Preconditioner

Peskin U, Miller W H and Ediund A 1995 Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner J. Chem. Phys. 103 10 030... 

For wet ESPs, consideration must be given to handling wastewaters. For simple systems with innocuous dusts, water with particles collected by the ESP may be discharged from the ESP system to a solids-removing clarifier (either dedicated to the ESP or part of the plant wastewater treatment system) and then to final disposal. More complicated systems may require skimming and sludge removal, clarification in dedicated equipment, pH adjustment, and/or treatment to remove dissolved solids. Spray water from an ESP preconditioner may be treated separately from the water used to wash the ESP collecting pipes so that the cleaner of the two treated water streams may be returned to the ESP. Recirculation of treated water to the ESP may approach 100 percent (AWMA, 1992). 

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... 

In the last decade, most new algorithms, schemes, solvers, and preconditioners have found their way into most commercial software packages. Multigrid solvers are also available. Furthermore, all CFD vendors have developed powerful pre- and post processing routines. 

Calculation of Energy Levels and Wave Functions Using an Efficient Preconditioner with the Inexact Spectral Transform Method. 

Time Evolution in Time-Dependent Fields and Time-Independent Reactive-Scattering Calculations via an Efficient Fourier Grid Preconditioner. 

Step 2. Compute the preconditioner for the amphtude equations, given by the diagonal linear terms of the amphtude equations (Eqs. (38) and (39)). 

The simplest implementation is to add only one new basis vector at a time, and to use a diagonal preconditioner... 

This defines the standard Davidson method. A very common improvement is the Davidson-Liu method, which uses several vectors at a time, and a different, diagonal, preconditioner for each root ... 

Furthermore, the preconditioner can be improved First separate out a small subset of preselected configurations, and diagonalize A in this subspace. Split up... 

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... 

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.

Preconditioning involves modification of the target linear system Ax = -b through application of a positive-definite preconditioner M that is closely related to A. The modified system can be written as... 

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... 

In the PCG process of the inner loop, Hessian/vector multiplications (Hd) and linear solutions of the system Mz — r for the preconditioner M are required repeatedly (see the linear PCG Algorithm [A3]). The products Hd can generally be computed satisfactorily by the following finite-difference design of gradients, at the expense of only one additional gradient evaluation per inner iteration ... 

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... 

When the preconditioner is constructed from a natural separability of the problem into terms of differing complexity, it may not necessarily be positive-definite, as required for straightforward implementation of PCG. A very useful technique for optimization has been a replacement of the standard Cholesky factorization of positive-definite systems by a modified Cholesky (MC) al-gorithm.5 137139 The MC process detects indefiniteness during the factorization itself and produces a decomposition for... 

Overall, a combination of effective truncation criterion, preconditioner, factorization of the preconditioner, and calculation of Hessian/vector products can produce a very powerful TN algorithm. Perhaps more than others, TN methods require problem tailoring for best performance. 

We assume that the function value and gradient are evaluated together in an operations (additions and multiplications), where n is the problem size and a is a problem-dependent number. The Hessian can then be computed in (a/2)n(n + 1) operations. When a sparse preconditioner M is used, we denote its number of nonzeros by m and the number of nonzeros in its Cholesky factor, L, by /. (We assume here that M either is positive-definite or is factored by a modified Cholesky factorization.) Thus M can be computed in about (a/2)nm operations. As discussed in the previous section, it is advantageous to reorder the variables a priori to minimize the fill-in for M. Alternatively, a precon-... 

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... 

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