T. F. Coleman and A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, Technical report TR95--1557, Computer Science Department, Cornell University, November 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....we analyze the problem of the estimation of a Jacobian matrix from a graph theoretic view point. In particular, we show how the known sparsity structure can be exploited in computing rows and columns. Methods for computing the Jacobian matrix by partitioning rows and columns have been studied in [11, 12]. The main results of the paper is given in Section 2. We begin Section 2 by considering the partitioning of the rows and the columns. We give examples where either row or column partitioning alone may not be able to take full advantage of the known sparsity. A more general partitioning problem is ....

Coleman, T. F. and Verma, A., The Efficient Computation of Sparse Jacobian Matrices Using Automatic Differentiation, Tech. Report CTC95TR225, Cornell Theory Center, Cornell University, November 1995.

....144 mesh in Table 6, computing the 306 Theta 306 Jacobian block J (5) average bandwidth of 10) required that we first assemble the 44064 Theta 44064 Jacobian matrix J. A very promising topic for future research is to make use of recent advances in using automatic differentiation (AD) [51], 52] to compute the symmetry adapted Jacobian blocks J ( in a much more direct manner. It can be shown that the orthogonal blocks J ( can be directly computed as J ( Q T f u (49) where f u is an n Theta m matrix whose columns are defined as f u ....

....as f u i = lim h 0 f (u hQ ( i) Gamma f (u) h (i = 1; 2; m ) 50) Note that eq: 50) only requires that f be computed in the standard coordinate system. Equations (49) and (50) can be used for the basis of a numerical differentiation procedure, as discussed in [51] and [52] to efficiently compute the orthogonal blocks J ( 5.4 Symmetry Boundary Conditions As discussed above, the symmetry adapted Jacobian matrices are found by first computing the Jacobian of the full matrix and then performing similarity transformations. For large problems, computing ....

T. F. Coleman and A. Verma. The efficient computation of sparse jacobian matrices using automatic differentiation. Tech. Report TR95-1557, Computer Science Department, Cornell University, November 1995.

....The purpose of this paper is to show how it is possible to dramatically lower the cost of computing J by exploiting structure and sparsity in the application of AD. Recently, techniques for the efficient determination of sparse Jacobian matrices J , via AD, have been developed [Averick1994a] [Coleman1995a], Hossain1995a] The bi coloring approach of Coleman and Verma [Coleman1995a] as discussed in Section 2, rests on the observation that is is usually possible to define thin matrices V; W such that the nonzero This research was partially supported by the Applied Mathematical Sciences Research ....

....the cost of computing J by exploiting structure and sparsity in the application of AD. Recently, techniques for the efficient determination of sparse Jacobian matrices J , via AD, have been developed [Averick1994a] Coleman1995a] Hossain1995a] The bi coloring approach of Coleman and Verma [Coleman1995a], as discussed in Section 2, rests on the observation that is is usually possible to define thin matrices V; W such that the nonzero This research was partially supported by the Applied Mathematical Sciences Research Program (KC 0402) of the Office of Energy Research of the U.S. Department of ....

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T. F. Coleman and A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, Technical report TR95--1557, Computer Science Department, Cornell University, November 1995.

....has only forward mode and we explore the use of reverse mode as well. When computing a sparse Jacobian matrix using AD, graph coloring algorithms can be used to significantly reduce the amount of work. The algorithm described in [2] illustrates the use of one sided coloring. Two sided coloring [3] combines the powers of both forward and reverse mode, by constructing thin matrices V and W so that the Jacobian J can be determined from the pair (JV , W T J) Coleman and Verma [4, 6] show how sparsity and structure can be exploited to compute Jacobian and Hessian matrices efficiently using ....

T. F. Coleman and A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, Tech. Rep. TR95-1557, Department of Computer Science, Cornell University, November 1995. To appear in SIAM Journal on Scientific Computing.

....expensive. The purpose of this paper is to show how it is possible to dramatically lower the cost of computing J by exploiting structure and sparsity in the application of AD. Recently, techniques for the efficient determination of sparse Jacobian matrices J , via AD, have been developed [1, 9, 13]. The bi coloring approach of Coleman and Verma [9] as discussed in x2, rests on the observation that is is usually possible to define thin matrices V; W such that the nonzero elements of J can be readily extracted from the pair To appear: Proceedings of the Second SIAM International ....

