Indefinite System Research Articles

Analysis of (shifted) piecewise quadratic polynomial collocation for nonlocal diffusion model

The piecewise quadratic polynomial collocation is used to approximate the nonlocal model, which generally leads to a nonsymmetric indefinite system (Chen et al. (2021) [5]). In this case, the discrete maximum principle is not satisfied, which might be trickier for the stability analysis of the high-order numerical schemes (D'Elia et al. (2020) [10]; Leng et al. (2021) [26]). Here, we present a modified (shifted-symmetric) piecewise quadratic polynomial collocation for solving the linear nonlocal diffusion model, which leads to a symmetric positive definite system and satisfies the discrete maximum principle. Using Faulhaber's formula and Riemann zeta function, the perturbation error for symmetric positive definite system and nonsymmetric indefinite system are given. Then rigorous convergence analysis for the nonlocal models are provided under the general horizon parameter δ=O(hβ), with β≥0. More concretely, the global error is O(hmin⁡{2,1+β}) if δ is not set as a grid point, while it recovers O(hmax⁡{2,4−2β}) when δ is set as a grid point. We also prove that the shifted-symmetric scheme is asymptotically compatible, which has the global error O(hmin⁡{2,2β}) as δ,h→0. The numerical experiments (including two-dimensional case) are performed to verify the convergence.

HyKKT: a hybrid direct-iterative method for solving KKT linear systems

We propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBLT factorization where is a lower triangular matrix and is or block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBLT factorization of the full system.

Three-dimensional finite-difference finite-element frequency-domain wave simulation with multi-level optimized additive Schwarz domain-decomposition preconditioner: A tool for FWI of sparse node datasets

Efficient frequency-domain full-waveform inversion (FWI) of long-offset node data can be designed with a few discrete frequencies, which lead to modest data volumes to be managed during the inversion process. Moreover, attenuation effects can be straightforwardly implemented in the forward problem without the computational overhead. However, 3D frequency-domain seismic modeling is challenging because it requires solving a large and sparse linear indefinite system for each frequency with multiple right-hand sides (RHSs). This linear system can be solved by direct or iterative methods. The former allows efficient processing of multiple RHSs but may suffer from limited scalability for very large problems. Iterative methods equipped with a domain-decomposition preconditioner provide a suitable alternative to process large computational domains for sparse-node acquisition. We have investigated the domain-decomposition preconditioner based on the optimized restricted additive Schwarz (ORAS) method, in which a Robin or perfectly matched layer condition is implemented at the boundaries between the subdomains. The preconditioned system is solved by a Krylov subspace method, whereas a block low-rank lower-upper decomposition of the local matrices is performed at a preprocessing stage. Multiple sources are processed in groups with a pseudoblock method. The accuracy, the computational cost, and the scalability of the ORAS solver are assessed against several realistic benchmarks. In terms of discretization, we compare a compact wavelength-adaptive 27-point finite-difference stencil on a regular Cartesian grid with a P3 finite-element method on h-adaptive tetrahedral mesh. Although both schemes have comparable accuracy, the former is more computationally efficient, the latter being beneficial to comply with known boundaries such as bathymetry. The scalability of the method, the block processing of multiple RHSs, and the straightforward implementation of attenuation, which further improves the convergence of the iterative solver, make the method a versatile forward engine for large-scale 3D FWI applications from sparse node data sets.

An integrated lot-sizing policy for the inventory management of constrained multi-level supply chains: null-space method

ABSTRACT The null-space method (NSM) for solving the nonlinear programming (NLP) is used to transform an indefinite system into a symmetric positive definite one of a smaller dimension. In this paper, an NSM optimises the constrained NLP model of the inventories in multi-level supply chains (SCs). The presented NSM can give a direct method with a predictable level of fill without pivoting, decreasing the number of taken iterations to find the optimum solution. We transform the nonlinear equations into an NLP model, which NSM can solved. We also investigate the suitability of using null-space-based factorisations to derive sparse direct methods. Accordingly, an integrated lot-sizing model of the multi-level SC is designed and then optimised using the presented NSM. The paper's objectives are to find the optimum number of stockpiles and the economic period length for inventories. Some numerical examples demonstrate the applicability of the presented NSM to optimise the integrated lot-sizing policy of the multi-level SCs. The presented NSM shows satisfactory performance in optimum solutions, the number of iterations, infeasibility, optimality error, and complementarity.

