Nonsingular Systems Research Articles

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.

Truncated Newton full waveform inversion method for the human brain imaging

Abstract It has been shown that full-waveform inversion (FWI) method can be a competitive alternative for medical imaging problems. It offers high-resolution results while delivering the advantages of being fast, safe, portable, and affordable, compared to magnetic resonance imaging (MRI) and x-ray computed tomography. However, to the best of our knowledge, FWI applications in medical imaging only use the first-order derivative information. In this case, the high parameter contrasts between different tissues of human body and multi-scatterings problems may lead FWI to local minima. Thus, we present a competitive truncated Newton method for high-resolution imaging of the human brain. This truncated Newton method, based on the efficient linear solver MINRES-QLP, can make full use of multiple scattering wavefield information. Compared with the truncated Newton method based on conjugate gradient, the MINRES-QLP iterative method presents various advantages when solving linear systems, such as the capacity to handle both non-singular and singular systems, less computational cost, and efficiency even for ill-conditioned systems. Numerical experiments for imaging the brain with and without the skull are conducted. Numerical results indicate that, compared with the truncated Newton method based on conjugate gradient, the truncated Newton method based on MINRES-QLP exhibits computational efficiency while maintaining the same level of accuracy.

Multi-sensor information fusion filtering for delayed nonlinear stochastic singular systems with packet disorders

The fusion filtering problem is investigated for a class of time-varying nonlinear singular systems with packet disorders (PDs). The key idea is to transfer the original singular systems with multiple sensors to full-order augmented nonsingular systems by utilizing the matrix transformation method. The phenomenon of PDs accompanied with random transmission delays (RTDs) is considered, which exists in the channel from sensor to filter. A set of independent and identically distributed random variables obeying known probability distribution are introduced to model the RTDs. A centralized fusion filter is designed, which employs the rounding function of the mathematical expectation of the RTDs. Furthermore, the nonlinear function is linearized by utilizing the method of Taylor series expansion. Then, the desired upper bounds of the estimation error covariance are guaranteed and then locally minimized through appropriately designing the corresponding filter gains. Finally, a numerical example with comparisons is carried out to verify the feasibility of the proposed fusion filtering method.

Fast simulation of planar Clifford circuits

A general quantum circuit can be simulated classically in exponential time. If it has a planar layout, then a tensor-network contraction algorithm due to Markov and Shi has a runtime exponential in the square root of its size, or more generally exponential in the treewidth of the underlying graph. Separately, Gottesman and Knill showed that if all gates are restricted to be Clifford, then there is a polynomial time simulation. We combine these two ideas and show that treewidth and planarity can be exploited to improve Clifford circuit simulation. Our main result is a classical algorithm with runtime scaling asymptotically as nω/2<n1.19 which samples from the output distribution obtained by measuring all n qubits of a planar graph state in given Pauli bases. Here ω is the matrix multiplication exponent. We also provide a classical algorithm with the same asymptotic runtime which samples from the output distribution of any constant-depth Clifford circuit in a planar geometry. Our work improves known classical algorithms with cubic runtime.A key ingredient is a mapping which, given a tree decomposition of some graph G, produces a Clifford circuit with a structure that mirrors the tree decomposition and which emulates measurement of the corresponding graph state. We provide a classical simulation of this circuit with the runtime stated above for planar graphs and otherwise ntω−1 where t is the width of the tree decomposition. Our algorithm incorporates two subroutines which may be of independent interest. The first is a matrix-multiplication-time version of the Gottesman-Knill simulation of multi-qubit measurement on stabilizer states. The second is a new classical algorithm for solving symmetric linear systems over F2 in a planar geometry, extending previous works which only applied to non-singular linear systems in the analogous setting.

$p$-Nonsingular systems of equations over solvable groups

$p$-Nonsingular systems of equations over solvable groups

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

Exponential Stability and Relative Controllability of Nonsingular Conformable Delay Systems

In this paper, we investigate a delayed matrix exponential and utilize it to derive a representation of solutions to a linear nonsingular delay problem with permutable matrices. To begin with, we present a novel definition of α-exponential stability for these systems. Subsequently, we put forward several adequate conditions to ensure the α-exponential stability of solutions for such delay systems. Moreover, by constructing a Grammian matrix that accounts for delays, we provide a criterion to determine the relative controllability of a linear problem. Additionally, we extend our analysis to nonlinear problems. Lastly, we offer several examples to verify the effectiveness of our theoretical findings.

