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.

Krylov subspace method based on data preprocessing technology

The performance of adaptive beamforming techniques is limited by the nonhomogeneous clutter scenario. An augmented Krylov subspace method is proposed, which utilizes only a single snapshot of the data for adaptive processing. The novel algorithm puts together a data preprocessor and adaptive Krylov subspace algorithm, where the data preprocessor suppresses discrete interference and the adaptive Krylov subspace algorithm suppresses homogeneous clutter. The novel method uses a single snapshot of the data received by the array antenna to generate a cancellation matrix that does not contain the signal of interest （SOI） component, thus, it mitigates the problem of highly nonstationary clutter environment and it helps to operate in real-time. The benefit of not requiring the training data comes at the cost of a reduced degree of freedom （DOF） of the system. Simulation illustrates the effectiveness in clutter suppression and adaptive beamforming. The numeric results show good agreement with the proposed theorem.

Adaptive coherence estimator based on the Krylov subspace technique for airborne radar

A novel adaptive detector for airborne radar space-time adaptive detection （STAD） in partially homogeneous environments is proposed. The novel detector combines the numerically stable Krylov subspace technique and diagonal loading technique, and it uses the framework of the adaptive coherence estimator （ACE）. It can effectively detect a target with low sample support. Compared with its natural competitors, the novel detector has higher proba- bility of detection （PD）, especially when the number of the training data is low. Moreover, it is shown to be practically constant false alarm rate （CFAR）.

Second-Order Krylov Subspace and Arnoldi Procedure

We report our recent work on a second-order Krylov subspace and the corresponding second-order Arnoldi procedure for generating its orthonormal basis. The second-order Krylov subspace is spanned by a sequence of vectors defined via a second-order linear homogeneous recurrence relation with coefficient matrices A and B and an initial vector u. It generalizes the well-known Krylov subspace K n(A;v), which is spanned by a sequence of vectors defined via a first-order linear homogeneous recurrence relation with a single coefficient matrix A and an initial vector v. The applications are shown for the solution of quadratic eigenvalue problems and dimension reduction of second-order dynamical systems. The new approaches preserve essential structures and properties of the quadratic eigenvalue problem and second-order system, and demonstrate superior numerical results over the common approaches based on linearization of these second-order problems.

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.

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.

DOA estimation of high-dimensional signals based on Krylov subspace and weighted l_1-norm

With the extensive application of large-scale array antennas,the increasing number of array elements leads to the increasing dimension of received signals,making it difficult to meet the real-time requirement of direction of arrival(DOA)estimation due to the computational complexity of algorithms.Traditional subspace algorithms require estimation of the covariance matrix,which has high computational complexity and is prone to producing spurious peaks.In order to reduce the computational complexity of DOA estimation algorithms and improve their estimation accuracy under large array elements,this paper proposes a DOA estimation method based on Krylov subspace and weighted l_1-norm.The method uses the multistage Wiener filter(MSWF)iteration to solve the basis of the Krylov subspace as an estimate of the signal subspace,further uses the measurement matrix to reduce the dimensionality of the signal subspace observation,constructs a weighted matrix,and combines the sparse reconstruction to establish a convex optimization function based on the residual sum of squares and weighted l_1-norm to solve the target DOA.Simulation results show that the proposed method has high resolution under large array conditions,effectively suppresses spurious peaks,reduces computational complexity,and has good robustness for low signal to noise ratio(SNR)environment.

The Convergence of Krylov Subspace Methods for Large Unsymmetric Linear Systems

The convergence problem of many Krylov subspace methods, e.g., FOM, GCR, GMRES and QMR, for solving large unsymmetric (non-Hermitian) linear systems is considered in a unified way when the coefficient matrix A is defective and its spectrum lies in the open right (left) half plane. Related theoretical error bounds are established and some intrinsic relationships between the convergence speed and the spectrum of A are exposed. It is shown that these methods are likely to converge slowly once one of the three cases occurs: A is defective, the distribution of its spectrum is not favorable, or the Jordan basis of A is ill conditioned. In the proof, some properties on the higher order derivatives of Chebyshev polynomials in an ellipse in the complex plane are derived, one of which corrects a result that has been used extensively in the literature.

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.

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.

Global quasi-minimal residual method for the Sylvester equations

In this paper, a global quasi-minimal residual （QMR） method was presented for solving the Sylvester equations. Some properties were investigated with a new matrix product for the global QMR method. Numerical results with the global QMR and GMRES methods compared with the block GMRES method were given. The results show that the global QMR method is less time-consuming than the global GMRES （generalized minimal residual） and block GMRES methods in some cases.

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.

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the Orr- Somrnerfeld and Squire （OSS） equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a re- orthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes （N-S） equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is pro- posed. The algorithm is then applied to a one-dimensional base flow （the plane Poiseuille flow） and a two-dimensional base flow （the plane Poiseuille flow with a low-speed streak） in a channel. For one-dimensional cases, the effects of the spanwise width of the chan- nel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is character- ized by high- and low-speed streaks and the corresponding streamwise vortical structures. The lift-up mechanism that induces the transient growth of perturbations is discussed. The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one- and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the cross- check method.

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.

Researted FOM for Multiple Shifted Linear Systems

The seed method is used for solving multiple linear systems A (i)x (i) =b (i) for 1≤i≤s, where the coefficient matrix A (i) and the right-hand side b (i) are different in general. It is known that the CG method is an effective method for symmetric coefficient matrices A (i). In this paper, the FOM method is employed to solve multiple linear sy stems when coefficient matrices are non-symmetric matrices. One of the systems is selected as the seed system which generates a Krylov subspace, then the resi duals of other systems are projected onto the generated Krylov subspace to get t he approximate solutions for the unsolved ones. The whole process is repeated u ntil all the systems are solved.

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.

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.

An incomplete generalized minimum backward perturbation algorithm for large nonsymmetric linear systems

This paper gives the truncated version of the generalized minimum backward error algorithm(GMBACK)—the incomplete generalized minimum backward perturbation algorithm(IGMBACK)for large nonsymmetric linear systems.It is based on an incomplete orthogonalization of the Krylov vectors in question,and gives an approximate or quasi-minimum backward perturbation solution over the Krylov subspace.Theoretical properties of IGMBACK including finite termination,existence and uniqueness are discussed in details,and practical implementation issues associated with the IGMBACK algorithm are considered.Numerical experiments show that,the IGMBACK method is usually more efficient than GMBACK and GMRES,and IMBACK,GMBACK often have better convergence performance than GMRES.Specially,for sensitive matrices and right-hand sides being parallel to the left singular vectors corresponding to the smallest singular values of the coefficient matrices,GMRES does not necessarily converge,and IGMBACK,GMBACK usually converge and outperform GMRES.

A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES

A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.

A RELAXED HSS PRECONDITIONER FOR SADDLE POINT PROBLEMS FROM MESHFREE DISCRETIZATION＊

In this paper, a relaxed Hermitian and skew-Hermitian splitting （RHSS） preconditioner is proposed for saddle point problems from the element-free Galerkin （EFG） discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS pre- conditioner to accelerate the convergence of some Krylov subspace methods （like GMRES） is also studied. Theoretical analyses show that the eigenvalues of the RHSS precondi- tioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.