Adaptive Filter based on Absolute Average Error Adaptive Algorithm for Modeling System

Adaptive identification of the bandpass finite impulse response (FIR) filtering system is proposed through this paper using variable step-size least mean square (VSS-LMS) algorithm called absolute average error-based adjusted step-size LMS as an adapted algorithm. This algorithm used to design an adaptive FIR filter by calculating the absolute averaged value for the recently assessed error with the previous one. Then, the step size has been attuned accordingly with consideration of the slick transition of the step size from bigger to smaller to score an achievement through high convergence rate and low steady-state misadjustment over the other algorithms used for the same purpose. The simulation results through the computer demonstrate remarkable performance compared to the traditional algorithm of LMS and another VSS-LMS algorithm (normalized LMS) which used in this paper for the designed filter. The powerful of the filter has been served in the identification system, bandpass filter has been chosen to be identified in the proposed adaptive system identification. It reports conceivable enhancements in the modeling system concerning the time of convergence, which is well-defined as a fast and steady-state adjustment defined with a low level. The designed filter identified the indefinite system with less than 10 samples; meanwhile, other algorithms were taking more than 20 samples for identification. Alongside the fine behavior of preserving the tradeoff between miss adjustment and the capability of tracking, the fewer calculations and computations regarding the algorithm requirement make the applied real-time striking.

Solution of the Analysis Problem of a Machine Assembly Distributed System with Time-Dependent Heat Transfer Coefficient

It is well known that temperature is a factor that significantly influences the accuracy of machine tools. Compensation enables machine errors to be reduced, even for a moderately accurate machine tool. The first step in compensation is to estimate the thermal characteristics of the machine part. Thermal models with distributed parameters provide high accuracy of estimation. In these thermal models, the environmental thermal fluctuations influencing the temperature may be taken into account as the time-dependent heat-transfer coefficient. The finite elements method facilitates simulation of the machine system geometry, but is computationally expensive. One approach is to use the simplified thermal model at an early stage of development, which allows the investigation of the temperature field and the possible influence of the environment at any point of the model. In this article, it is proposed to use the spectral method based on the expansion of the temperature function in a Fourier series to analyze the thermal model distributed along the axial coordinate presented in PDE form. To maintain the similarity of thermal processes and the model, the dimension parameters of the model should be chosen such that the Biot and Fourier coefficients would be the same for the model and the machine part. The proposed method allows the PDE to be represented as an indefinite system of linear algebraic equations for the coefficients of the Fourier series, which are the amplitudes of the space–time modes of the temperature function. The solution has the advantage of an analytical solution because it provides information about the model’s temperature at any point.

Particular properties of estimation of partial capacitances of insulation of three core power cables by applying aggregate measurements

This paper presents a description of specific properties of determining the values of partial capacitances of insulation gaps in power cables with paper insulation for various ways of forming and solving the system of linear algebraic equations. Possible ways of inspection the insulation of three core power cables for the estimation of values of partial capacitances by applying aggregate measurements which are based on various ways of connection of emittance meter to tested sample of power cable are given. Estimation of partial capacitances by the direct solution of a system of linear algebraic equations, by minimizing the root mean square error of solving an overdetermined system of equations by the least squares method, as well as by finding a normal solution of an indefinite system of equations by the pseudo-inverse matrix, is also considered. It is shown that minimization of the root mean square error by the least squares method and the direct solution of system of equations show quite similar results for the case of estimation of partial capacitances by means of aggregate measurements, at the same time the solution of an indefinite system of equations by the method of a pseudo-inverted matrix allows to reproduce rather accurately only 3 out of 6 values of partial capacitances. The uneven effect of frequency on the electrical capacitance of the insulation gaps between the cores of the power cable and between its cores and the sheath is shown. It was proposed to use the frequency dependence of the electrical capacitance of insulation gaps as an informative parameter about the technical state of insulating gaps between the cores of the power cable and between its cores and its sheath.
 
 Keywords: root mean square error; least squares method; system of linear algebraic equations; dielectric losses; dielectric permittivity.

