlinear terms in the CMs. We generate synthetic experimental data using the Giesekus and Phan-Thien Tanner CMs and consider two different scenarios. In the complete information scenario, we assume that the shear stress, along with the first and second normal stress differences, is measured. This leads to a sparse linear regression problem that can be solved efficiently using l1 regularization. In the partial information scenario, we assume that only shear stress data are available. This leads to a more challenging sparse nonlinear regression problem, for which we propose a greedy two-stage algorithm. In both scenarios, the proposed methods fit and interpolate the training data remarkably well. Predictions of the inferred CMs extrapolate satisfactorily beyond the range of training data for oscillatory shear. They also extrapolate reasonably well to flow conditions like startup of steady and uniaxial extension that are not used in the identification of CMs. We discuss ramifications for experimental design, potential algorithmic improvements, and implications of the non-uniqueness of CMs inferred from partial information.

Application of model predictive control scheme for piezoelectric ceramic micro-displacement

Abstract Currently, in the domain of micro-control technology, many operations need to meet the standard of nanometer-level positioning. As a high-precision micro-displacement element, piezoelectric ceramics are widely used in the field of micro-control due to their characteristics such as no need for transmission devices, high resolution, large output force, small size, and fast response speed. However, the inherent hysteresis nonlinearity of piezoelectric ceramics can lead to decreased displacement accuracy, oscillation, and even system instability. Therefore, eliminating the influence of nonlinear characteristics and improving displacement accuracy possess significant practical importance. In this paper, a model predictive control scheme is designed, which achieves faster response speed and higher accuracy compared to traditional control methods such as Proportional Integral Derivative control. Relevant simulations and tests have been conducted. In this scheme, hysteresis nonlinearity is first modeled and then compensated through feedforward, thereby simplifying the displacement control of piezoelectric ceramics into a more manageable linear problem. Subsequently, a linear model predictive control method is adopted to control the entire system. Both simulation and experimental results have verified the effectiveness of this control method.

Acoustics of wet porous media with evaporation/condensation

This paper investigates acoustic wave propagation in wet rigid-frame porous media accounting for evaporation and condensation. At equilibrium, the solid walls are covered by a thin water film, and water vapor in the air is at its temperature-dependent saturation pressure. Small acoustic perturbations cause water to vaporize or condense, which together with the reversibility of the phase change, lead to a linear problem where the usual local poro-acoustics physics is enriched with the (i) Clapeyron relation linking liquid-wall temperature, vapor pressure, and latent heat of vaporization, (ii) latent heat transfer in the solid frame, (iii) diffusion equation for water vapor in air, and (iv) water vapor's equation of state. The equilibrium temperature highly influences the vapor concentration and the physical parameters of saturated moist air. Using the two-scale asymptotic homogenization method, it is shown that the dynamic permeability is determined similarly to classical porous media, while the effective compressibility is modified by evaporation/condensation and the equilibrium temperature. This modification is influenced by vapor mass and heat flows associated with phase changes through a local fully coupled heat transfer and water vapor diffusion problem, with specific boundary conditions at the gas–water interface. The analysis identifies dimensionless parameters and characteristic frequencies defining the upscaled model's features. Depending on equilibrium temperature, the theory qualitatively and quantitatively determines the characteristics of acoustic waves propagating through the media. The results are illustrated and discussed with analytically developed models.

The operational matrices for Elliptic Partial Differential Equations with mixed boundary conditions

Abstract The purpose of this research is to implement the orthogonal polynomials associated with operational matrices to get the approximate solutions for solving two-dimensional elliptic partial differential equations (E-PDEs) with mixed boundary conditions. The orthogonal polynomials are based on the Standard polynomial (xi ), Legendre, Chebyshev, Bernoulli, Boubaker, and Genocchi polynomials. This study focuses on constructing quick and precise analytic approximations using a simple, elegant, and potent technique based on an orthogonal polynomial representation of the solution as a double power series. Consequently, a linear partial differential equation is transformed into a linear algebraic system which is solved by the Mathematica®12. Approximate solutions can be found if the answers are polynomials in and of itself. Three applications involving well-known linear problems Laplace, Poisson, and Helmholtz equations have been solved by using the proposed methods, and a comparison of the approaches has been provided. Furthermore, the computation of the error norm L∞ has been done to show the accuracy of the suggested approaches. The results clearly demonstrate how precise, efficient, and dependable the proposed methods are in obtaining rough solutions to the problem. Bernoulli was one of the best methods in most examples.

