Computing: brute force

All of the above methods use approximations to obtain low-dimensional embeddings of the data and so alleviate the computational bottlenecks associated with spectral dimensionality reduction. One strand of research does not however attempt to find an approximate solution to the problem, but rather uses advances in parallel computing to use computational brute force to obtain a low-dimensional embedding of large scale datasets. 

It can be said that these three main strategies have been applied equally and very often in combination. Basically, the first approach implies the use of a faster computer or a parallel architecture. To some extent it sounds like a brute force approach but the exponential increase of the computer power observed since 1970 has made the hardware solution one of the most popular approaches. The Chemical Abstracts Service (CAS) [10] was among first to use the hardware solution by distributing the CAS database onto several machines. 

If it is known that a drug must bind to a particular spot on a particular protein or nucleotide, then a drug can be tailor-made to bind at that site. This is often modeled computationally using any of several different techniques. Traditionally, the primary way of determining what compounds would be tested computationally was provided by the researcher s understanding of molecular interactions. A second method is the brute force testing of large numbers of compounds from a database of available structures. 

The algebraic/iterative and the brute force methods are numerical respectively computational techniques that operate on the chosen mathematical model. Raw residuals r are weighted to reflect the relative reliabilities of the measurements. 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

To calculate the inverse of a matrix by this procedure is equally tedious and probably more work than solving a set of equations by the brute-force high-school technique. However, the procedure is readily converted into computer code and this is now the only way recommended for matrix inversions. 

Molecules, in general, have some nontrivial symmetry which simplifies mathematical analysis of the vibrational spectrum. Even when this is not the case, the number of atoms is often sufficiently small that brute force numerical solution using a digital computer provides the information wanted. Of course, crystals have translational symmetry between unit cells, and other elements of symmetry within a unit cell. For such a periodic structure the Hamiltonian matrix has a recurrent pattern, so the problem of calculating its eigenvectors and eigenvalues can be reduced to one associated with a much smaller matrix (i.e. much smaller than 3N X 3N where N is the number of atoms in the crystal). 

The variables are updated and the step is repeated until convergence is achieved. Errors can be propagated either linearly in a matrix form by computing the derivatives of the unknowns with respect to the observations or by Monte-Carlo (brute force) techniques (e.g., Albarede 1995). 

One may follow the brute-force approach of enumerating fully all possible combinations of 0 - 1 variables for the elements of the y vector. Unfortunately, such an approach grows exponentially in time with respect to its computational effort. For instance, if we consider one hundred 0 - 1 y variables then we would have 2100 possible combinations. 

In practice, the number of trial orientations and positions for the phasing model is enormous, so a brute-force search is impractical, even on the fastest computers. The procedure is greatly simplified by separating the search for the best... 

Clearly, computational resources are not yet to the point that brute force methods will suffice for high-precision calculations. Physical and chemical intuition play an important role in constructing appropriate trial wave functions and, despite the complexity and size of the basis sets used in these calculations, can improve the convergence of the basis significantly. It is important, therefore, to understand the nature of an atomic or molecular system in terms of its physical as well as its more formal mathematical properties. 

In the previous section we briefly described the ab initio QC methods and the problems arising when they are applied to the modeling of complex systems. These problems cannot be considered as merely technical ones even if the computer power is sufficient and the required solution of the many electron problem can be obtained by brute force, the problem of the status of the result produced by the uncertainty introduced by poorly defined junction between the quantum and classical regions may still be important. Pragmatically, however, the resource requirements may have already... 

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