Solution Of Linear Systems Research Articles

Solutions of linear systems of moment differential equations via generalized matrix exponentials

A generalized exponential matrix based on the construction of kernel operators for generalized summability is defined and analyzing its main properties, generalizing the classical exponential matrix and fractional exponential matrix.This object serves as a practical tool to express the solutions of linear systems of moment differential equations in a compact manner, in the spirit of the classical exponential matrix.

Existence of solution of a nonlinear elliptic coupled system

This paper is concerned with the proof of the existence of solutions for a non linear elliptic coupled system in Sobolev space, more precisely such systems involve non linear partial differential equations of type elliptic.

Pre-twisted calculus and differential equations

In this paper we introduce the φA-differentiability for functions f:U⊂Rk→Rn, where U is an open set, A is the linear space Rn endowed with a unital associative commutative algebra product, and φ:U⊂Rk→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.

State-space solution of singular linear continuous-time systems using the conformable derivative and Sumudu transform

The aim of this work is the application of the Sumudu transform for solving singular continuous-time linear systems based on the conformable derivative operator. Thanks to the interesting properties of the conformable Sumudu transform that we have established, a new approach is developed. Through academic and real examples, our method is compared to the existing ones, where the applicability and the accuracy of the developed process are shown.

Multiple-Frequency Preconditioned Iterative Inverse Source Solutions

Inverse source formulations relate unknown equivalent radiation sources to given field observations, which are commonly obtained from measurements. After discretization, the linear operator equation is advantageously solved in the form of an appropriate positive semidefinite normal system of equations. Several preconditioning strategies for the iterative solution of such linear systems of equations are investigated in this article. Observation point density-based preconditioners balance the effect of the constraint equations and speed up the solution process for globally irregular sample location distributions. Reduced-accuracy approximations of the radiation operator based on propagating plane-wave expansions of the pertinent Green’s functions can speed up the operator evaluation considerably. They are used within inner–outer iterative solvers or, even more efficiently, within a two-step iterative solution of the inverse source problem. Multiple-frequency inverse source solutions are considerably accelerated by start vector estimations from already available solutions for other nearby frequencies, where in particular a modified Galerkin approach is found to be very effective. The presented accelerated iterative solution approaches achieve the same accuracy levels as its not accelerated counterparts. They are validated and evaluated for a variety of near-field (NF) and far-field (FF) observation data of different antennas, whereby the benefits of the introduced solution approaches are of particular importance for large antennas with many unknowns.

Correlation-based passive localization: Linear system modeling and sparsity-aware optimization.

In underwater passive localization, matched field processing has faced mismatch challenges for many years. To overcome the mismatch, data-derived replicas have gained increasing popularity due to their better localization performance than parameter-based ones, among which the cross correlation-based replicas showed enhanced robustness against the spectral differences between the library and target vessels. However, cross correlation-based matching localizations result in high sidelobes and require enough frequency samples to eliminate grating lobes. This issue has been widely reported in the previous literature but has yet to be theoretically analyzed. In this work, we revisit the conventional correlation-based matching procedures and formulate the ambiguity surface as a least-norm solution of an underdetermined linear system. This formulation permits understanding the sidelobes from an optimization perspective. Based on this view, the high sidelobe challenge and requirement of measurements are systematically tackled by recasting the matching problem as a sparsity-aware optimization problem. The superiority of the proposed optimization approach is showcased through simulated data in the waveguide, microphone data in the air, and SWellEx-96 data. The linear system modeling is also extended to a distributed sensor network, profiting from the spatial gain brought by various azimuthal directions with respect to the source.

