Nonlinear iterative

Equation (9.53) for the desired molecular field is nonlinear, typically solved iteratively. For this molecular-field approach to become practical, an alternative to this nonlinear iterative calculation is required. A natural idea is that a useful approximation to this molecular field might be extracted from simulations with available generic force fields. Then with a satisfactory molecular field in hand, the more ambitious quasichemical evaluation of the free energy can be addressed, presumably treating the actual binding interactions with chemical methods specifically. This is work currently in progress. 

The nonlinear iterative partial least-squares (NIPALS) algorithm, also called power method, has been popular especially in the early time of PCA applications in chemistry an extended version is used in PLS regression. The algorithm is efficient if only a few PCA components are required because the components are calculated step-by-step. 

NIPALS Nonlinear iterative partial least-squares... 

Wold, H. Soft Modeling by Latent Variables the Nonlinear Iterative Partial Least Squares Approach," Ed. J. Gani, in Perspective in Probability and Statistics - Papers in Honor of M. S. Bartlett, Academic Press, London 1975, pp. 117-142... 

SIMCA uses the NIPAI>S (Nonlinear Iterative PArtial teast uares) algorithm for principal component abstraction ( ). Due to the simplicity of the algorithm and the ease of programming it for use... 

The nonlinear iterative methods described here are based on the linear relaxation methods developed in Sections III. C.l and III.C.2 of Chapter 3. Initially, the correction term was set equal to zero in regions where o(k) was nonphysical. To illustrate this, we may rewrite the point-simultaneous equation [Chapter 3, Eq. (23)] with a relaxation parameter that depends on the estimate d(k) ... 

Wold, H., Soft modeling by latent variables the nonlinear iterative partial least squares approach, in Perspectives in Probability and Statistics, Papers in Honor of M.S. Bartlett, Gani, J., Ed., Academic Press, London, 1975. 

The concentration of any of these species depends on the total concentration of dissolved aluminum and on the pH, and this makes the system complex from the mathematical point of view and consequently, difficult to solve. To simplify the calculations, mass balances were applied only to a unique aluminum species (the total dissolved aluminum, TDA, instead of the several species considered) and to hydroxyl and protons. For each time step (of the differential equations-solving method), the different aluminum species and the resulting proton and hydroxyl concentration in each zone were recalculated using a pseudoequilibrium approach. To do this, the equilibrium equations (4.64)-(4.71), and the charge (4.72), the aluminum (4.73), and inorganic carbon (IC) balances (4.74) were considered in each zone (anodic, cathodic, and chemical), and a nonlinear iterative procedure (based on an optimization method) was applied to satisfy simultaneously all the equilibrium constants. In these equations (4.64)-(4.74), subindex z stands for the three zones in which the electrochemical reactor is divided (anodic, cathodic, and chemical). 

A straightforward method for PCA is the NIPALS (nonlinear iterative partial least squares) algorithm it is quickly implemented and can be applied to large datasets [58, 79],... 

Wold H. Soft modelling with latent variables The nonlinear iterative partial least squares approach. In Gani J, editor, Perspectives in probability and statistics Papers in honour of M.S. Barlett. London Academic Press, 1975. p. 114-42. 

Wold H., Lyttkens E.. Nonlinear iterative partial least squares (NIPALS) estimation procedures in Bull. Intern. Statist. Inst. Proc., 37th session, London -15 1969. 

The simplest method for PCA used in analytics is the iterative nonlinear iterative partial least squares (NIPALS) algorithm explained in Example 5.1. More powerful methods are based on matrix diagonalization, such as SVD, or bidiagonalization, such as the partial least squares (PLS) method. 

Equations 12.45, 12.46, 12.47, and 12.48 form a complete set of governing equations which are strongly coupled to each other. Therefore, these equations can be solved by nonlinear iterative procedures [133, 134, 198] and efficient second-order algorithms [1, 71,72,132]. 

A high fidelity model of a fuel cell may include many dependent variables such as concentration, temperature, velocity, potential, pressure, and other fields described in the model. For this reasOTi, a segregation of the solution procedure is required by solving a smaller set of equations fuUy coupled and then iterating over all possible smaller sets in one nonlinear iteration. To obtain faster convergence, variables such as the potential in the electrodes and in the electrolyte should then be solved in the same set, due to their tight coupling and presence in the... 

During the calibration step, the PLS technique assumes that the spectral data set X can be decomposed in the form of Equation 6.16. PLS factors are then computed with the help of iterative numerical procedures, such as the popular nonlinear iterative partial least squares (NIPALS) algorithm, as described in standard texts [46,74,75]. The PLS factors can be regarded as rotations of the PCA factors computed in... 

As in Example 6-1, we consider three botmdary value problems for steady gas flows in homogeneous, isotropic media. Two are easily posed and solved, but the third requires nonlinear iteration. 

---