On the convergence of randomized and greedy relaxation schemes for solving nonsingular linear systems of equations

We extend results known for the randomized Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We obtain convergence results for a whole range of parameters describing the probabilities in the randomized method or the greedy choice strategy in the Gauss-Southwell-type methods. We identify those choices which make our convergence bounds best possible. Our main tool is to use weighted ℓ1-norms to measure the residuals. A major result is that the best convergence bounds that we obtain for the expected values in the randomized algorithm are as good as the best for the deterministic, but more costly algorithms of Gauss-Southwell type. Numerical experiments illustrate the convergence of the method and the bounds obtained. Comparisons with the randomized Kaczmarz method are also presented.

Nonsingular transformations that are ergodic with isometric coefficients and not weakly doubly ergodic

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are ergodic with isometric coefficients but are not weakly doubly ergodic; hence these two notions are not equivalent for infinite measure. We also give type IIIλ examples of such systems, for 0<λ≤1. We prove that under certain hypotheses, systems that are weakly mixing are ergodic with isometric coefficients and along the way we give an example of a uniformly rigid topological dynamical system along the sequence (ni) that is not measure theoretically rigid along (ni) for any nonsingular ergodic finite measure.

On Reeb components of nonsingular polynomial differential systems on the real plane

Denote by r(n) the maximal number of Reeb components that a nonsingular polynomial differential system of degree n on the real plane can have. It is known that r(0)=r(1)=0, r(2)=r(3)=2 and n−1≤r(n)≤n for n≥4. In this paper we prove that r(n)=n for all n≥4. This completely solves, for nonsingular systems, a problem posed by Chicone and Tian in 1982. We prove our result by explicitly constructing nonsingular Hamiltonian systems of degree n presenting n Reeb components for any n≥4.

Finite time stability for nonsingular impulsive first order delay differential systems

This primer article focuses on the representation of solutions and finite-time stability of impulsive first-order delay differential systems. We define delayed matrix function with impulses and use variation of parameters to obtain a representation of solutions of linear systems with impulse effects. The famous classical Grownwall inequalities and properties of delayed matrix exponential with impulses are used to develop sufficient conditions for finite-time stability. In the end, we provide some examples to support the results.

Krieger’s type of nonsingular Poisson suspensions and IDPFT systems

Given an infinite countable discrete amenable group Γ \Gamma , we construct explicitly sharply weak mixing nonsingular Poisson Γ \Gamma -actions of each Krieger’s type: I I I λ III_\lambda , for λ ∈ [ 0 , 1 ] \lambda \in [0,1] , and I I ∞ II_\infty . The result is new even for Γ = Z \Gamma =\mathbb {Z} . As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli Γ \Gamma -actions and infinite direct product of finite type systems of each possible Krieger’s type.

A generalized Shift-HSS splitting method for nonsingular saddle point problems

&lt;abstract&gt;&lt;p&gt;In this paper, we propose a generalized shift-HSS (denoted by SFHSS) iteration method for solving nonsingular saddle point systems with nonsymmetric positive definite (1, 1)-block sub-matrix, and theoretically verify its convergence property. In addition, we discuss the algebraic properties of the resulted SFHSS preconditioner and estimate the sharp eigenvalue bounds of the related preconditioned matrix. Finally, numerical experiments are given to support our theoretical results and reveal that the new method is feasible and effective.&lt;/p&gt;&lt;/abstract&gt;

The modified Uzawa methods for solving singular linear systems

For nonsingular linear systems, Yun in 2013 studied three variants of the Uzawa method; see Yun (2013) [18]. These methods contain the Uzawa-AOR method and the Uzawa-SAOR method as special cases. On the basis of the Uzawa-AOR method and the Uzawa-SAOR method, Li in 2017 proposed two modified Uzawa methods by constructing the coefficient matrix of AOR method into a new form of the product of lower triangular matrix and upper triangular matrix; see Li (2017) [26]. In this paper, we present two modified Uzawa methods for solving singular linear systems. The semi-convergence of these methods is analyzed by using the techniques of singular value decomposition and Moore-Penrose inverse. The numerical results are used to verify the theoretical results.