Bernoulli wavelet method for numerical solution of linear system of Fredholm integral equation of the second kind

One of the key tools for many fields of applied mathematics is the integral equations. Integral equations are widely utilized in many models, atmosphere–ocean dynamics, fluid mechanics, mathematical physics, and many other physical and engineering disciplines. A new numerical strategy based on the Bernoulli wavelet is introduced to solve system of Fredholm integral equations of second kind. In this paper, the Bernoulli wavelets are first built. Second, the system of Fredholm integral equations has been reduced into an algebraic system. In order to demonstrate the viability, and accuracy of the suggested Bernoulli wavelet approach, some numerical examples are offered at the end. The derived numerical results are examined with those from other numerical techniques and with exact solutions, demonstrating the superiority of the proposed method over those techniques. The novelty of proposed technique is that it can be extended for the numerical solution of two dimensional integral equations and differential equations appearing in engineering models, however some modifications will be required.

Resolviendo ecuaciones diferenciales ordinarias con Symbolic Math Toolbox™ (Matlab) y SymPy (Python)

This paper shows solutions of ordinary differential equations (EDOS) obtained by using two symbolic packages: Symbolic Math Toolbox™ (Matlab) and SymPy (Python). The basic instructions to obtain solutions of both packages are explained step by step, through a group of examples from a traditional ordinary differential equations course. Differential equations that are solved with methods such as: separable variables, linear equations, indeterminate coefficients, variation of parameters, power series, Laplace transform, and numerical solutions are included. By means of the symbolic computation carried out with these packages it is possible to obtain the solution of linear systems, as well as the visualization of the direction field of a differential equation or of a non-linear system of differential equations. The main contribution of this work consists in providing the reader with a practical guide that allows him to start the study of differential equations assisted by Symbolic Math Toolbox™ or SymPy. Among the benefits of using these computational tools in teaching and/or learning practices, it is shown how the use of symbolic or numerical computation saves us effort in the computation of tedious calculations; focusing attention on important ideas and concepts such as: the relationship between the mathematical model and its physical counterpart, asymptotic behavior and qualitative analysis of the solutions.

Hyers–Ulam and Hyers–Ulam–Rassias Stability for Linear Fractional Systems with Riemann–Liouville Derivatives and Distributed Delays

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems.

Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization

Obtaining lightweight and accurate approximations of discretized objective functional Hessians in inverse problems governed by partial differential equations (PDEs) is essential to make both deterministic and Bayesian statistical large-scale inverse problems computationally tractable. The cubic computational complexity of dense linear algebraic tasks, such as Cholesky factorization, that provide a means to sample Gaussian distributions and determine solutions of Newton linear systems is a computational bottleneck at large-scale. These tasks can be reduced to log-linear complexity by utilizing hierarchical off-diagonal low-rank (HODLR) matrix approximations. In this work, we show that a class of Hessians that arise from inverse problems governed by PDEs are well approximated by the HODLR matrix format. In particular, we study inverse problems governed by PDEs that model the instantaneous viscous flow of ice sheets. In these problems, we seek a spatially distributed basal sliding parameter field such that the flow predicted by the ice sheet model is consistent with ice sheet surface velocity observations. We demonstrate the use of HODLR Hessian approximation to efficiently sample the Laplace approximation of the posterior distribution with covariance further approximated by HODLR matrix compression. Computational studies are performed which illustrate ice sheet problem regimes for which the Gauss–Newton data-misfit Hessian is more efficiently approximated by the HODLR matrix format than the low-rank (LR) format. We then demonstrate that HODLR approximations can be favorable, when compared to global LR approximations, for large-scale problems by studying the data-misfit Hessian associated with inverse problems governed by the first-order Stokes flow model on the Humboldt glacier and Greenland ice sheet.

