Jacobian sparse

It is not hard to change the variables to make the Jacobian sparse, and it is simple to run a simulation using DAEPACK rather than DASPK or VODE, to see whether the sparsity-aware software is less demanding of computer... 

Schubert, L. K., "Modification of a Quasi-Newton Method for Nonlinear Equations with Sparse Jacobian", Math. Comp. (1970) 2 27-30. 

The main reason for slow computations is the dimensions of the sparse 160 x 160 Jacobian. Although other routines are available through the work of Hindmarsh (1, 2) to handle banded Jacobians or Jacobians with certain structures there are no commercial routines that could be used for the particular Jacobian that arises in the modeling of the low pressure pyrolysis of polydispersed coal particles. 

Fill and solve the sparse Jacobian for a new set of temperatures, T/s, and component vapor and liquid rates, l s and s, by solving the independent equations during one pass through the Newton-Baphson procedure. 

The expression is easily coded, since T<0>, Eq. 41, and the r 1 are known. It simplifies for substitution on a principal plane or axis and for symmetrically equivalent multiple substitution, because several of the elements of the matrix T will then vanish. It is clear from Eq. 45 that the derivatives are nonvanishing only for those atoms a that have actually been substitued in the particular isotopomer s. Therefore, the Jacobian matrix X generated from these derivatives is, in general, a sparse matrix. 

Jacobian is not usually calculated at each iteration, and not even at every timestep. Further time is saved by using sparse matrix techniques to take advantage of the fact that the Jacobian usually possesses many zero elements (cf. equation (2.52) for example). Sparse matrix techniques are similarly used in solving equation (2.78) once the Jacobian has been found. Finally, the integration routine will seek to lengthen the timestep to the maximum extent consistent with a defined accuracy criterion, to take advantage of the strong stability properties of the implicit method. 

As originally proposed, the sparsity of the jacobian matrix is destroyed by Broyden s method. Two procedures (or modifications) which preserves the sparsity of the jacobian matrices are presented. The procedures are demonstrated by use of simple algebraic examples and applied to the solution of distillation problems whose jacobian matrices are sparse. 

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. 

Although the inverse of J0 appears in Eq. (5-31), it should be noted that the explicit expression of Jo 1 need never be developed only the LU factorization is required. If J0 is sparse, its inverse Jo 1 is not necessarily sparse, but its factorization L0U0 is sparse. Thus, throughout the remainder of the development, inverses are shown but the actual numerical solutions are to be found by use of the LU factorizations rather than the inverses of the jacobian matrices. 

Schubert proposed a modification of Broyden s method which takes advantage of the fact that in the case of sparse jacobian matrices, most of the elements... 

Thus, if J0 is sparse and a sparse factorization is available, Broyden s procedure can be implemented while effectively maintaining the sparsity of the jacobian. The algorithm for effecting these calculations in an efficient manner is developed as follows. Suppose that it is desired to solve... 

In many practical problems, the Jacobian J and thus the matrix G are sparse and structured. It is therefore indispensable to use a dedicated factorization that exploits such a feature. 

Buzzi-Ferraris and Manenti (2014, Vol. 3) have shown that the Jacobian of a nonlinear system can be calculated by simultaneously varying several variables when the Jacobian is sparse. If Equation 2.240 is adopted to evaluate a Jacobian matrix, which is supposed to be ftill, then the vector is the null array except for position k where the element is equal to 1. In this case, the system is called N times to evaluate the derivatives of the functions with respect to the N variables. Consider the sparse Jacobian matrix shown in Figure 2.11, where the symbol x represents a nonzero element. 

When the system is sparse, the total number of calk necessary to evaluate the Jacobian matrix can be drastically reduced. 

When several processors are available and parallel computing can be exploited, it k possible to update the Jacobian J in a much more efficient way for both the cases of sparse and dense matrices. 

The objects of the classes BzzOdeSparseStiff and BzzOdeSparseStif-fObJect exploit the devices broached in Section 2.16.2.4 when the Jacobian matrix is sparse. 

It is well known that solvers that exploit the sparse structure of its Jacobian are more efficient in terms of CPU computation time than solvers that do not Unfortunately, those solvers are not always available or applicable to the specific studied case. 

Actually, when the problem does not have a fully structured Jacobian (see Figure 5.1), several very high-performance algorithms designed for sparse and well-structured systems cannot be used. 

The Jacobian is still highly sparse, even though the few elements in the ellipses corrupt the compact tridiagonal structure. 

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