T. Steihaug and A. S. Hossain. Graph coloring and the estimation of sparse Jacobian matrices with segmented columns. Technical Report 72, Department of Informatics, University of Bergen, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the product y T F 0 (x) when the vector y is initialized to jth coordinate vector. An excellent account of the recent developments in Automatic differentiation can be found in [5] JACOBIAN ESTIMATION 3 Much of the research on the efficient estimation of sparse Jacobian and Hessian matrices [1, 2, 4, 6, 7, 9, 10] uses divided differences to obtain estimates of the nonzeros. In this approach one forms groups of columns that are structurally orthogonal i.e. columns that do not have nonzero in the same row position. The estimates of the nonzeros in those columns are then obtained from a divided difference ....

....orthogonality in an effort to reduce the number of function evaluation. Coleman and Mor e [2] showed that forming groups of structurally orthogonal columns can be seen as a graph coloring problem. Their observation led to several coloring heuristics with improved performance. Steihaug and Hossain [10] have suggested a technique where coloring heuristics from Coleman and Mor e [2] are used on blocks of rows of the Jacobian matrix. This property of structural orthogonality can also be utilized when we use AD to obtain the nonzero elements. Here we initialize the vector x such that x j = 1 for ....

Steihaug, Trond and Hossain, A. K. M. S. (1992), Graph coloring and the estimation of sparse Jacobian matrices using row and column partitioning, Report 72, Department of Informatics, University of Bergen.

....separability and other structure, even when the basic reverse mode has a lower operations count, as is the case for gradients. Fortunately, much of the excellent research that has been conducted regarding the estimation of sparse Jacobians and Hessians by differencing (see, e.g. 10] 9] 18] [20]) carries over to computational differentiation. The main difference is that, instead of approximating Jacobian vector products by divided differences, one obtains them without any truncation errors by the forward mode of automatic differentiation. Moreover, since the reverse mode yields vector ....

....also refer to f and F as horizontal expansions of f . Analogously f and F may be called vertical expansions of f . Conversely we may refer to f as horizontal and vertical contraction of f and F or f and F , respectively. 2. 4 Relation to Multicoloring It has often been observed [20] that for a partitioned vector function f(x) f (1) x) f (2) x) # : IR n IR m 1 m 2 the partial Jacobians J (i) i f (i) j 0 for i = 1; 2 may satisfy (J) J (1) J (2) Then the CPR approach should be applied to obtain J (1) and J (2) separately, ....

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Trond Steihaug and A. K. M. Shahadat Hossain, Graph coloring and the estimation of sparse Jacobian matrices using row and column partitioning, Report 72, Department of Informatics, University of Bergen, 1992.

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T. Steihaug and A. S. Hossain. Graph coloring and the estimation of sparse Jacobian matrices with segmented columns. Technical Report 72, Department of Informatics, University of Bergen, 1997.

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T. Steihaug and A. S. Hossain. Graph coloring and the estimation of sparse Jacobian matrices with segmented columns. Technical Report 72, Department of Informatics, University of Bergen, 1997.

....vertices of an associated graph. The coloring obtained is dependent on the order in which the vertices are considered during the coloring procedure. An often cited example [4] shows that it may be advantageous to consider a more general problem where segments of columns are grouped together. In [18], we have proposed new techniques to estimate sparse Jacobian matrices. In this approach both columns and rows are grouped together and the resulting segments of columns are partitioned in an effort to reduce the number of function evaluations. A graph coloring formulation of this partitioning ....

....are structurally orthog onal if they do not have nonzeros in the same row position. Segmented columns A(wi,j) and A(wp, q) i p and j q, are structurally orthogonal if A(wi, j) and A(wi, q) are structurally orthogonal and A(wp,j) and A(wp, q) are structurally orthogonal. It can be shown that [18] a group of orthogonal 3 segmented columns can be esti mated with one extra function evaluation. If segmented columns can be grouped such that each segmented column is included in exactly one group and segmented columns in the same group are orthogonal then we have a consistent partition of ....

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T. Steihaug and A. S. Hossain. Graph coloring and the estimation of sparse Jacobian matrices with segmented columns. Technical Report 72, Department of Informatics, University of Bergen, 1997.

....ordering of the nonzero elements of A. Lemma 1 shows that the substitution ordering saves one AD pass provided that p #. This result also indicates that the problems where the Jacobian matrix has a relatively few nonzeros per row but the chromatic number of its intersection graph is large [15] are good candidates for a substitution scheme. If each row of the compressed matrix has at least two consecutive zero elements, then the matrix formed by C 1 , C 2 , C 2 , C 3 , C p 1 , C p will have at least one zero element in every row, and the process could be repeated. ....

....is at most p # p (# 1) #. Let us assume that a consistent partition of the columns of A contains p = k(# 1) groups, where k # 1. Then we save k forward passes. We illustrate the above substitution method on an example. Consider a 15 6 matrix A with the following sparsity pattern [15]. # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # (1.4) The column intersection graph G(A) is a graph on 6 vertices, and the chromatic number is 6, hence any consistent column partition ....

Trond Steihaug and A.K.M. Shahadat Hossain. Graph coloring and the estimation of sparse Jacobian matrices with segmented columns. Technical Report 72, Department of Informatics, University of Bergen, 1997.

.... a research cooperation between the Norwegian Academy of Science and Den norske stats oljeselskap a.s. Statoil) y Department of Informatics, University of Bergen, Hyteknologisenteret, N 5020 Bergen , Norway (Trond.Steihaug ii.uib.no and shahadat ii.uib.no) 1 Revised May 1997. Revision of [7] 1 2 TROND STEIHAUG AND A. K. M. SHAHADAT HOSSAIN The segmented column approach described in this paper can be combined with a block iterative method similar to the techniques in Coleman [8] and Dennis and Steihaug [9] An example of solving unconstrained optimization problem using approximate ....

Trond Steihaug and A. K. M. Shahadat Hossain, Graph coloring and the estimation of sparse Jacobian matrices using row and column partitioning , Report 72, Department of Informatics, University of Bergen, 1992.

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