Algebraic Multigrid Preconditioner for Statically Condensed Systems Arising from Lowest-Order Hybrid Discretizations

We address the numerical solution of linear systems arising from the hybrid discretizations of second-order elliptic partial differential equations. Such discretizations hinge on a hybrid set of degrees of freedom (DoFs), respectively, defined in cells and faces, which naturally gives rise to a global hybrid system of linear equations. Assuming that the cell unknowns are only locally coupled, they can be efficiently eliminated from the system, leaving only face unknowns in the resulting Schur complement, which is also called the statically condensed matrix. We propose in this work an algebraic multigrid (AMG) preconditioner specifically targeting condensed systems corresponding to lowest-order discretizations (piecewise constant). Like traditional AMG methods, we retrieve geometric information on the coupling of the DoFs from algebraic data. However, as the condensed matrix only gives information on the faces, we use the uncondensed version to reconstruct the connectivity graph between elements and faces. An aggregation-based coarsening strategy mimicking a geometric coarsening or semicoarsening can then be set up to build coarse levels. Numerical experiments are performed on diffusion problems discretized by the hybrid high-order method at the lowest order. Our approach uses a K-cycle to precondition an outer flexible Krylov method. The results demonstrate similar performances, in most cases, compared to a standard AMG method and a notable improvement on anisotropic problems with Cartesian meshes.

Quantum state preparation using tensor networks

Quantum state preparation is a vital routine in many quantum algorithms, including solution of linear systems of equations, Monte Carlo simulations, quantum sampling, and machine learning. However, to date, there is no established framework of encoding classical data into gate-based quantum devices. In this work, we propose a method for the encoding of vectors obtained by sampling analytical functions into quantum circuits that features polynomial runtime with respect to the number of qubits and provides accuracy, which is better than a state-of-the-art two-qubit gate fidelity. We employ hardware-efficient variational quantum circuits, which are simulated using tensor networks, and matrix product state representation of vectors. In order to tune variational gates, we utilize Riemannian optimization incorporating auto-gradient calculation. Besides, we propose a ‘cut once, measure twice’ method, which allows us to avoid barren plateaus during gates’ update, benchmarking it up to 100-qubit circuits. Remarkably, any vectors that feature low-rank structure—not limited by analytical functions—can be encoded using the presented approach. Our method can be easily implemented on modern quantum hardware, and facilitates the use of the hybrid-quantum computing architectures.

A two‐step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations

AbstractFor solving large sparse systems of linear equations, we construct a paradigm of two‐step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive‐definite matrix class. This two‐step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two‐step matrix splitting iteration methods. This result provides systematic treatment for the two‐step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems.

Stein-based preconditioners for weak-constraint 4D-var

Algorithms for data assimilation try to predict the most likely state of a dynamical system by combining information from observations and prior models. Variational approaches, such as the weak-constraint four-dimensional variational data assimilation formulation considered in this problem, can ultimately be interpreted as a minimization problem. One of the main challenges of such a formulation is the solution of large linear systems of equations which arise within the inner linear step of the adopted nonlinear solver. Depending on the adopted approach, these linear algebraic problems amount to either a saddle point linear system or a symmetric positive definite (SPD) one. Both formulations can be solved by means of a Krylov method, like GMRES or CG, that needs to be preconditioned to ensure fast convergence in terms of the number of iterations. In this paper we illustrate novel, efficient preconditioning operators which involve the solution of certain Stein matrix equations. In addition to achieving better computational performance, the latter machinery allows us to derive tighter bounds for the eigenvalue distribution of the preconditioned linear system for certain problem settings. A panel of diverse numerical results displays the effectiveness of the proposed methodology compared to current state-of-the-art approaches.

