Newton s gradient method

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

Natural cause and effect 175 Naturally occurring oscillations 126 Negative feedback 158 Nelder-Mead search algorithm 108 Newton s gradient method 108 Nitrogen 572 Non-equilibrium... 

One valid alternative to the Levenberg-Marquardt method is the dogleg method, also known as Powell s hybrid method (Rabinowitz, 1970). Once again, this couples the Newton and gradient methods. The original version of Powell s method was close to the tmst region concept. Powell proposed a strategy for the modification of parameter d subject to both the successes and failures of the procedure. 

The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton s method requires second derivative information but is veiy efficient, while quasi-Newton retains most of the benefits of Newton s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. 

It is noted that the Rosenbrock function given by the next equation has been used to test the performance of various algorithms including modified Newton s and conjugate gradient methods (Scales, 1986)... 

Newton s method makes use of the second-order (quadratic) approximation of fix) at x and thus employs second-order information about fix), that is, information obtained from the second partial derivatives of fix) with respect to the independent variables. Thus, it is possible to take into account the curvature of fix) at x and identify better search directions than can be obtained via the gradient method. Examine Figure 6.9b. 

Find the minimum of the following objective function by (a) Newton s method or (b) Fletcher-Reeves conjugate gradient... 

Usually, the values of the transport coefficients for a gas phase are extremely sensitive to pressure, and therefore predictive methods specific for high-pressure work are desired. On the other hand, the transport properties of liquids are relatively insensitive to pressure, and their change can safely be disregarded. The basic laws governing transport phenomena in laminar flow are Newton s law, Fourier s law, and Fick s law. Newton s law relates the shear stress in the y-direction with the velocity gradient at right angles to it, as follows ... 

In the one-dimensional search methods there are two principle variations some methods employ only first derivatives of the given function (the gradient methods), whereas others (Newton s method and its variants) require explicit knowledge of the second derivatives. The methods in this last category have so far found very limited use in quantum chemistry, so that we shall refer to them only briefly at the end of this section, and concentrate on the gradient methods. The oldest of these is the method of steepest descent. 

Rawlings and co-workers proposed to carry out parameter estimation using Newton s method, where the gradient can be cast in terms of the sensitivity of the mean (Haseltine, 2005). Estimation of one parameter in kinetic, well-mixed models showed that convergence was attained within a few iterations. As expected, the parameter values fluctuate around some average values once convergence has been reached. Finally, since control problems can also be formulated as minimization of a cost function over a control horizon, it was also suggested to use Newton s method with relatively smooth sensitivities to accomplish this task. The proposed method results in short computational times, and if local optimization is desired, it could be very useful. 

Newton s method includes the quadratic nature of the energy hypersurface in the search direction computation. Consider the Taylor s series for the change in gradient given by... 

The number of line searches, energy and gradient evaluations are given in Table VI for both the Newton and Quasi-Newton methods. Table VI clearly indicates that the use of an exact inverse Hessian requires less points to arrive at the optimum geometry. However in Table VI we have not included the relative computer times required to form the second derivative matrix. If this is taken into account, then the Newton s method with its requirement for an exact Hessian matrix is considerably slower than the quasi-Newton procedures. 

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

Derivative methods and derivative approximation methods use steepest ascent/descent (or Cauchy s method), conjugate gradients, Newton s, or quasi-Newton methods. 

For the optimization of Hartree-Fock wave functions, it is usually sufficient to apply the SCF scheme described in Sec. 3.1. By contrast, the optimization of MCSCF wave functions requires more advanced methods (e.g., the quasi-Newton method or some globally convergent modification of Newton s method, which involves, directly or indirectly, the calculation of the electronic Hessian as well as the electronic gradient at each iteration) [45]. 

The setup of the initial configuration, the application of the interaction potential, and the periodic boundary conditions are identical in MD and MC methods. In the case of MD, a Boltzmann distribution of velocities appropriate to the temperature is also assigned to the atoms. The atoms move under the gradients (the negative of the gradients gives the force that is actually used in the MD simulation) of the potential for a time step (At) according to Newton s laws of motion (F = ma = -dVIdr) to obtain a new set of coordinates, and the process is repeated for N time steps to obtain a simulation time of NAt.42... 

Quasi-Newton methods attempt to achieve the very fast convergence of Newton s method without having to calculate the Hessian matrix explicitly. The idea is to use gradients to successively build up an approximation to the inverse Hessian. For Newton s method, new directions are taken as... 

Newton s method may be seen as a gradient-based method in a space where... 

The Levenberg-Marquardt method is able to move between Newton s method and the gradient method. This feature will be discussed later conversely, it is now important to consider this method for removing the problem of a nonpositive definite Hessian matrix. 

This criterion should be used only rarely and only if the Newton prediction is unsatisfactory. In fact, the new function evaluation is on the gradient direction that must be used to unlock Newton s method. 

The gradient method may be efficiently coupled with Newton s method since it is quite complementary to it. Newton s method is quite efficient in the final steps of the search while the gradient method is efficient in the initial ones. 

---