Newton iterations

Compute a new set of values of the T) tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas gorithm is obtained. If we set gj equal to Eq. (13-72), i.e., its residual, the hnearized equations to be solved simultaneously are... 

The Newton iteration is initiated by providing reasonable guesses for all unknowns. However, these can be generated from guesses of just T, Tv, and one interstage value of Fj or Lj. Remaining values of Tj are obtained by linear interpolation. By assuming constant molal over-... 

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

However, because of the strong nonllnearltles In the reactor fiow problem, continuation procedures must be used to obtain a good Initial guess for the Newton Iteration. A simple first order continuation scheme falls at fiow transition points (bifurcations) where the Jacobian, G, becomes singular. To circumvent this problem an arclength continuation scheme discussed by Keller (26.27) and Chan (2S.) Is used which leads to the Infiated system ... 

Following Chan (2. ) a difference approximation Is used to compute the second derivatives of G. A Newton Iteration Is then applied to the equation... 

The vertical reactor simulations reported In this paper typically Involved 14,000 unknowns and took 25 CPU seconds per Newton Iteration on a Cray-2. The tracing of a complete family of solutions for one parameter (e.g. susceptor temperature) cost approximately 25 CPU minutes. The latter number underscores the advantage of using supercomputers to understand the structure of the solution space for physical problems which often Involve many parameters. 

Newton Iteration Compute Xi by Newton iterations on the reduced KKT system. 

Note that as the line search process continues and the total step from the initial point gets larger, the number of Newton iterations generally increases. This increase occurs because the linear approximation to the active constraints, at the initial point (0.697,1.517), becomes less and less accurate as we move further from that point. 

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

With this implicit boundary-condition specification, the eigenvalue computation can be easily incorporated into a Newton iteration to solve the entire problem. Time marching and adaptive meshing can also be incorporated [158,159]. 

Fig. 15.7 Conceptual illustration of the behavior of a Newton iteration on a nonlinear, stiff system of algebraic equations. A contour map of a norm of the residual vector F is plotted. The curvature represents nonlinear behavior, and the elongation represents disparate scaling, or stiffness. The desired solution of the problem is represented by the X the current iteration is marked by a dot. The elliptical contours represent residuals of the local linearization at the current iterate.

The modified Newton iteration, and the reason that damping is effective, can be explained in physical terms. Chemical-kinetics problems often have an enormous range of characteristic scales—this is the source of stiffness, as discussed earlier. These problems are also highly nonlinear. 

K. With a series of continuations, solutions were computed at increasing temperatures up to 1400 K. In all cases the solution was obtained directly by Newton iteration, with no need for time stepping to assist convergence. Just in the vicinity of the turning point (around 1100 K), it is apparent that at least three of the solutions depart significantly from the true physical solution. 

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

Rewrite the equation in residual form so that the Newton iteration can be formulated to seek x such that f(x) = 0. 

The effect is to keep the iteration matrix (Jacobian) banded, which considerably improves the efficiency of the Newton iteration that is used to solve the discrete problem. This procedure is equivalent to solving a simple first-order differential equation,... 

The burner-face temperature is an element in the dependent-variable vector and determined through the Newton iteration just as is the temperature at any other mesh point. Even though the implicit imposition of boundary conditions has relatively little benefit for the simple example just shown, it has great benefit in more complex boundary conditions that are frequently needed in chemically reacting flow problems. For example, as will be discussed later, surface chemistry can result in boundary conditions that are far too difficult to impose explicitly. 

The Jacobian of the system is a square matrix, but importantly, because the residuals at any mesh point depend only on variables at the next-nearest-neighbor mesh point, the Jacobian is banded in a block-tridiagonal form. Figure 16.10 illustrates the structure of the Jacobian in the form used by the linear-equation solution at a step of the Newton iteration,... 

Because the T matrix is symmetric and has real elements, it turns out that the three roots A i, A.2, and A3 must be real numbers. For any arbitrary stress state the cubic equation is usually most conveniently solved by numerical iteration. For example, one could graph the determinant as a function of A, observing the approximate A values of the zeros. Then, taking the approximate values of A as starting iterates, a Newton iteration could be used to determine each of the exact roots. The three roots are the principal stresses. 

Quasi-Newton methods may be used instead of our full Newton iteration. We have used the fast (quadratic) convergence rate of our Newton algorithm as a numerical check to discriminate between periodic and very slowly changing quasi-periodic trajectories the accurate computed elements of the Jacobian in a Newton iteration can be used in stability computations for the located periodic trajectories. There are deficiencies in the use of a full Newton algorithm, such as its sometimes small radius of convergence (Schwartz, 1983). Several other possibilities for continuation methods also exist (Doedel, 1986 Seydel and Hlavacek, 1986). The pseudo-arc length continuation was sufficient for our calculations. 

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

Newton iteration starts with an approximation xo of a root x of / and goes along the tangent line to the curve at (xo, /(xo)) until it intersects the x axis. This intersection is labeled xi. ft is an improved iterative approximation for the actual root, and the process continues leading from xi to X2 via the tangent to / at xi, etc. until the difference between successive iterates becomes negligible, see Figure 1.1 for our trial polynomial equation p x) = x3 — 2x2 +4 = 0 and the start xo = —2. 

In order to distinguish between the n components of each iterate, marked by subscripts, we mark the iterates in R themselves by superscripts. Thus fully written out the ith Newton iterate z in Rn is the vector... 

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