Micro-iterative method

One promising approach to the problem of effectively reducing the number of direct Cl matrix-vector products is the approximate Hamiltonian operator method of Werner and coworkers - described in Section III. This is an extended micro-iterative method in which the Hamiltonian operator is allowed to be approximated during the solution of the wavefunction correction vector within an MCSCF iteration. 

In complete active space self-consistent field (CASSCF) calculations with long configuration expansions the most expensive part is often the optimization of the Cl coefficients. It is, therefore, particularly important to minimize the number of Cl iterations. In conventional direct second-order MCSCF procedures , the Cl coefficients are updated together with the orbital parameters in each micro-iteration. Since the optimization requires typically 100-150 micro-iterations, such calculations with many configurations can be rather expensive. A possible remedy to this problem is to decouple the orbital and Cl optimizations , but this causes the loss of quadratic convergence. The following method allows one to update the Cl coefficients much fewer times than the orbital parameters. This saves considerable time without loss of the quadratic convergence behaviour. 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

The solution of the orbital corrections has usually been uncoupled from the CSF corrections within the micro-iterative procedure with this approach . This usually causes no problems for ground-state calculations, but it is expected to be detrimental for excited states, particularly those with negative eigenvalues of the orbital Hessian matrix at convergence. This could be addressed by using, for example, the PSCI iterative method within a micro-iteration for a fixed operator to solve simultaneously for and Another disadvantage, for large numbers of virtual orbitals, is that the space... 

The FastStokes program [6, 7] was used to compute the drag forces on the micro-resonator shown schematically in Fig. 2. FastStokes combines the iterative method GMRES [8] with the precorrected-FFT [3] technique for computing fast matrix-vector products. The discretized, or meshed, structure is shown in Fig. 3. Note that both the... 

In the previous subsections, we developed Newton s method for the optimization of Cl wave functions and energies. Newton s method requites only a few macro (outer) iterations, but, in each macro iteration (i.e. in each Newton step), a relatively large number of micro (inner) iterations are needed for solving the linear equations (11.5.3). Each micro iteration requires the multiplication of the Hamiltonian matrix by a trial vector (11.5.1). The total number of micro iterations needed for convergence may therefore become quite large with Newton s method. 

Technology-enhancedformative assessment This method, also denoted as TEFA is based on the A2L approach and was proposed by Beatty Geraee (2009). TEFA specifies an iterative eyele of question posing, student discussion prior to selection of answers, post-discussion based on the responses without revealing the correct one and finally, a summary, micro-lecture or closure is provided including metalevel communication. The content of the final closure is normally determined by the previous part of the cycle. 

The optimization an MCSCF wave function is carried out as a sequence of inner and outer iterations. The outer iterations are those discussed in Sections 12.3 and 12.4 - the individual iterations of the second-order methods carried out to converge the wave function. The inner iterations are those carried out to solve the linear equations (12.5.1) or the eigenvalue equations (12.5.2). The inner and outer iterations are also referred to as micro and macro iterations. 

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