A Robust Approximation of the Schur Complement Preconditioner for an Efficient Numerical Solution of the Elliptic Optimal Control Problems

In this paper, we consider the numerical solution of the optimal control problems of the elliptic partial differential equation. Numerically tackling these problems using the finite element method produces a large block coupled algebraic system of equations of saddle point form. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The solution of such systems is a major computational task and poses a greater challenge for iterative techniques. Thus they require specialised methods which involve some preconditioning strategies. The preconditioned solvers must have nice convergence properties independent of the changes in discretisation and problem parameters. Most well known preconditioned solvers converge independently of mesh size but not for the decreasing regularisation parameter. This work proposes and extends the work for the formulation of preconditioners which results in the optimal performances of the iterative solvers independent of both the decreasing mesh size and the regulation parameter. In this paper we solve the indefinite system using the preconditioned minimum residual method. The main task in this work was to analyse the 3 × 3 block diagonal preconditioner that is based on the approximation of the Schur complement form obtained from the matrix system. The eigenvalue distribution of both the proposed Schur complement approximate and the preconditioned system will be investigated since the clustering of eigenvalues points to the effectiveness of the preconditioner in accelerating an iterative solver. This is done in order to create fast, efficient solvers for such problems. Numerical experiments demonstrate the effectiveness and performance of the proposed approximation compared to the other approximations and demonstrate that it can be used in practice. The numerical experiments confirm the effectiveness of the proposed preconditioner. The solver used is robust and optimal with respect to the changes in both mesh size and the regularisation parameter.

Solvability and optimal stabilization controls of discrete-time mean-field stochastic system with infinite horizon

The paper addresses the optimal control and stabilization problems for the indefinite discrete-time mean-field system over infinite horizon. Firstly, we show the convergence of the generalized algebraic Riccati equations (GAREs) and establish their compact form GARE. By dealing with the GARE, we derive the existence of the maximal solution to the original GAREs along with the fact that the maximal solution is the stabilizing solution. Then, the maximal solution is employed to design the linear-quadratic (LQ) optimal controller and the optimal value of the control problem. Specifically, we deduce that under the assumption of exact observability, the mean-field system is L^{2}-stabilizable if and only if the GAREs have a solution, which is also the maximal solution. By semi-definite programming (SDP) method, the solvability of the GAREs is discussed. Our results generalize and improve previous results. Finally, some numerical examples are exploited to illustrate the validity of the obtained results.

A fast constrained image segmentation algorithm

Normalized cut or Ncut has been one of the most widely used models for image processing. A constraint can also be included in the framework of Ncut to represent a priori information for an effective image segmentation. This results in the so-called constrained Ncut problem. In this paper we present an observation that the constrained Ncut problem can be formulated as an indefinite system of equations under a mild condition on targeted segmentation results. We then show that the indefinite system can be effectively handled by the Augmented Lagrangian Uzawa iterative method together with a classical Algebraic Multigrid Method. Both mathematically and numerically, we demonstrate that the Augmented Lagrangian Uzawa method achieves a solution in one iteration. We show how the proposed method can be efficiently applied for the newly tested recursive two-way Ncut with constraints, i.e., a new constrained sequential segmentation as well. A number of numerical experiments are presented to confirm our theory and to show the superiority of the proposed method. In particular, numerical experiments show that the speed of our algorithm can be orders of magnitude faster than the previously proposed Projected Power Method, a significant improvement of conventional image segmentation algorithms.

Corona Virus; A Comprehensive Overview About Its Life Cycle and Pathogenecity

COVID-19, a rapidly spreading new strain of coronavirus, has affected more than 150 countries and received worldwide attention. The lack of efficacious drugs or vaccines against SARS-CoV-2 has further worsened the situation. Thus, there is an urgent need to boost up research for the development of effective therapeutics and affordable diagnostic against COVID-19. In this time of health crisis, it is the duty of scientific research community to provide alternative, effective and affordable strategies to vaccinate human bodies against viral infections-COVID-19 based on focused experimental approaches. Corona virus (CoV) is forward RNA virus on its surface, containing stick-shaped spikes. It is an undesirable broad genome of RNA, with an indefinite replication system. In mammals, birds, pigs and chickens, corona virus cause various infections. This triggers the pathogens of the upper respiratory tract, which can lead to death due to breathing illnesses. In this article the author briefly justify sudden occurrence of this highly pathogenic extreme lung disease and recently discovered in Middle East respiratory syndrome corona virus (MERS-CoV). This is review article are awareness about the CoVID-19 infection and its out break throughout the world. In this article brief history about life cycle, pathogenecity and transmittance of corona virus given.

