Matrix preconditioner

The matrix A is known as the preconditioner and has to be chosen such that the condition number of the transformed linear system is smaller than that of the original system. 

Krylov subspace methods (such as Conjugate Gradient CG, the improved BiCGSTAB, and GMRES) in combination with preconditioners for matrix manipulations aimed at enhanced convergence, and... 

The 9x9 matrix = o.l + 0.9 S,j has one eigenvalue close to 2. Use Davidson s method to find the eigenvalue and the corresponding eigenvector. Note In this case, the preconditioner (3.2.11) will be singular. Common practice is that whenever the preconditioning formula will require a divide by 0, it is replaced by 1. 

Figure 2 Sample matrix patterns for (a) block diagonal and (b-e) sparse unstructured. Pattern (b) corresponds to the Hessian approximation (preconditioner) for a potential energy model from the local energy terms (bond length, bond angle, and dihedral angle terms), and (c) is a reordered matrix pattern that reduces fill-in during the factorization. Pattern (d) comes from a molecular dynamics simulation of super-coiled DNA36 and describes pairs of points along a ribbonlike model of the duplex that come in close contact during the dynamics trajectory pattern (e) is the associated reordered structure that reduces fill-in.

The preconditioner is problem dependent and should be chosen in large-scale applications as a sparse approximation to H that can be factored rapidly. A Cholesky factorization of a positive-definite matrix M produces... 

In cases where iterative methods are employed to solve large, sparse linear systems, both the efficiency and robustness of these methods can be significantly improved by use of preconditioners. A preconditioner of a matrix A is a matrix such that has a smaller condition number than A. The... 

Preconditioning is a technique which improves the condition number of a matrix and thereby increases the convergence rate of Krylov subspace methods. Thus, if the preconditioner A4 is a symmetric, positive definite matrix, the original problem Ax = b can be solved indirectly by solving M Ax = M h. The the residual can then be written as ... 

The well-known domain decomposition (DD) methods, see e.g. Chan Mathew (1994), use data partition induced by a decomposition of the computational domain. This decomposition can be used for two purposes Firstly for parallel implementation of vector updates, inner products and matrix-vector multiplication, i.e. for parallel implementation of the CG method. Secondly, for a construction of efficient preconditioners. 

The conjugate gradient is appealing. This is especially true of the version explained later, which uses a preconditioner of the Hessian matrix, when the matrix is sparse, nonstructured, and very large. 

Input vector of right-hand side terms b, first guess x and preconditioner matrix M. Output solution x. 

Solve the previous system using the three-diagonal matrix obtained by A as a preconditioner matrix. The program is... 

This version solves a linear system or the equivalent minimization of a quadratic function using the PCG method where the preconditioner is the band matrix with the band dimension provided by the integer in the argument. 

Solve the previous linear system by using the Solve function and the band matrix with bandwidth 1 as the preconditioner. 

Therefore, although the Hessian is very sparse and its nonzero elements are known, it might be better, in the initial phase, to calculate the product Gp using the formula (4.18) and a limited number of iterations to estimate d. In this case, the Hessian is calculated only in the final phase and is kept constant for several iterations together with the possible preconditioner matrix M. 

Given the function (4.20), calculate the direction d using the Algorithm 4.4 in xq = 0.5 -with the diagonal matrix of the second derivatives as preconditioner. Next, compare the sequence of y obtained by the CG and PCG procedures. 

Unfortunately, there is no criterion regarding the selection of the best preconditioner for a generic matrix. 

The preconditioner M is a band matrix obtained from the Hessian. 

The preconditioner M is a matrix obtained during a partial factorization of the matrix A by Cholesky by limiting as possible the filling-up problem (Nocedal and Wright, 2000). 

If the structure of the Hessian matrix is not provided, the product Gp is performed using the approximation (4.18). In this case, the diagonal of the Hessian is used as preconditioner. 

If the iterative PCG method is adopted, the preconditioner M (see Chapter 4) must be the closest one to the coefficient matrix of the system, N GN. 

Here, C has to be chosen as suitable preconditioner to S but being easier to invert in an iterative way. The number of PSC cycles L can be fixed or chosen adaptively so as to achieve a prescribed tolerance for the residual. The task at this point is to construct a suitable preconditioner C, which would provide a sufficiently good approximation of S. Since in our case, the Reynolds numbers in the considered flow configurations are relatively small (laminar flows), and if the time steps are small enough to resolve to complex dynamical behavior, the (lumped) mass matrix Ml proves to be a reasonable approximation to the complete operator A, so that our basic iteration (13.19) for the PSC equation can be interpreted and implemented as a discrete projection scheme as follows ... 

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