State feedback H ∞ control for a class of affine nonlinear singular systems: Input restricting approach

Here, a new method for solving the H∞ control problem for a class of affine nonlinear singular systems is presented. The main idea is to restrict the control inputs and initial states to the sets of admissible inputs and consistent initial values, respectively. Consequently, the impulse-freeness of the system response is guaranteed. In this regard, a novel concept of “distance from the admissible inputs set” is introduced. Based on this concept, a set of sufficient conditions for the H∞ control problem's solvability and the corresponding H∞ controller is provided. The adopted approach can also be employed to solve the H∞ control problem for indefinite nonlinear singular systems as well as non-singular nonlinear systems with a restricted inputs set. Sufficient conditions for the solvability of and corresponding solution to these problems are also presented. Finally, the practicality of this method is illustrated using a numerical example.

Alternating stationary iterative methods based on double splittings

Matrix double splitting iterations are simple in implementation while solving real non-singular (rectangular) linear systems. In this paper, we present two Alternating Double Splitting (ADS) schemes formulated by two double splittings and then alternating the respective iterations. The convergence conditions are then discussed along with comparative analysis. The set of double splittings used in each ADS scheme induces a preconditioned system which helps in showing the convergence of the ADS schemes. We also show that the classes of matrices for which one ADS scheme is better than the other, are mutually exclusive. Numerical experiments confirm that the proposed ADS schemes have several computational advantages over the existing methods. Though the problems are considered in the rectangular matrix settings, the same problems are even new in non-singular matrix settings.

Time-varying output formation-containment control for homogeneous/heterogeneous descriptor fractional-order multi-agent systems

In this paper, the time-varying output formation-containment (TV-OFC) problem of homogeneous and heterogeneous descriptor fractional-order multi-agent systems (DFO-MASs) is investigated. The results obtained in previous works for nonsingular integer-order multi-agent systems can be regarded as particular cases of our results. This control aims at making leaders keep time-varying shape to move and followers move inside the convex hull spanned by leaders simultaneously. First, both the output feedback control strategy and observer-based control algorithm are designed to address the TV-OFC problem of heterogeneous DFO-MASs, where the virtual leader provides the reference trajectory for leaders to achieve the formation tracking. Then, sufficient conditions with less computation complexity are presented and the methods to determine gain matrices of the schemes are proposed for homogeneous and heterogeneous DFO-MASs, so as to achieve the TV-OFC. Finally, numerical examples are given to demonstrate the effectiveness of theoretical conclusions.

Neural network approach for solving nonsingular multi-linear tensor systems

The main propose of this paper is to develop two neural network models for solving nonsingular multi-linear tensor system. Theoretical analysis shows that each of the neural network models ensures the convergence performance. For possible hardware implementation of the proposed neural network models, based on digital circuits, we adopt the Euler-type difference rule to discretize the corresponding Gradient neural network (GNN) models. The computer simulation results further substantiate that the models can solve a multi-linear system with nonsingular tensors.

Robust generalized observer design for uncertain one-sided Lipschitz systems

This paper develops a novel generalized observer design approach for the uncertain descriptor systems with one-sided Lipschitz nonlinearities, parametric uncertainties, and external perturbations. The nonlinearities, uncertainties, and external perturbations are considered both in input and output equations to consider a matter-of-fact observer design. A generalized structure for the observer is employed to deal with a large number of systems, considering both non-singular and singular systems. The proposed observer scheme is based on non-strict and strict linear matrix inequalities (LMIs), which are derived using the concepts of generalized Lyapunov theory, uncertainty bounds, quadratic inner-boundedness, one-sided Lipschitz condition, matrix transformations, and L2 gain minimization criteria. The proposed state filtering approach is robust for disturbances with asymptotically stable estimation error dynamics under zero external perturbations and attenuation of disturbance effects to keep the estimation error within prescribed limits. In comparison to the conventional observer designs for the one-sided Lipschitz systems, the presented scheme is based on a generalized observer and can deal with parametric uncertainties. A numerical simulation example and an application example of spring-mass-damper system are provided to verify the effectiveness of the suggested filtering schemes.

Asymptotics of the Solution to the Minimization Problem of the Integral Quadratic Performance Index on Trajectories of a Quasi-Linear System

The problem of minimizing an integral quadratic performance index on the trajectories of a quasi-linear dynamic system with a small parameter multiplying the nonlinearities subject to terminal constraints is considered. Asymptotic approximations of the optimal control in this problem in the form of open loop control and feedback are constructed. The computations are reduced to solving the unperturbed linear-quadratic problem, integrating systems of linear differential equations, and finding the roots of nonsingular linear algebraic systems.