Uzawa methods for a class of block three‐by‐three saddle‐point problems

SummaryIn this work, we consider numerical methods for solving a class of block three‐by‐three saddle‐point problems, which arise from finite element methods for solving time‐dependent Maxwell equations and some other applications. The direct extension of the Uzawa method for solving this block three‐by‐three saddle‐point problem requires the exact solution of a symmetric indefinite system of linear equations at each step. To avoid heavy computations at each step, we propose an inexact Uzawa method, which solves the symmetric indefinite linear system in some inexact way. Under suitable assumptions, we show that the inexact Uzawa method converges to the unique solution of the saddle‐point problem within the approximation level. Two special algorithms are customized for the inexact Uzawa method combining the splitting iteration method and a preconditioning technique, respectively. Numerical experiments are presented, which demonstrated the usefulness of the inexact Uzawa method and the two customized algorithms.

ОСЯГНЕННЯ ФЕНОМЕНА СМЕРТІ ЯК ЕКЗИСТЕНЦІЙНА ІДЕНТИЧНІСТЬ ОСОБИСТОСТІ

Modern philosophical anthropology reveals more and more complicated structure of a person’s inner world. While in the classic paradigm when reviewing the essence of a human the attention was paid to the importance of the person’s essential features, that is to the existence of certain circumstances (social determinants, divine origin, etc…), contemporary science considers it as something that splits into many epihpenomas, stratifies in its own ambiguity, which brings it to losing meaningfulness of the personal existence, takes away its objective reality and comprehension of its own existence and its self-completeness.In such circumstances, it is relevant to study the phenomena of human existence, which could update its life, «turn it face to face» with its life and itself. A human is the only creature, whose being is not determined by the fact of birth («to be a human» is not the same as «to belong to biological species»). It is transcendence that goes beyond the limits of the biological sphere and opens the existential dimension of the personality and forms the «humaneness in a human being».The essence of a person acts like «indefinite-definite»: while he / she is alive nothing is known for sure about him/ her and when death comes, it’s impossible to change anything. Life consists of many events, phenomena of existence, «singularities», «temporal identities», which together form a single integrity. Each of the «here-and-now» phenomena of life can affect a personal «I» as a whole and change it.In this sense, self-consciousness can be considered as the analogy of the fractal, as eye visualization of the idea of «chaotic» complexity, endless formation and incompleteness of many socio-cultural phenomena, that is, it is considered a stochastic system. The essence of the personality in the existential sense is constant transcendence, going out of the bounds of certainty, openness to the world, constant dialogue, etc. In this dialogue, the existence means both the beginning, or the «starting point», and a prospect, or an opportunity. Death is the end of any projects and opportunities.«We do not know where and when death awaits us, so let’s live as if it were near...» It is the comprehension of the phenomenon of death, death here and now that can annually determine its own certainty, opens to the person its own self in the unity, integrity of the manifestations of the phenomena. Therefore, the comprehension of the phenomenon is death (unavoidable and possible here and now) comes into collision with «here and now essence», «a meeting dialogue» with one of the «temporary self-independence». This is the dialogue between what you are now and what you would like to be. An essential feature of an individual, an existential identity consists of a set of «temporary» identities, each of which can become primordial and final. A person as «I-fractal» unfolds possible projects, self-identification. So, in this sense, the attainment of one’s own mortality can become an existential birth, a transition to higher levels of being. Human as «I-fractal» is a stochastic, indefinite system until its completion, and hence the change of completed Self-independence can be over at any time.

