Modeling One Dimensional Two-Cell Model with Tumor Interaction Using Krylov Subspace Methods

A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.

Evaluation of different Krylov subspace methods for simulation of the water faucet problem

In this study,a one-dimensional two-phase flow four-equation model was developed to simulate the water faucet problem.The performance of six different Krylov subspace methods,namely the generalized minimal residual(GMRES),transpose-free quasi-minimal residual,quasi-minimal residual,conjugate gradient squared,biconjugate gradient stabilized,and biconjugate gradient,was evaluated with and without the application of an incomplete LU(ILU)factorization preconditioner for solving the water faucet problem.The simulation results indicate that using the ILU preconditioner with the Krylov subspace methods produces better convergence performance than that without the ILU preconditioner.Only the GMRES demonstrated an acceptable convergence performance under the Krylov subspace methods without the preconditioner.The velocity and pressure distribution in the water faucet problem could be determined using the Krylov subspace methods with an ILU preconditioner,while GMRES could determine it without the need for a preconditioner.However,there are significant advantages of using an ILU preconditioner with the GMRES in terms of efficiency.The different Krylov subspace methods showed similar performance in terms of computational efficiency under the application of the ILU preconditioner.

Subspace-Based Blind Channel Estimation Method for MIMO-OFDM Systems

A subspace-based blind channel estination algo rithm for MIMO-OFDM systems is proposed. This algorithm exploits the cyclostationarity introduced by cyclic prefix of OFDM to estimate the channel parameters. The proposed new algorithm is found to be outperforming the other algorithm with respect to convergence rate and achievable mean square error and robustness to channel order over determination.

KRYLOV SUBSPACE PROJECTION METHOD AND ITS APPLICATION ON OIL RESERVOIR SIMULATION

Krylov subspace projection methods are known to be highly efficient for solving large linear systems. Many different versions arise from different choices to the left and right subspaces. These methods were classified into two groups in terms of the different forms of matrix H-m, the main properties in applications and the new versions of these two types of methods were briefly reviewed, then one of the most efficient versions, GMRES method was applied to oil reservoir simulation. The block Pseudo-Elimination method was used to generate the preconditioned matrix. Numerical results show much better performance of this preconditioned techniques and the GMRES method than that of preconditioned ORTHMIN method, which is now in use in oil reservoir simulation. Finally, some limitations of Krylov subspace methods and some potential improvements to this type of methods are further presented.

Subspace Methods in Multi-Parameter Seismic Full Waveform Inversion

In full waveform inversion(FWI)high-resolution subsurface model param-eters are sought.FWI is normally treated as a nonlinear least-squares inverse problem,in which the minimum of the corresponding misfit function is found by updating the model parameters.When multiple elastic or acoustic properties are solved for,simple gradient methods tend to confuse parameter classes.This is referred to as parameter cross-talk;it leads to incorrect model solutions,poor convergence and strong dependence on the scaling of the different parameter types.Determining step lengths in a subspace domain,rather than directly in terms of gradients of different parameters,is a potentially valuable approach to address this problem.The particular subspace used can be defined over a span of different sets of data or different parameter classes,provided it involves a small number of vectors compared to those contained in the whole model space.In a subspace method,the basis vectors are defined first,and a local min-imum is found in the space spanned by these.We examine the application of the sub-space method within acoustic FWI in determining simultaneously updates for velocity and density.We first discuss the choice of basis vectors to construct the spanned space,from linear updates by distinguishing only the contributions of different parameter classes towards nonlinear updates by adding the contributions of higher-order pertur-bations of each parameter class.The numerical character of FWI solutions generated via subspace methods involving different basis vectors is then analyzed and compared with traditional FWI methods.The subspace methods can provide better reconstructions of the model,especially for the velocity,as well as improved convergence rates,while the computational costs are still comparable with the traditional FWI methods.

Some Results on the Range-Restricted GMRES Method

In this paper we reconsider the range-restricted GMRES (RRGMRES) method for solving nonsymmetric linear systems. We first review an important result for the usual GMRES method. Then we give an example to show that the range-restricted GMRES method does not admit such a result. Finally, we give a modified result for the range-restricted GMRES method. We point out that the modified version can be used to show that the range-restricted GMRES method is also a regularization method for solving linear ill-posed problems.

IMPROVED COVARIANCE DRIVEN BLIND SUBSPACE IDENTIFICATION METHOD

An improved covariance driven subspace identification method is presented to identify the weakly excited modes. In this method, the traditional Hankel matrix is replaced by a reformed one to enhance the identifiability of weak characteristics. The robustness of eigenparameter estimation to noise contamination is reinforced by the improved Hankel matrix, in combination with component energy index （CEI） which indicates the vibration intensity of signal components, an alternative stabilization diagram is adopted to effectively separate spurious and physical modes. Simulation of a vibration system of multiple-degree-of-freedom and experiment of a frame structure subject to wind excitation are presented to demonstrate the improvement of the proposed blind method. The performance of this blind method is assessed in terms of its capability in extracting the weak modes as well as the accuracy of estimated parameters. The results have shown that the proposed blind method gives a better estimation of the weak modes from response signals of small signal to noise ratio （SNR）and gives a reliable separation of spurious and physical estimates.

Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full. description, of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invaxiant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators.

SVD-MPE: An SVD-Based Vector Extrapolation Method of Polynomial Type

An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.

奇异鞍点问题中广义位移分裂迭代方法的半收敛性分析

Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(2,2)-block.In this paper,we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite(1,1)-block and symmetric positive semidefinite(2,2)-block,prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix.Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use.

AINV AND BILUM PRECONDITIONING TECHNIQUES

It was proposed that a robust and efficient parallelizable preconditioner for solving general sparse linear systems of equations, in which the use of sparse approximate inverse (AINV) techniques in a multi-level block ILU (BILUM) preconditioner were investigated. The resulting preconditioner retains robustness of BILUM preconditioner and has two advantages over the standard BILUM preconditioner: the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner.

A Model-free Approach to Fault Detection of Continuous-time Systems Based on Time Domain Data

In this paper, a model-free approach is presented to design an observer-based fault detection system of linear continuoustime systems based on input and output data in the time domain. The core of the approach is to directly identify parameters of the observer-based residual generator based on a numerically reliable data equation obtained by filtering and sampling the input and output signals.

A Two-Level Preconditioning Technique for Wire Antennas Attached with Dielectric Objects

An iterative solution of linear systems is studied,which arises from the discretization of a wire antennas attached with dielectric objects by the hybrid finite-element method and the method of moment （hybrid FEM-MoM）.It is efficient to model such electromagnetic problems by hybrid FEM-MoM,since it takes both the advantages of FEM＇s and MoM＇s ability.But the resulted linear systems are complicated,and it is hard to be solved by Krylov subspace methods alone,so a two-level preconditioning technique will be studied and applied to accelerate the convergence rate of the Krylov subspace methods.Numerical results show the effectiveness of the proposed two-level preconditioning technique.

Invariant Subspace and Exact Solutions to the Generalized Kudryashov-Sinelshchikov Equation

In this research,invariant subspaces and exact solutions for the governing equation are obtained through the invariant subspace method,and the generalized second-order Kudryashov-Sinelshchikov equation is used to describe pressure waves in a liquid with bubbles.The governing equations are classified and transformed into a system of ordinary differential equations,and the exact solutions of the classified equation are obtained by solving the system of ordinary differential equations.The method gives logarithmic,polynomial,exponential,and trigonometric solutions for equations.The primary accomplishments of this work are displaying how to obtain the exact solutions of the classified equations and giving the stability analysis of reduced ordinarydifferential equations.

A QMR-Type Algorithm for Drazin-Inverse Solution of Singular Nonsymmetric Linear Systems

In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.

Subspace Search Method for Quadratic Programming With BoxConstraints

A subspace search method for solving quadratic programming with box constraints is presented in this paper. The original problem is divided into many independent subproblem at an initial point, and a search direction is obtained by solving each of the subproblem, as well as a new iterative point is determined such that the value of objective function is decreasing. The convergence of the algorithm is proved under certain assumptions, and the numerical results are also given.

A New Subspace Correction Method for Nonlinear Unconstrained Convex Optimization Problems

This paper gives a new subspace correction algorithm for nonlinear unconstrained convex optimization problems based on the multigrid approach proposed by S. Nash in 2000 and the subspace correction algorithm proposed by X. Tai and J. Xu in 2001. Under some reasonable assumptions, we obtain the convergence as well as a convergence rate estimate for the algorithm. Numerical results show that the algorithm is effective.

VIBRATION CONTROL OF FLUID-FILLED PRISMATIC SHELL WITH ACTIVE CONSTRAINED LAYER DAMPING TREATMENTS

Active constrained layer damping （ACLD） combines the simplicity and reliability of passive damping with the light weight and high efficiency of active actuators to obtain high damping over a wide frequency band. A fluid-filled prismatic shell is set up to investigate the validity and efficiency of ACLD treatments in the case of fluid-structure interaction. By using state subspace identification method, modal parameters of the ACLD system are identified and a state space model is established subsequently for the design of active control laws. Experiments are conducted to the fluid-filled prismatic shell subjected to random and impulse excitation, respectively, For comparison, the shell model without fluid interaction is experimented as well. Experimental results have shown that the ACLD treatments can suppress vibration of the fluid-free and fluid-filled prismatic shell effectively. Under the same control gain, vibration attenuation is almost the same in both cases.

SUBSPACE SEARCH METHOD FOR A CLASS OF LEAST SQUARES PROBLEM

Presents information on a study which described a subspace search method for solving a class of least squares problem. Derivation of the algorithm; Convergence results; Modification of algorithm and applications.