....it is possible to dramatically lower the cost of computing J by exploiting structure and sparsity in the application of AD. Recently, techniques for the efficient determination of sparse Jacobian matrices J , via AD, have been developed [1, 9, 13] The bi coloring approach of Coleman and Verma [9], as discussed in x2, rests on the observation that is is usually possible to define thin matrices V; W such that the nonzero elements of J can be readily extracted from the pair To appear: Proceedings of the Second SIAM International Workshop on Computational Differentiation, Sante Fe, ....

[Article contains additional citation context not shown here]

T. F. Coleman and A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, Tech. Rep. TR95-1557, Computer Science Department, Cornell 12 University, November, 1995.

....of the section, we demonstrate the usefulness of AD in stiff solvers. In particular, we present three case studies in the next three subsections. 4.1 Ease and Effectiveness of Forming Jacobians Automatically For most large scale problems, the Jacobian J is sparse. Recent work by Coleman and Verma [4, 3] provide techniques to exploit sparsity and structure of Jacobian matrices effectively using AD. The sparsity structure of the 10 Jacobian matrix can be automatically detected via AD. On the other hand, for sparse finite differencing, the sparsity structure must be provided by the user. This ....

....as shown in Figure 2 0 10 20 30 40 50 0 5 10 15 20 25 30 35 40 45 50 nz = 148 Fig. 2. The sparsity structure of Jacobian matrix To compute the above Jacobian matrix of size n by finite differencing requires n function evaluations, but using the Bi Partitioning technique of Coleman and Verma [4], which employs AD, only 3 function evaluations are required. The timing results are tabulated in Table 1. The timings were done for ode23s solver in the MATLAB ODE suite. The times required by the AD solution are much less, and the ratio increases with the problem size. However, both the methods ....

T. F. Coleman and A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, Tech. Rep. TR95-1557, Computer Science Department, Cornell University, November, 1995.

....the calculation a kind of microscopic structure exploitation. We discuss how this is done in the next section. In principle, what we are exploiting is specific sparsity structure that is inherent in the finite difference scheme. A general approach for exploiting sparsity in AD is described in [2]. 4 Exploiting the stencil structure The finite difference method that we used in the 1 D can be written as indicated in (4) which we rewrite here u k 1 = F (c; u k ; u k Gamma1 ) with u Gamma1 = u 0 = 0: This shorthand notation does not reveal the stencil structure given by the ....

T. Coleman and A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, SIAM Journal on Scientific Computing (1998), 19, pp. 12101233.

....possible to have first and second derivatives automatically computed given a code that computes the objective function. The difficulty is one of computational cost: straightforward application of automatic differentiation tools may be inordinately expensive for large problems. Results obtained in [1, 6] show that for the related sparse Jacobian problem, the cost can be dramatically reduced if sparsity is exploited. In principle similar techniques [5] can be applied to the sparse Presented at the International Conference on Nonlinear Programming, Beijing, September, 1996. This research was ....

.... Newton equations, JE 0 ffi x ffi y 1 ffi y 2 1 A = 0 0 0 GammaF (x) 1 A ; 4) where JE = 2 4 Gamma J I 0 A x y 2 GammaI A Gamma J x 0 Gamma J y 3 5 : 5) The point here is that the extended Jacobian matrix JE is sparse and clearly sparse AD techniques, e.g. [1, 6, 9] can be applied with respect to FE (x; y) 0 y 1 Gamma F (x) A(x)y 2 Gamma y 1 Gamma F (x; y 2 ) 1 A (6) 3 to efficiently determine JE . For example, the work required by the bi coloring technique developed in [6] is of order Delta (FE ) Delta (F ) where is a ....

[Article contains additional citation context not shown here]

T. F. Coleman and A. Verma, The efficient computation of sparse Jacobian matrices using automatic differentiation, Tech. Report TR95--1557, Computer Science Department, Cornell University, November 1995.

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