Graph Sparsification by Universal Greedy Algorithms

Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications, such as simplification of social networks, least squares problems, numerical solution of symmetric positive definite linear systems and etc. In this paper, inspired by the well-known sparse signal recovery algorithm called orthogonal matching pursuit (OMP), we introduce a deterministic, greedy edge selection algorithm called universal greedy approach (UGA) for graph sparsification. For a general spectral sparsification problem, e.g., positive subset selection problem from a set of $m$ vectors from $\mathbb{R}^n$, we propose a nonnegative UGA algorithm which needs $O(mn^2+ n^3/\epsilon^2)$ time to find a $\frac{1+\epsilon/\beta}{1-\epsilon/\beta}$-spectral sparsifier with positive coefficients with sparsity $\le\lceil\frac{n}{\epsilon^2}\rceil$, where $\beta$ is the ratio between the smallest length and largest length of the vectors. The convergence of the nonnegative UGA algorithm will be established. For the graph sparsification problem, another UGA algorithm will be proposed which can output a $\frac{1+O(\epsilon)}{1-O(\epsilon)}$-spectral sparsifier with $\lceil\frac{n}{\epsilon^2}\rceil$ edges in $O(m+n^2/\epsilon^2)$ time from a graph with $m$ edges and $n$ vertices under some mild assumptions. This is a linear time algorithm in terms of the number of edges that the community of graph sparsification is looking for. The best result in the literature to the knowledge of the authors is the existence of a deterministic algorithm which is almost linear, i.e. $O(m^{1+o(1)})$ for some $o(1)=O(\frac{(\log\log(m))^{2/3}}{\log^{1/3}(m)})$. Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches.

A 3D frequency-domain electromagnetic solver employing a high order compact finite-difference scheme

Three-dimensional (3D) electromagnetic (EM) modeling is an important tool for geophysical applications. In EM induction methods the modified Helmholtz equation is often used to describe scattered or residual electric fields in three dimensions. Throughout this paper, a high order compact finite difference scheme for the solution of that equation for vertical magnetic dipole source (VMD) is presented. The approximation of the residual electric field intensity using a fourth order compact finite difference (FD) discretizer is achieved with the solution of a block linear system, the coefficient matrices are large and sparse with a particular structure, implying the application of a matrix-free Krylov subspace method for an efficient numerical solution. The proposed solver is being examined using a number of test problems in uniform and non-uniform grid spacing where numerical and analytical solutions for the homogeneous half-space cases are being compared.

A finite convergence algorithm for solving linear-quadratic network games with strategic complements and bounded strategies

We propose a new algorithm for solving a class of linear-quadratic network games with strategic complements and bounded strategies. The algorithm is based on the sequential solution of linear systems of equations and we prove that it finds the exact Nash equilibrium of the game after a finite number of iterations. The new algorithm is then applied to a social network model of juvenile delinquency which has been investigated recently where we also consider random perturbations of some data. Experimental results show the efficiency of the algorithm in solving large scale problems.

Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications

In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the cover order and the cover length, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with non-negativity constraints on variables for both homogeneous and nonhomogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including linear programming (LP) problems and non-negative least squares (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem.

Traveling Wave Method

A survey of the development of the traveling wave method for one-dimensional media is presented. The main results and changes in the statement of the problem of representing solutions of linear systems of partial differential equations in terms of “traveling waves” (more precisely, in terms of a system of wave transport equations) are presented. It is shown that as the study of systems becomes more complicated, the problem of representing the solution by the traveling wave method turns out to be applicable not only for hyperbolic systems but also for systems containing (even implicitly) both parabolic and elliptic components and thereby approaches the general problem of decomposition of an arbitrary system of linear equations into a system of first-order equations with a main part of the canonical type and with a subordinate linear part.

A rate-of-injection model for predicting single and double injection with or without fusion.

Models used to predict the instantaneous injected fuel mass are of varied interest in automotive applications, including for providing inputs to CFD calculations or for engine control. While multiple injection strategies are now commonly used in diesel engines, the overall approach may be susceptible to injection fusion, which is defined as two successive injections that are partly or totally coupled due to the short time interval between each event. In this work, a new model to predict the instantaneous mass flow rate from a diesel injector is proposed based on the analytical solution of a first-order linear dynamic system exposed to an impulsion. Experiments are also conducted to quantify the main injection characteristics of a solenoid indirect-action injector under different injection pressures, backpressures and injection durations, representing a total of 33 different conditions. From these results, a model is proposed and validated against experimental data using a single injection strategy. Then, the model is enhanced to predict split injection with and without injection fusion. Successful comparisons are realized between the model and the experiment. The model is then used to successfully simulate a piezoelectric injector experiencing different levels of fusion available in the literature so as to illustrate the universality of the proposed approach.