Iterative multi-dimensional

Stone, H.L., "Iterative Solution of Implicit Approximations of Multi-Dimensional Partial Differential Equations", SIAM J. Numerical Analysis, 5, 530-558 (1968). 

In the multi-dimensional case, the simplest generalization of this procedure is to carry out the process iteratively. Thus, for LiOH, for example, we might first find a parabolic minimum for the OH bond, then for the LiO bond, then for the LiOH bond angle (in each case holding... 

We have already had a brief introduction to the iterative solutions of equations on the form f(x)=0. We implicitly assumed that both f and x were complex- or real-valued. However, most of it carries over directly to multi-dimensional cases. Some new nomenclature has to be presented, and in practice the computational complexity and the convergence problems may be more severe, but the basics are precisely the same. We will assume here that the problem has been mapped onto TV1. None of the problems concerning the parametrization, i.e. how to select a proper mapping of x onto 7 , will be considered here our problem will be assumed to have the already parametrized form f(x)=0, with both x and f in 7Jn. We begin by studying some general iterative procedure... 

Operations performed on the data can be computation-intensive, such as some matrix operations and repetitive multi-dimensional Fourier transforms, and other iterative calculations can proceed for thousands of steps. 

Like PCA, NLM or multi-dimensional scaling, is a method for visualizing relationships between objects, which in medicinal chemistry context often are compounds, but could equally be a number of measured activities." It is an iterative minimization procedure which attempts to preserve interpoint distances in multi-dimensional space in a 2D or 3D representation. Unlike PCA, however, the axes are not orthogonal and are not clearly interpretable with respect to the original variables. However, it can be valuable in cases where the first two or three PCs are influenced by outliers (extreme data points) or only explain a small percentage of the original data. NLM has been used to cluster aromatic and aliphatic substituents," " for example. 

At the vehicle systems level, there are thousands of interacting parameters that can influence the performance of a fuel cell vehicle. ADVISOR makes the task of quantifying the impacts of these parameters easier. However, studies that go beyond one or two parameters are difficult to do through manual iteration. Optimization tools have been incorporated into ADVISOR by NREL to automate the process of multi-dimensional analysis of fuel cell hybrid vehicles. 

Stone H.L. 1986. Iteration solution of implicit approximations of multi-dimensional partial differential equations. J. Soc. Ind. Appl. Math. 5(6) pp.530-558. 

Nonlinear optimization is one of the crucial topics in the numerical treatment of chemical engineering problems. Numerical optimization deals with the problems of solving systems of nonlinear equations or minimizing nonlinear functionals (with respect to side conditions). In this article we present a new method for unconstrained minimization which is suitable as well in large scale as in bad conditioned problems. The method is based on a true multi-dimensional modeling of the objective function in each iteration step. The scheme allows the incorporation of more given or known information into the search than in common line search methods. 

The describing equations are expressed in terms of the simple models, and are rearranged in a novel way so that a complete solution for the primitive variables is possible. In most cases this is achieved by converging a N-dimensional inner iteration loop, where N is the number of stages. In multi-stage applications, this inner loop is particularly amenable to solution by... 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

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