A class of second-order nonlocal indefinite impulsive differential systems

We consider the second-order nonlocal impulsive differential system \t\t\t{−x″=a(t)xy+ω(t)f(x),0<t<1,t≠tk,−y″=b(t)x,0<t<1,t≠tk,Δx|t=tk=Ik(x(tk)),k=1,2,…,n,Δy|t=tk=Jk(y(tk)),k=1,2,…,n,x(0)=∫01h(t)x(t)dt,x′(1)=0,y(0)=∫01g(t)y(t)dt,y′(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -x''=a(t)xy+\\omega (t)f(x), \\quad 0< t< 1, t\\neq t_{k}, \\\\ -y''=b(t)x, \\quad 0< t< 1, t\\neq t_{k}, \\\\ \\Delta x|_{t=t_{k}}=I_{k}(x(t_{k})), \\quad k=1,2,\\ldots,n, \\\\ \\Delta y|_{t=t_{k}}=J_{k}(y(t_{k})), \\quad k=1,2,\\ldots,n, \\\\ x(0)=\\int_{0}^{1}h(t)x(t)\\,dt, \\qquad x'(1)=0, \\\\ y(0)=\\int_{0}^{1}g(t)y(t)\\,dt, \\qquad y'(1)=0, \\end{cases} $$\\end{document} where the weight functions a(t), b(t), and omega (t) change sign on [0,1], and g(t)not equiv 0 and h(t)not equiv 0 on [0,1]. By constructing a cone K_{1}times K_{2}, which is the Cartesian product of two cones in space PC[0,1], and applying the well-known fixed point theorem of cone expansion and compression in K_{1}times K_{2}, we obtain conditions for the existence and multiplicity of positive solutions of a nonlocal indefinite impulsive differential system. An example is given to illustrate the main results.

The generalized C-to-R method for solving complex symmetric indefinite linear systems

Recently, Axelsson et al. proposed the so called C-to-R method to solve the complex symmetric definite linear systems (see Axelsson et al., Numer Algor. 2014;66:811–841). In this paper, a generalized C-to-R (GCTR) iterative method is presented for solving the complex symmetric indefinite linear system. The convergent analysis of GCTR method is given, which shows the GCTR method is convergent under the assumptions of and has smaller convergence factor than some existing methods. This GCTR method also can result in an efficient preconditioner, which also has tighter clustering than some existing preconditioners. Numerical results are presented, which show full robustness and effectiveness of our proposed method.

A comparative study of null‐space factorizations for sparse symmetric saddle point systems

SummaryNull‐space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null‐space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null‐space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.

A Geometric Frequency-Domain Wave Propagation Formulation for Fast Convergence of Iterative Solvers

The frequency-domain wave propagation problem is notoriously difficult to solve through iterative methods because it leads to a symmetric but indefinite linear system. For this reason, direct methods are usually employed at the expense of great memory usage. Convergence of iterative methods, however, could be obtained by regularizing the wave equation. We introduce such regularization in discrete geometric approach framework on polyhedral grids. Moreover, we extend the regularization to the impedance boundary condition.

Preconditioned steepest descent-like methods for symmetric indefinite systems

This paper addresses the question of what exactly is an analogue of the preconditioned steepest descent (PSD) algorithm in the case of a symmetric indefinite system with an SPD preconditioner. We show that a basic PSD-like scheme for an SPD-preconditioned symmetric indefinite system is mathematically equivalent to the restarted PMINRES, where restarts occur after every two steps. A convergence bound is derived. If certain information on the spectrum of the preconditioned system is available, we present a simpler PSD-like algorithm that performs only one-dimensional residual minimization. Our primary goal is to bridge the theoretical gap between optimal (PMINRES) and PSD-like methods for solving symmetric indefinite systems, as well as point out situations where the PSD-like schemes can be used in practice.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

AbstractWe consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.

A Robust Multilevel Method for the Time-harmonic Maxwell Equation with High Wave Number

A robust multilevel preconditioner for the time-harmonic Maxwell equation with high wave number is presented in this paper. We choose a stable continuous interior penalty finite element method as a coarse correction space and design different smoothers for the indefinite algebraic system in different mesh levels. The multilevel method is performed as a preconditioner in the outer GMRES iteration. To give quantitative insight of our algorithm we use local Fourier analysis to analyze the convergence property of the proposed multilevel method. Numerical results are presented to confirm the efficiency of our multilevel method.