Modification of the dimensional doma method for linear elliptic equations

Currently, one of the most common methods for the numerical solution of boundary value problems are difference methods (grid method, finite difference method). These methods are well suited for solving problems in regular domains. In areas of complex structure, their use leads to a number of difficulties both in the construction of difference schemes, and in the implementation on a computer. One of the methods to overcome these difficulties is the method of fictitious regions. There are four interrelated directions in the development of the method of fictitious regions: study of various ways to continue the original problems in a fictitious area; obtaining unimprovable estimates in stronger metrics; extension of the class of problems on the application of the method of fictitious areas; construction of effective difference schemes for solving auxiliary problems constructed by the method of fictitious areas. For the first time, the fictitious domain method for the Navier-Stokes equation was studied in the work of A.N. Bugrova, Sh. Smagulov, Smagulov Sh. This article discusses the method of additional areas, which is an analogue of the method of fictitious areas for solving linear boundary value problems, but in the auxiliary problem of this method there is no small perimeter. This method is based on the variational principle of solving linear boundary value problems of elliptic type in an arbitrary region. The existence of a solution is proved by the Galerkin uniqueness method.

Existence of extremal solutions for a class of fractional integro-differential equations

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

CALCULATION OF FLAT SHELLS BY THE BOUNDARY ELEMENT METHOD

This article describes the iterative process of solving the problem of the stress-strain state of a long flexible cylindrical panel. Stress-strain state and the study of the stability of flat shells taking into account the physical, mechanical properties of the material, the process of deformation change. It is based on sequential approximation methods and the boundary element method (MGE). The results of algorithmic weighting are presented in the form of a table showing what value each band represents. The equations are determined by solving twelve unknown values of functions at the ends of the segment. Linear problems of elasticity theory and plate theory fundamental solutions have a simple form, so the method is widely used here. For flat shells, the matrix of fundamental solutions is determined by complex volumetric expressions, and for flat shells - by special functions. Therefore, there is little research on solving problems in the theory of flat shells by the boundary element method. In this regard, an urgent topic of research is the development of methods of boundary integral equations for solving linear and nonlinear problems in the theory of flat shells based on the application of fundamental solutions defined by simple analytical expressions. The scientific novelty of the work consists in the development of a methodology for assessing the reliability of thin-walled spatial structures using the boundary element method.

Метод R-функцій для розв’язання задачі масообміну циліндричного тіла з твірною у вигляді кривої Ламе з в’язкою нестисливою рідиною

The problem of stationary mass exchange of a cylindrical body with viscous incompressible fluid flow is reduced to solving the equation for concentration with the corresponding boundary conditions on the surface of the body and at infinity. Applying the constructive apparatus of the R-functions theory allows us to accurately take into account the geometry of the domain, as well as the boundary conditions together with the condition at infinity. The R-function method was proposed by V. L. Rvachev, Ukraine National Academy of Sciences academician. For boundary value problems of mathematical physics, this method allows constructing the so-called structure of the solution of the boundary value problem, i.e. a bundle of functions that accurately takes into account all boundary conditions of the problem and depends on some uncertain components. The choice of these uncertain components is performed in such a way as to satisfy (in a certain sense) the equation of the problem. For this, one can use standard numerical methods of mathematical physics (the Ritz method, the Galerkin method, the least squares method, collocations method, etc.). It should be noted that the geometry of the region is taken into account exactly, i.e. in particular, there is no replacement of curvilinear sections of the boundary with polygonal lines inscribed in them, as it is done, for example, in the finite element method. The purpose of this article is to apply the R-functions method to the problem of mass exchange of a cylindrical body formed by the Lame curve, with a viscous incompressible fluid flowing around it. To achieve this goal, a complete structure of the solution of a linear boundary value problem for concentration is constructed by the R-functions theory methods, and a numerical algorithm based on the Galerkin method for approximating indefinite components in the problem structure is used. The degree of rigor and the conditions for using the considered equations of fluid motion are not considered in the work; these equations are considered as mathematical models subject to numerical algorithmization

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method.

On a case of -linear conjugation problem on the unit circle

The R -linear conjugation problem on the unit circle with rational coefficients is solved. The problem is equivalently reduced to the vector–matrix C -linear conjugation problem for which the canonical matrix is effectively constructed. The applied method is based on the two-fold application of G.N.Chebotarev's approach to the factorization of rational matrix functions.

