Jacobian update

Jacobian updating is not performed through a generic quasi-Newton method but by exploiting the specific structure of the equations. Many derivatives can be easily calculated analytically, in fact. 

The development of an SC procedure involves a number of important decisions (1) What variables should be used (2) What equations should be used (3) How should variables be ordered (4) How should equations be ordered (5) How should flexibility in specifications be provided (6) Which derivatives of physical properties should be retained (7) How should equations be linearized (8) If Newton or quasi-Newton hnearization techniques are employed, how should the Jacobian be updated (9) Should corrections to unknowns that are computed at each iteration be modified to dampen or accelerate the solution or be kept within certain bounds (10) What convergence criterion should be applied ... 

Like Newton s method, the Newton-Raphson procedure has just a few steps. Given an estimate of the root to a system of equations, we calculate the residual for each equation. We check to see if each residual is negligibly small. If not, we calculate the Jacobian matrix and solve the linear Equation 4.19 for the correction vector. We update the estimated root with the correction vector,... 

ODE solver. Relative to non-stiff ODE solvers, stiff ODE solvers typically use implicit methods, which require the numerical inversion of an Ns x Ns Jacobian matrix, and thus are considerably more expensive. In a transported PDF simulation lasting T time units, the composition variables must be updated /Vsm, = T/At 106 times for each notional particle. Since the number of notional particles will be of the order of A p 106, the total number of times that (6.245) must be solved during a transported PDF simulation can be as high as A p x A sim 1012. Thus, the computational cost associated with treating the chemical source term becomes the critical issue when dealing with detailed chemistry. 

The subroutine between lines 4792 - 4806 provides divided difference approximation of the appropriate segment of the Jacobian matrix, stored in the array G(NY,NP). In some applications the efficiency of the minimization can be considerably increased replacing this general purpose routine by analytical derivatives for the particular model. In that case, however, Y(NY) should be also updated here. 

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]. 

Figure 9.24 presents the predicted velocity field for an isothermal flow. Here, the charge and the mold surfaces have the same temperature of 150°C. Using the velocity field, Osswald et.al [12] moved the nodes by a small time At. A new velocity field was computed using the updated mesh. These steps were repeated until the mesh became too distorted, leading to unreasonable values in the velocity field. Excessive element distortion is often detected when the determinant of the Jacobian matrix, J, is less or equal to zero. 

The quasi-Newton methods. In the Newton-Raphson method, the Jacobian is filled and then solved to get a new set of independent variables in eveiy trial. The computer time consumed in doing this can be very high and increases dramatically with the number of stages and components. In quasi-Newton methods, recalculation of the Jacobian and its inverse or LU factors is avoided. Instead, these are updated using a formula based on the current values of the independent functions and variables. Broyden s (119) method for updating the Jacobian and its inverse is most commonly used. For LU factorization, Bennett s (120) method can be used to update the LU factors. The Bennett formula is... 

Tomich recommends filling and inverting the Jacobian only once and using the Broyden method (Sec. 4.2.6) to update the inverse. This reduces computer time per trial but increases the number of column trials or passes through the procedure and for 6ome columns may also decrease reliability of the method. It is the author 6 experience that for many columns, solution is easier to reach when the Jacobian is filled and inverted in each trial and Brcyden s method is not used. 

Get a new set of stripping factors by solving the energy balances and specification equations as the independent functions of the quasi-Newton technique of Broyden (Sec. 4.2.6). The derivatives of the Jacobian matrix are generated numerically and must include steps 4, 5, and 6 each time the independent variables are perturbed, The Jacobian is not recalculated after the first trial in the loop but is instead updated by Broyden s equation. 

Vickery and Taylor (81) used a Naphtali-Sandholm method containing all of the MESH equations and variables [M2C + 3) equations] with the variables represented by x. H is the Jacobian from the Naphtali-Sandholm method solution of the known problem, G(x) = 0, This is numerically integrated from t = 0 to t - 1, finding a H, at each Step and updating H when the solution is reached at each step, With Hj. and H, known, dxjdt is solved, and with step size t, a new set of values for the independent variables x is found by Euler s rule... 

C Scalar used in a quasi-Newton update of a Jacobian, Sec, 4.2.6,... 

The deferred updating of the Jacobian matrix PD evidently works very well 23 updates sufficed for the 231 integration steps. Here 563 evaluations of the vector f were needed besides those used in updating PD this corresponds to about two Newton iterations per integration step. 

In the preceding equations, the superscripts (-) and (-t) indicate the values before and after the measurement update has occurred, respectively. F and H are the Jacobian matrices of the functions f and h relative to x. ... 

