Approximate preconditioner

It should be noted that a seemingly worse approximate preconditioner sometimes improves convergence. This happens e.g. when a Slater determinant basis is used, and the reason is that the diagonal approximation breaks the spin symmetry. If the preconditioner is replaced by... 

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... 

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. 

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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