Partially observable optimal control using exponential cost criterion

We consider a partially observable optimal control model in which the system and observation noises are correlated, and the cost criterion is in the form of an exponential of an integral. A new minimum principle criterion is derived and applied to compute the explicit solution of linear exponential Gaussian optimisation problem with correlated noises.

Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov–Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and C∞ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

An outcome space algorithm for solving general linear multiplicative programming

The following article presents and corroborates an outcome space branch-and-bound algorithm for solving the general linear multiplicative programming problem (GLMPP). In this new algorithm, GLMPP is transformed into its equivalent problem by introducing auxiliary variables. To compute the tight upper bound for the optimal value of GLMPP, the linear relaxation programming problem is assembled by using bound and hull linear approximations. Furthermore, the global convergence and computational complexity of the algorithm are demonstrated. Finally, the soundness and advantage of the proposed algorithm are validated by solving a demonstrative linear multiplicative problem.

A New Approach to the Solution of Facility Layout Problems with Filled Function Method

Facility layout problems are generally solved by stochastic methods in the literature. The large number of iterations used in these methods is quite costly in terms of solution time. In this study, in order to get rid of this disadvantage, the facility layout problem was solved using the filled function method, which is known to be very successful in solving global optimization problems. In order to demonstrate the effectiveness of the filled function method, the classical linear facility layout problem was handled in a non-linear manner and the problem was deliberately made more difficult. In order to use the filled function method, the facility layout problem was transformed into an unconstrained and multimodal (including more than one local minima) global optimization problem by using the hyperbolic smoothing technique and the penalty function method. Thus, in this first study in the literature where a deterministic method is combined with the solution of the facility layout problem, it is shown that the non-convex facility layout problem can be solved with the filled function method with very few iterations and short solution times.

Policy with guaranteed risk-adjusted performance for multistage stochastic linear problems

Policy with guaranteed risk-adjusted performance for multistage stochastic linear problems

Capture of two coordinated evaders in a linear pursuit problem on time scales

In a finite-dimensional Euclidean space, we consider the problem of pursuit of two evaders by a group of pursuers, described by a linear system with a simple matrix on a given time scale. It is assumed that the evaders use the same control. The pursuers employ quasistrategies based on information about the initial positions and control history of the evaders. The set of admissible controls for each participant is a ball of unit radius centered at the origin, and the terminal sets are the origin. The goal of the group of pursuers is to capture the two evaders. In the study, we use the method of resolving functions as a base one, which allows us to obtain sufficient conditions for the solvability of the approach problem in a certain guaranteed time. In terms of the initial positions and parameters of the game, a sufficient condition for capturing the evaders is obtained.

A novel global algorithm for solving linear multiplicative problem by integrating linear combination rule and branch-and-bound framework

A novel global algorithm for solving linear multiplicative problem by integrating linear combination rule and branch-and-bound framework

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

A new perspective on denoising based on optimal transport

Abstract In the standard formulation of the classical denoising problem, one is given a probabilistic model relating a latent variable $\varTheta \in \varOmega \subset{\mathbb{R}}^{m} \; (m\ge 1)$ and an observation $Z \in{\mathbb{R}}^{d}$ according to $Z \mid \varTheta \sim p(\cdot \mid \varTheta )$ and $\varTheta \sim G^{*}$, and the goal is to construct a map to recover the latent variable from the observation. The posterior mean, a natural candidate for estimating $\varTheta $ from $Z$, attains the minimum Bayes risk (under the squared error loss) but at the expense of over-shrinking the $Z$, and in general may fail to capture the geometric features of the prior distribution $G^{*}$ (e.g. low dimensionality, discreteness, sparsity). To rectify these drawbacks, in this paper we take a new perspective on this denoising problem that is inspired by optimal transport (OT) theory and use it to study a different, OT-based, denoiser at the population level setting. We rigorously prove that, under general assumptions on the model, this OT-based denoiser is mathematically well-defined and unique, and is closely connected to the solution to a Monge OT problem. We then prove that, under appropriate identifiability assumptions on the model, the OT-based denoiser can be recovered solely from information of the marginal distribution of $Z$ and the posterior mean of the model, after solving a linear relaxation problem over a suitable space of couplings that is reminiscent of standard multimarginal OT problems. In particular, due to Tweedie’s formula, when the likelihood model $\{ p(\cdot \mid \theta ) \}_{\theta \in \varOmega }$ is an exponential family of distributions, the OT-based denoiser can be recovered solely from the marginal distribution of $Z$. In general, our family of OT-like relaxations is of interest in its own right and for the denoising problem suggests alternative numerical methods inspired by the rich literature on computational OT.