Tree code method

The quantum-chemical tree code (QCTC) [M. Challacombe and E. Schwegler, J. Chem. Phys., 106, 5526 (1997)] is a modification of the classical tree-code method. The QCTC method allows calculation of the matrix elements of the Coulomb matrix J for large molecules in a time that is proportional to the number of basis functions b this calculation is 0 b), and one says that the calculation exhibits linear scaling with size of the molecule. Challacombe and Schwegler used the QCTC method to do an ab initio SCE MO calculation on the 698-atom monomer of the P53 protein at a fixed geometry (obtained from a protein data bank) using the 3-21G basis set (3836 basis functions). They then calculated the molecular electrostatic potential (Section 15.7) of the P53 monomer. (The P53 protein is a tetramer and acts as a tumor suppressor. Mutations in the gene for this protein are found in half of human cancers.)... 

Giese, T. J., and York, D. M. (2008). Extension of adaptive tree code and fast multipole methods to high angular momentum particle charge densities,/. Comput Chem. 29(12), 1895-1904. 

As mentioned in Sect. 2.5, in principle the FMM, multigrid methods, or tree codes can handle this situation, but they are much too slow for the normally only small number of charges involved, and error estimates are not easy to obtain. Also, a modified Ewald method in which the summation of the reciprocal-space vectors was modified [70], similar to the one used by Kawata and Mikami [71] exists, but also here the approximations made seem hard to control which render the method rather useless. 

FMM = fast tnultipole method RBM = recursive bisection method TC = tree code. 

A problem similar to the evaluation of J[p, p ] is encountered in simulations of systems of classical particles (point charges), where the particle-particle interaction also has a quadratic O(N ) scaling. For this problem, three algorithms with linear or near-linear scaling have been introduced recently the Fast Multipole Method (FMM), Tree Codes (TC), and the Recursive Bisection Method (RBM). The success of these three methods has prompted its application to the 7[p, p ] problem. ... 

FTAP The Eault Tree Analysis Erogram (Willie, 1978, Section 6.4, and di.stribution disk) is a cutset generation code developed at the University of California, Berkeley Operation Research I iter. FTAP is unique in offering the user a choice of three processing methods to >wn. 

Fault tree or equivalent analysis is key to PSA. Small logical structures may be evaluated by hand using the iciples of Chapter 2 but at some point computer support eeded. Even for simple structures, uncertainty analysis VIonte Carlo methods requires a computer. However, t of the codes are proprietary or a fee is charged for their... 

As discussed in Section 9.2, the Excel Solver uses a BB algorithm to solve MILP problems. It uses the same method to solve MINLP problems. The only difference is that for MINLP problems the relaxed subproblems at the nodes of the BB tree are continuous variable NLPs and must be solved by an NLP method. The Excel Solver uses the GRG2 code to solve these NLPs. GRG2 implements a GRG algorithm, as described in Chapter 8. 

Implementing the reordering algorithm is somewhat complicated, since the sorting requires comparison of sequences element-by-element. See (28) for a good implementation. Constructing the code for a tree of n vertices requires 0(n) time with this implementation. We can expect to find no faster algorithm, since any method must inspect the entire tree. 

We test the component for planarity. If it is planar, we encode it using one of the methods in Section 7. If it is not planar, we encode it using the partitioning algorithm of Section 1+. We use the codes for components as labels in the decomposition tree, and encode the tree (and thus the entire graph) using the method of Section 5. 

Fig. 2.10 Phylogeny of 88 full sequences coding for fungal and protistan cytochrome c peroxidases (CcP). The reconstructed tree obtained from the NJ-method of the MEGA package [13] is presented. A nearly identical tree was obtained with the ProML method of the PHYLIP...

JPRED [159, 160], or CODE [161] by systematically optimizing the performance on a training set by using decision-tree methods or machine-learning approaches such as support-vector machines [162]. 

Many methods are available to identify hazards—from historical data, from codes and standards, from the observations of learned people, and from the use of one of several analytical methods such as the more simple What-If system through to the more complex Fault Tree Analysis. And the literature, of which there is a great deal, speaks extensively of those methods. 

---