Calculate the Lagrange multipliers H = —Z T yf after updating the reduced Jacobian... 

The values of Sj,Ej are calculated from Equations 13.37 and 13.38 (combined with Equation 13.4) at current values of the variables, and the derivatives are calculated by finite difference methods. Equations 13.41 and 13.42 are solved for AT] and Al] by inverting the Jacobian matrix, the matrix of partial derivatives of Sj and Ej with respect to Tj and 1]. The independent variables Tj and 1] are then updated for the next iteration ... 

In the classical Newton-Raphson technique, the Jacobian matrix is inverted every iteration in order to compute the corrections AT] and Al]. The method of Tomich, however, uses the Broyden procedure (Broyden, 1965) in subsequent iterations for updating the inverted Jacobian matrix. 

Once the Jacobian matrix is solved for the corrections Aw, the straight Newton-Raphson method could be applied to update the variables for the next iteration ... 

The next objective is to update Sji, Rf,R, and Qj to satisfy Equations 13.51 and 13.52. In the most general case all these parameters are variable, bringing the total number of variables to 4 /. The equations to be solved are N energy balances (Equation 13.51) and 3N specifications (Equation 13.52). The Newton method is used by numerically calculating the Jacobian matrix then inverting it to determine the corrections to the variables. The Jacobian elements are the partial derivatives of each of the residuals of Equations 13.51 and 13.52 with respect to each of the variables ... 

The 78 equality constraints in the complete model were thus reduced to 6 nonlinear equations as the genetic algorithm, NSGA-II-aJG is not effective in handling multiple equality constraints. Its inadequateness was also observed even when the equations had been reduced to 6 equations. Hence, the Broyden s update and finite-difference Jacobian function (DNEQBF) of the IMSL Library was embedded in the objective evaluation to solve the nonlinear equations 10.1 to 10.6. 

Tomich25 was the first to apply Broyden s method (developed in chap. 15) to the solution of distillation problems. Broyden s method is based on the use of numerical approximations of the partial derivatives appearing in the jacobian matrix. The approach proposed by Broyden permits the inverse of the jacobian matrix to be updated each trial after the first through the use of Householder s formula.6 Thus, it is necessary to invert the jacobian matrix only once. Since approximate values for the partial derivatives are used, procedure 2 generally requires more trials than does procedure 1. However, since the evaluation of the partial derivatives and the inversion of the jacobian matrix are not generally required after the first trial of procedure 2, it requires less computer time per trial than does procedure 1. 

Instead of applying Householder s formula, the calculation of an inverse of the jacobian may be avoided altogether by use of the algorithm proposed by Bennett for updating the LU factors of the jacobian matrix. Example 4-9 will show that fewer numerical operations are required to compute the LU factors than are required to compute the inverse of a matrix. Bennett s algorithm is applied to the Broyden equations as follows. 

Broyden s algorithm consists of successively updating of the jacobian matrix of the Newton-Raphson equations by use of the correction matrix xCy7, that is,... 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

Two methods have been proposed for retaining the desirable characteristics of Broyden s method and eliminating the undesirable characteristic of the loss of sparsity of the jacobian matrix through the use of inverses. In both of these modifications of Broyden s method, the necessity for the development of analytical expressions for the partial derivations is eliminated. To initiate the calcula-tional procedure in each of these modified versions of Broyden s method, the partial derivatives appearing in the jacobian matrix are evaluated numerically, and the jacobian matrix is updated in subsequent trials through the use of functional evaluations. The first modified form of Broyden s method is the one proposed by Gallun and Holland,9 and the second modification is the one proposed by Schubert.21... 

As shown in Chap. 4, Broyden proposed the following formula for updating the jacobian matrix Jfc to obtain Jfc+1... 

In this algorithm, Broyden s method is applied by updating the jacobian matrices by use of Householder s formula.13 Let J0 be the initial approximation of the jacobian matrix with which the iterative procedure is started. Then... 

After Ax0 has been used to find Xj as described above, the updated jacobian matrix Ji is found as follows... 

Broyden. This method solves directly the equation 3.6. It is worthy to note that the updating of the Jacobian takes place by algebraic computations on matrix elements and not by matrix inversion. It may be used for the convergence of multiple tear streams, design specifications, or mixed tear streams and design specifications. 

To this point, we have assumed that the activity coefficients y(- and y,- as well as the activity of water aVJ are known values. In fact, these values vary with m,. Our strategy is to ignore this variation while calculating the Jacobian matrix and then update the activity coefficients and water activity after each step in the iteration. 

---