Local preconditioner

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 Hessians of potential energy functions, for example, separate naturally into local terms (among atom pairs involved in bonds, bond angles, and dihedral angles) and nonlocal terms (among nonbonded atom pairs). The number of local terms increases linearly with n, whereas the nonlocal terms increase as n2. Thus, a preconditioner from the local terms is a good choice that has performed well in practice.23-82... 

Can display local quadratic convergence Requires construction and factorization of preconditioner Performance may be slow for highly nonlinear functions when directions of negative curvature are detected repeatedly... 

The TN code in CHARMM uses a preconditioner from the local chemical interactions (bond length, bond angle, and dihedral-angle terms). This sparse matrix is rapid to compute and was found to be effective in practice. Other possibilities of preconditioners in general contexts have also been developed, such as a matrix derived from the BFGS update (defined in Section 6.1). ... 

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